8 Mathematical Analysis, Differential Equations & Applications

Kyrgyz-Turkish Manas University Publication: 331 Conference Series: 46

M A D E A – 9

INTERNATIONAL CONFERENCE

Mathematical Analysis, Differential Equations & Applications

Kyrgyzstan–Turkey–Ukraine

ABSTRACTS

June 21–25, 2021 Bishkek, Kyrgyz Republic MATHEMATICAL ANALYSIS,DIFFERENTIAL EQUATION &APPLICATIONS - MADEA 9 KYRGYZ-TURKISH MANAS UNIVERSITY,BISHKEK,KYRGYZ REPUBLIC,JUNE 21-25, 2021

M 34 Mathematical Analysis, Differential Equations & Applications (MADEA-9) International Conference Abstracts – Bishkek: KTMU. 2021 – 88 p.

ISBN 978-9967-9294-4-9 e-ISBN 978-9967-9294-5-6

The book contains the abstracts of the participants of the International scientiﬁc conference “Mathematical Analysis, Differential Equation & Applications MADEA–9”. The conference is organized by Kyrgyz-Turkish Manas University, Atatürk University, Mersin University, Institute of Mathematics of NAS of Ukraine and Taras Shevchenko National University of Kyiv at the Kyrgyz-Turkish Manas University in Bishkek, Kyrgyz Republic on June 21-25, 2021. This is the ninth international scientiﬁc MADEA conference devoted to the study of important problems of mathematical analysis, differential equations, their applications and related topics.

• The ﬁrst conference was held at National Juriy Fedkovich University of Chernivtsi (Cher- nivtsi, Ukraine) on August 26-30, 2003.

• The second was held at Mersin University (Mersin, Turkey) on September 07-11, 2004.

• The third was held at Uzhgorod National University (Uzhgorod, Ukraine) on September 18-23, 2006.

• The fourth was held at Eastern Mediterranean University (Famagusta, North Cyprus) on September 12-15, 2008.

• The ﬁfth was held at Sunny Beach, Bulgaria on September 15-20, 2010.

• The sixth was held at Mersin University (Mersin, Turkey) on September 04-09, 2012.

• The seventh was held at Institute of Mathematics and Mechanics NAS of Azerbaijan (Baku, Azerbaijan) on September 08-13, 2015.

• The eighth was held at the Kyrgyz-Turkish Manas University (Bishkek, Kyrgyz Republic) on June 17-23, 2018.

“Matematiksel Analiz, Diferansiyel Denklemler ve Uygulamaları Sempozyumu Bildiri Özetleri”. Kırgızistan-Türkiye Manas Üniversitesi Yönetim Kurulu’nun 28.05.2021 tarih ve 2021-07.33 Nolu kararı ile basılmı¸stır. KTMÜ Cengiz AYTMATOV Kampüsü, Cal, Bi¸skek.

ISBN 978-9967-9294-4-9 e-ISBN 978-9967-9294-5-6

2 Mathematical Analysis, Differential Equation & Applications - MADEA 9 Kyrgyz-Turkish Manas University, Bishkek, Kyrgyz Republic, June 21-25, 2021

Organizers and Partners:

Kyrgyz-Turkish Manas University, Bishkek, Kyrgyz Republic

Atatürk University, Erzurum, Turkey

Mersin University, Mersin, Turkey

Institute of Mathematics of the National Academy of Sciences of Ukraine, Kyiv, Ukraine

Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

Editors:

Ceylan A., Abdullayev F. G., Kopuzlu A., Shevchuk I. O., Shidlich A. L.

3 MATHEMATICAL ANALYSIS,DIFFERENTIAL EQUATION &APPLICATIONS - MADEA 9 KYRGYZ-TURKISH MANAS UNIVERSITY,BISHKEK,KYRGYZ REPUBLIC,JUNE 21-25, 2021

Organizing Committee

Honorary Chairmen:

Prof. Dr. CEYLAN A. Rector of Kyrgyz-Turkish Manas University Prof. Dr. KULMYRZAYEV A. Co-Rector of Kyrgyz-Turkish Manas University Prof. Dr. ÇOMAKLI Ö. Rector of Atatürk University Prof. Dr. ÇAMSARI A. Rector of Mersin University Acad. Prof. Dr. SAMOILENKO A. M. Director of Institute of Mathematics of NAS of Ukraine

Chairmen:

ABDULLAYEV F.G. (Kyrgyz-Turkish Manas University, Kyrgyz Republic) KOPUZLU A. (Atatürk University, Turkey)

Vise-Chairmen:

CENGIZ N., (Head of Dep. of Mathematics of Atatürk University) KAMALI M. (Dep. of Mathematics of Kyrgyz-Turkish Manas University) ROMANYUK A.S. (Head of Dep. IM NAS of Ukraine) SHEVCHUK I.O. (Head of Dep. Taras Shevchenko National University of Kyiv, Ukraine)

International Scientiﬁc Committee:

Aasma A. (Estonia), Abbas M. (South Afrika), Amirov R. (Turkey), Andrievskii V.V. (USA), Aripov M. (Uzbekistan), Aslanov H. (Azerbaijan), Avkhadiev F. (Russia), Azamov A. (Uzbek- istan), Banakh T. (Ukraine), Bayraktar M. (Turkey), Bilalov B. (Azerbaijan), Blatt H.P. (Ger- many), Borubaev A. (Kyrgyz Republic), Buslaev V.(Russia), Celebi O. (Turkey), Chubarikov V.N. (Russia), Daulbaev M. (Kazakstan), Delgado J.M. (Spain), Deniz E. (Turkey), Farkov Y.A. (Rus- sian), Gauthier P. (Canada), Goginava U. (Georgia), Golberg A. (Israel), Guliyev V. (Azerbaijan), Ilolov M. (Tajikistan), Ismailov V. (Azerbaijan), Karakaya V. (Turkey), Karandjulov L. (Bulgaria), Kerimov N.B. (Azerbaijan), Kocinacˇ L. (Serbia), Kopotun K. (Canada), Kunyang W. (China), Leviatan D. (Israel), Mahmudov N. (North Cyprus), Mardanov M. (Azerbaijan), Marinets V.V. (Ukraine), Özdemir M. (Turkey), Petryshyn R.I. (Ukraine), Recke L. (Germany), Rontó A. (Czech Rep.), Ronto M. (Hungary), Shabozov M. (Tadjikistan), Shapiro M. (Mexico), Shoikhet D. (Is- rael), Skopina M. (Russia), Softova L. (Italy), Starovoytov A. (Belarus), ¸SerbetciA. (Turkey), Temirgaliyev N. (Kazakstan), Troﬁmchuk S.I. (Chile), Vuorinen M. (Finland), Zajaç J. (Poland).

4 Mathematical Analysis, Differential Equation & Applications - MADEA 9 Kyrgyz-Turkish Manas University, Bishkek, Kyrgyz Republic, June 21-25, 2021

International Organizing Committee:

Antoniouk A. (Ukraine), Asanov A. (Kyrgyz Republic), Ayupov Sh. (Uzbekistan), Boichuk A.A. (Ukraine), Chekeev A. (Kyrgyz Republic), Çelik E. (Turkey), Deg¸er U. (Turkey), Iskandarov S. (Kyrgyz Republic), Kadıoglu E. (Turkey), Küçük A.(Turkey), Küçükaslan M. (Turkey), Orhan H. (Turkey), Makarov V.L. (Ukraine), Perestyuk M.O. (Ukraine), Re¸sidoglu˘ Kh. (Turkey), Sava¸sE. (Turkey), Savchuk V.V. (Ukraine), Serdyuk A.S. (Ukraine), Suba¸sıM. (Turkey), Shidlich A.L. (Ukraine), Stanzhitskii O.M. (Ukraine), ¸Sim¸sekD. (Turkey), Tunç T.(Turkey), Ya¸sarE. (Turkey).

Local Organizing Committee

Co-Chairmen of Local Organizing Committee:

OMURALIYEV A., Head of Dep. of Appl. Math. & Infor. (Manas Univ.) URDALETOVA A., Head of Dep. of Math. (Manas Univ.).

Local Organizing Committee:

Abdyldayeva E., Adanbayeva N., Akmatbekova A., Atınta¸sI., Imashkyzy M., Ismailova R., Kara- sheva T., Matanova K., Muhametjanova G.

Secretaries of Conference:

Abylayeva E., Esengul kyzy P.

Contents

Fahreddin Abdullayev. Bernstein-Walsh type inequalities for higher order derivatives of algebraic polynomials ...... 12 Fahreddin Abdullayev, Stanislav Chaichenko, Andrii Shidlich. Jackson type inequalities in the Musielak-Orlicz type spaces ...... 13 Fahreddin Abdullayev, Viktor Savchuk. Fejér-type positive operator based on Takenaka–Malmquist system on unit circle ...... 14 Elmira Abdyldaeva. On solvability of the synthesis problem of optimal control of systems with distributed parameters ...... 15 Elena Afanas’eva. Absolute continuity of quasisymmetric mappings in metric spaces ...... 16 Azat Akmatbekova, Gulshat Muhametjanova. Organization of virtual laboratory work of students in physics in the distance learning system ...... 17 Merve Aktay, Murat Özdemir. On φ− weak Pata contractions in metric spaces . 17 Mahabat Amanalieva. Forecasting the development of tourist regional infrastructures ...... 18 Ruhul Amin, Sahadat Hossain. New concepts of R1 separation in fuzzy bitopological spaces in quasi-coincidence sense ...... 18 Avyt Asanov, Kalyskan Matanova. Approximate solution of the system of Volterra- Stieltjes linear integral equations of the second kind with the generalized trapezoid rule ...... 19 Ismet˙ Altınta¸s,Arzıgul Imankulova.˙ On elementary soft compact spaces . . . . . 20 Ismet˙ Altınta¸s,Hatice Parlatıcı. An invariant of regular isotopy for disoriented diagrams ...... 20 Alexandr Belyaev. Poincaré method in KAM-theory ...... 21 Bilal Bilalov. On the Fredholmness of the Dirichlet problem for a second-order elliptic equation in grand-Sobolev spaces ...... 22 Viktoriia Bilet, Oleksiy Dovgoshey. Metric spaces with finite asymptotic clusters of pretangent spaces ...... 23 Hans-Peter Blatt. Maximal convergence and interpolation on unconnected sets . . 24 Oleksandr Boichuk, Oleksandr Pokutnyi, Viktor Feruk, Dmytro Bihun. Min- imizing of the functionals on Hopfield networks ...... 24 Belkacem Chaouchi, Lakhdar Benaissa. An abstract approach for the study of a boundary value problem for the biharmonic equation set in a iingular domain 25 Asylbek Chekeev. On new results in category of uniform spaces ...... 26 Asylbek Chekeev, Azim Abdiev. Realcompactifications of A-spaces ...... 26 Asylbek Chekeev, Tumar Kasymova. On Dieudonne τ-completeness of a space of compact subsets ...... 27

7 MATHEMATICAL ANALYSIS,DIFFERENTIAL EQUATION &APPLICATIONS - MADEA 9 KYRGYZ-TURKISH MANAS UNIVERSITY,BISHKEK,KYRGYZ REPUBLIC,JUNE 21-25, 2021

Asylbek Chekeev, Ainaz Kasymova, Elnura Zhusupbekova. On m-reflective uniform spaces and their embeddings in topological groups ...... 28 Md. Lalin Chowdhury, Ruhul Amin. New notion in R1 supra fuzzy topological space ...... 29 Sergii Chuiko, Maryna Dziuba. Boundary value problem for a system of matrix differential-algebraic equations with pulse perturbation ...... 29 Sergii Chuiko, Yaroslav Kalinichenko. Linear Noetherian boundary-value problem for a system of linear difference-algebraic equations ...... 30 Sergii Chuiko, Vlada Kuzmina. Boundary-value problems for systems of integral- differential equations of Fredholm type ...... 31 Sergii Chuiko, Olga Nesmelova. Nonlinear boundary-value problems for degenerate differential-algebraic systems ...... 32 Feng Dai. Entropy numbers and Marcinkiewicz-type discretization ...... 33 Roman Dmytryshyn. Approximation of function of several variables by multidimensional A-fractions with independent variables ...... 34 Tzanko Donchev, Alina Lazu. Nonlinear evolution inclusions with time lag . . . 34 Olgun Durmaz, Bu¸sraAkta¸s,Halit Gündogan.˘ Introduction to ordering Dual numbers ...... 35 Olga Dyuzhenkova, Mykola Dudkin. Singularly nonsymmetric finite rank perturbations H−2-class of self-adjoint operators ...... 36 German Dzyubenko, Victoria Voloshyna. Degrees of coconvex approximation of periodic functions ...... 37 Idris˙ Ellik, Ugur˘ Deger.˘ An approximation problem in the weighted Orlicz space . 38 Oksana Fedunyk-Yaremchuk, Svitlana Hembars’ka. Approximative characteristics of the Nikol’skii-Besov-type classes of periodic functions of one and several variables ...... 38 Nalin Fonseka, Ratnasingham Shivaji, Byungjae Son, Keri Spetzer. Elliptic boundary value problems where a parameter affects both the equation and the boundary conditions ...... 39 Valery Gaiko. Limit cycles of multi-parameter polynomial dynamical systems . . 40 Atanaska Georgieva, Mira Spasova. Application of Natural decomposition method to solve nonlinear Volterra fuzzy integro-differential equations ...... 41 Sertac Gokta¸s,Emrah Yılmaz, Ayse Cigdem Yar. Some properties of multiplicative Hermite equation ...... 41 Anatoly Golberg, Elena Afanas’eva. Mapping theory from metric point of view . 42 Cevahir Doganay˘ Gün. Bernstein-Walsh type estimations for the derivatives of arbitrary algebraic polynomials ...... 43 Layan El Hajj. A convexity problem for a semi-linear PDE ...... 43 Snezhana Hristova. Ulam type stability for fractional differential equation with generalized proportional fractional derivatives ...... 44 Samandar Iskandarov, Abdibait Baigesekov. On the influence of integral perturbations of the Volterra-Stiltyes type to the boundedness of solutions of a second-order linear differential equations on the semi-axis ...... 44 Samandar Iskandarov, Atahan Khalilov. On the method of Lyapunov functionals for linear Volterra integro-differential equation of first order with a delay on the half-axis ...... 45

8 Mathematical Analysis, Differential Equation & Applications - MADEA 9 Kyrgyz-Turkish Manas University, Bishkek, Kyrgyz Republic, June 21-25, 2021

Raihanul Islam, Ruhul Amin, Sudipto Kumar Shaha. On T2 separation axioms in fuzzy soft topological spaces ...... 46 Ekrem Kadıoglu. Historical development of univalent functions summary . . . . 46 Muhammet Kamali. Belong to a subclass of analytic functions Fekete-Szegö inequality ...... 46 Tamara Karasheva. New aspects of the physical worldview in the course concepts of modern natural science ...... 47 Olha Karpenko, Victoria Mogylova, Tetiana Dobrodzii. About the relation between the existence of bounded solutions of the differential equations and equations on time scales ...... 48 Leena Kathuria, Shashank Goel, Nikhil Khanna. Some Results on Fourier–Boas- Like Wavelets ...... 49 Yurii Kharkevych, Inna Kal’chuk. Approximation of function from the class r Wβ,∞ by integrals of Abel-Poisson type ...... 49 Ljubiša D.R. Kocinac,ˇ Dragan Djurciˇ c.´ Combinatorics of double sequences . . . 50 Kirill Kopotun. Polynomial approximation with Jacobi and other doubling weights 50 Yulia Kozachenko, Viktor Savchuk. Strong rectangular Fejér means for bounded holomorhic functions in polydisk ...... 51 Grzegorz Kuduk, Michael Symotyuk. Nonlocal problem with integral conditions for homogeneous system of partial differential equations of second order ...... 52 Dilip Kumar. Some properties of fractional Boas transforms of wavelets ...... 53 Tugçe˘ Kunduracı, Ceren Sultan Elmalı, Tamer Ugur.˘ Ditopologies associated with texture graphs ...... 53 Naiyer Mohammadi Lanbaran, Ercan Celik. Using Fuzzy-Rough subset evaluation for feature selection and naive Bayes to classify the Parkinson’s disease 54 Nazim Mahmudov. Multi-delayed perturbation of Mittag-Lefﬂer type functions . 54 Oksana Motorna, Igor Shevchuk. An addendum to Jackson inequality ...... 55 Julia Myslo, Mykhaylo Pahirya. An estimate of the remainder of the interpolation continued C–fraction ...... 56 Burak Ogul,˘ Dagıstan˘ ¸Sim¸sek. Dynamical Behavior of Rational Difference Equa- xn−4 tion xn+ = ...... 57 1 ±1±xnxn−1xn−2xn−3xn−4 Asan Omuraliev, Ella Abylaeva. Asymptotics of the solution of parabolic problems with nonsmooth boundary functions ...... 58 Asan Omuraliev, Peyil Esengul kyzy. Asymptotics of the solution of a parabolic system ...... 59 Mukhtarbai Otelbaev, Nurbek Kakharman. Solution estimates for one class of elliptic and parabolic nonlinear equations ...... 60 N. Pelin Özkartepe, Cevahir Doganay˘ Gün. On some approximate properties of the p-Bieberbach polynomials in closed regions ...... 61 Nataliia Parﬁnovych. The best non-symmetric approximation of classes of differentiable functions by splines ...... 62 Tamara Petrova, Irina Petrova. About generalization of pointwise interpolation estimates of a monotone approximation of functions having a fractional derivative of arbitrary order ...... 62 Kateryna Pozharska. Sampling recovery in the reproducing kernel Hilbert space setting ...... 63

9 MATHEMATICAL ANALYSIS,DIFFERENTIAL EQUATION &APPLICATIONS - MADEA 9 KYRGYZ-TURKISH MANAS UNIVERSITY,BISHKEK,KYRGYZ REPUBLIC,JUNE 21-25, 2021

Mohd Qasim. Approximation by modified Lupa¸s-Stancuoperators based on (p,q)- integers ...... 64 Mohd Qasim. Generalized Jain-Beta operators and their approximation properties 64 Habib ur Rehman, Murat Ozdemir, Poom Kumam. Accelerated explicit Tseng’s extragradient methods for solving different classes of variational inequalities 65 Anatolii Romanyuk, Sergii Yanchenko. Entropy numbers of the Nikol’skii– Besov classes in the space of quasi-continuous functions ...... 65 Olga Rovenska, Viktor Savchuk. An extremal problem for Cesàro means on some class of holomorphic functions ...... 66 Lakhlifa Sadek, Hamad Talibi Alaoui. The extended nonsymmetric block Lanc- zos methods for solving large-scale differential Lyapunov equations . . . . 67 Sabina Sadigova. On Riemann problem in weighted Smirnov classes with power weight ...... 68 Arif Salimov, Tarana Sultanova. Lift problems of differential geometric objects . 68 Hatice Kübra Sarı, Abdullah Kopuzlu. On topological structures of the simple undirected graphs ...... 69 Zhyldyz Sarkelova, Taalaibek Omurov. Inverse loaded two-speed transfer problems of Katz type in an unbounded domain with a collision integral . . . . 69 Ekrem Savas. Deferred almost sequences spaces ...... 70 Viktor Savchuk, Maryna Savchuk. An extremal problem for the invariant differential operators on class of Cauchy type integrals ...... 71 Anatolii Serdyuk, Ulyana Hrabova. Order estimates of the uniform approximations by Zygmund sums on the classes of convolutions of periodic functions 72 Anatolii Serdyuk, Andrii Shidlich. Jackson type inequalities in Besicovitch- Stepanets spaces ...... 73 Anatolii Serdyuk, Igor Sokolenko. On asymptotic equations for the widths of classes of the generalized Poisson integrals ...... 74 Anatolii Serdyuk, Tetiana Stepaniuk. Uniform approximations by Fourier sums on classes ...... 75 Nitin Sharma. Woven Frames in quaternionic Hilbert spaces ...... 76 Mykyta Shchehlov. Pointwise estimate of deviation of Kriakin polynomial from a function, continuous on a segment ...... 76 Olga Shvai. Mathematical modeling as a method of cognition ...... 77 Andriy Stanzhytskyi. The long time behavior of nonlinear stochastic functional- differential equations of neutral type in Hilbert spaces with non-Lipschitz nonlinearities ...... 78 Oleksandr Stanzhytskyi, Roza Uteshova, Meirambek Mukash. The averaging method for boundary value problems for differential equations with non- fixed impulsive moments ...... 78 Tuncay Tunç, Erdem Alper. On statistically convergence theorems for Lebesgue integrable functions ...... 79 Tuncay Tunç, Burcu Fedakar. On Stancu Generalization of Szasz-Mirakyan- Bernstein Operators ...... 80 Anarkul Urdaletova, Syrgak Kydyraliev. Solution of some systems of linear ordinary differential equations with variable coefficients ...... 80 Anarkul Urdaletova, Syrgak Kydyraliev, Elena Burova. Common course "Math- ematics" in the school, as a symbiosis of geometry and algebra ...... 81

10 Mathematical Analysis, Differential Equation & Applications - MADEA 9 Kyrgyz-Turkish Manas University, Bishkek, Kyrgyz Republic, June 21-25, 2021

Sergii Vakarchuk, Michailo Vakarchuk. On the widths of classes of complex variable analytic functions ...... 82 Viktoriia Voloshyna. On sign-preserving approximation of periodic functions by trigonometric polynomials ...... 83 List of participants ...... 84

11 MATHEMATICAL ANALYSIS,DIFFERENTIAL EQUATION &APPLICATIONS - MADEA 9 KYRGYZ-TURKISH MANAS UNIVERSITY,BISHKEK,KYRGYZ REPUBLIC,JUNE 21-25, 2021

ABSTRACTS*

BERNSTEIN-WALSHTYPEINEQUALITIESFORHIGHERORDER DERIVATIVES OF ALGEBRAIC POLYNOMIALS Fahreddin Abdullayev Kyrgyz-Turkish Manas University, Bishkek, Kyrgyz Republic; Mersin University, Turkey [email protected]; [email protected]

Let C be a complex plane and C := C∪{∞}; G ⊂ C be a bounded Jordan region with boundary L := ∂G such that 0 ∈ G; Ω := C \G = extL; ∆ := ∆(0,1) := {w : |w| > 1}. Let w = Φ(z) be the Φ(z) univalent conformal mapping of Ω onto ∆ such that Φ(∞) = ∞ and limz→∞ z > 0. For R > 1, we take LR := {z : |Φ(z)| = R}, GR := intLR and ΩR := extLR. Let ℘n denotes the class of all algebraic polynomials Pn(z) of degree at most n ∈ N, and let h(z) be some weight function. Let 0 < p ≤ ∞ and σ be the two-dimensional Lebesgue measure. For the Jordan region G, we introduce: ZZ 1/p kP k : = kP k := h(z)|P (z)|p d , 0 < p < , n p n Ap(h,G) n σz ∞ G

kPnk∞ : = kPnkA (1,G) := max|Pn(z)|, p = ∞, ∞ z∈G Well known Bernstein -Walsh Lemma [2] says that:

kP k ≤ |Φ(z)|n kP k . (1) n C(GR) n C(G) N. Stylianopoulos in [1] replaced the norm kP k with norm kP k on the right-hand side n C(G) n A2(G) of (1) and found a new version of the Bernstein-Walsh Lemma for the some regions as follows: √ n n+1 |Pn(z)| ≤ c kPnk |Φ(z)| , z ∈ Ω, d(z,L) A2(G) where d(z,L) := inf{|ζ − z| : ζ ∈ L}. In this work, we study forvmore general regions pointwise estimation in unbounded region Ω,

(m) for the derivative Pn (z) ,m = 0,1,2,..., in the following type:

(m) n+1 Pn (z) ≤ ηn(G,h, p,m,d(z,L),|Φ(z)| )kPnkp ,z ∈ Ω where ηn(G,h, p,d(z,L)) → ∞, as n → ∞, depending on the properties of the G, h. [1] Stylianopoulos N., Strong asymptotics for Bergman polynomials over domains with corners and applications, Constructive Approximation, 38 (1), 59-100 (2012). [2] Walsh J. L. Interpolation and Approximation by Rational Functions in the Complex Domain, AMS, 1960.

∗The abstracts are published in the author’s version without making any signiﬁcant changes by the editors.

12 Mathematical Analysis, Differential Equation & Applications - MADEA 9 Kyrgyz-Turkish Manas University, Bishkek, Kyrgyz Republic, June 21-25, 2021 JACKSONTYPEINEQUALITIES INTHE MUSIELAK-ORLICZ TYPE SPACES Fahreddin Abdullayev1, Stanislav Chaichenko2, Andrii Shidlich3 1Kyrgyz-Turkish Manas University, Bishkek, Kyrgyz Republic; Mersin University, Turkey 2Donbas State Pedagogical University, Sloviansk, Ukraine 3Institute of Mathematics of NAS of Ukraine, Kyiv, Ukraine [email protected], [email protected], [email protected]

Let M = {Mk(u)}k∈Z, u ≥ 0, be a sequence of nondecreasing convex functions, Mk(0) = 0, Mk(u) → ∞ as u → ∞. The modular space (or Musilak-Orlicz space) SM is the space of 2π- periodic Lebesgue summable functions f , defined on the real axis such that the following quantity (the Orlicz norm of f ) is finite: n o k f k∗ := sup λ | f (k)| : λ ≥ 0, M (λ ) ≤ 1 , M ∑ k b k ∑ k k k∈Z k∈Z 1 R 2π −ikx where fb(k) = 2π 0 f (x)e dx are the Fourier coefficients of the function f . Consider the set Φ of all continuous bounded nonnegative pair functions ϕ such that ϕ(0) = 0 and the Lebesgue measure of the set {t ∈ R : ϕ(t) = 0} is equal to zero. For a fixed function ϕ ∈ Φ, we define the generalized modulus of smoothness of a function f ∈ SM by the equality: n o ω ( f ,δ)∗ := sup sup λ |ϕ(kh) f (k)| : λ ≥ 0, M (λ ) ≤ 1 , δ ≥ 0. ϕ M ∑ k b k ∑ k k |h|≤δ k∈Z k∈Z Let M (τ), τ > 0, be a set of bounded nondecreasing functions µ that differ from a constant on [0,τ]. By Ω ( f ,τ, µ,u)∗ , u > 0, denote the average value of the generalized modulus of ϕ M smoothness ω ( f ,t)∗ of f with the weight µ ∈ (τ), i.e., ϕ M M Z u ∗ 1 ∗ τt Ωϕ ( f ,τ, µ,u) := ωϕ ( f ,t) dµ . M µ(τ) − µ(0) 0 M u For any function f ∈ , denote by E ( f )∗ its best approximation by the trigonometric poly- SM n M nomials of the order n − 1 in the space SM. Theorem. Assume that f ∈ SM, ϕ ∈ Φ, τ > 0, µ ∈ M (τ). Then for any n ∈ N µ(τ) − µ(0) τ ∗ E ( f )∗ ≤ Ω f ,τ, µ, , (1) n M ϕ In,ϕ (τ, µ) n M where Z τ kt In,ϕ (τ, µ) := inf ϕ dµ(t). k∈N:k≥n 0 n R τ If, in addition, ϕ is non-decreasing on [0,τ] and In,ϕ (τ, µ) = 0 ϕ(t)dµ(t), then inequality (1) can not be improved and therefore, ∗ En( f ) µ(τ) − µ(0) sup M = . Ω ( f ,τ, µ, τ )∗ R τ ϕ(t)dµ(t) f ∈SM ϕ n M 0 f 6=const

Acknowledgements. This work was partially supported by the Kyrgyz-Turkish Manas University (Bishkek / Kyrgyz Republic), project No. KTMÜ-BAP-2019.FBE.02.

13 MATHEMATICAL ANALYSIS,DIFFERENTIAL EQUATION &APPLICATIONS - MADEA 9 KYRGYZ-TURKISH MANAS UNIVERSITY,BISHKEK,KYRGYZ REPUBLIC,JUNE 21-25, 2021

FEJÉR-TYPEPOSITIVEOPERATORBASEDON TAKENAKA–MALMQUISTSYSTEMONUNITCIRCLE Fahreddin Abdullayev1, Viktor Savchuk 2 1 Kyrgyz-Turkish Manas University, Bishkek, Kyrgyz Republic; Mersin University, Turkey 2 Institute of Mathematics of NAS of Ukraine, Kyiv, Ukraine [email protected], [email protected]

Let D := {z ∈ C : |z| < 1} and T := {t ∈ C : |t| = 1}. For a given system of points a := ∞ {ak}k=0, ak ∈ D (the points ak are enumerated taking into account their multiplicity), the extended ∞ Takenaka–Malmquist system ϕ = {ϕk}k=−∞ is defined by p 2 1 − |ak| ϕk(z) := Bk(z), k ∈ Z+, and ϕ−k(t) = tϕk−1(t), k ∈ N, 1 − zak where  1, if k = 0, k−1 Bk(z) := z − a j ∏ , if k ∈ N.  j=0 1 − za j We let by TMS denote the set of all Takenaka–Malmquist systems. 1 If function f ∈ L (T), we associate with f the Fourier series f ∼ ∑k∈Zh f ,ϕkiϕk, on given system ∈ TMS, where h f , i := R f dm, k ∈ . ϕ ϕk T ϕk Z It is well known that ϕ is an orthonormal and complete system in L2(T) if and only if ∞ ∑ (1 − |ak|) = +∞. k=0 1 Consider the Fejér type operator σn,ϕ defined on L (T), as follows:  ix  f (e ), if n = 0,  Z y  2 ix sin γn(t)dt σn, ( f )(e ) := Z π ϕ 1 iy x  f (e ) dy, if n ∈ N 4πγ (x) 2 y − x  n −π sin 2 where n−1 2 1 1 − |ak| γn(t) := ∑ 2 . 2 k=0 1 − 2|ak|cos(t − argak) + |ak| ∞ In this talk, we discus the necessary and sufficient condition on a = {ak}k=0 in order that σn,ϕ ( f ) converges uniformly on T to f for every function f ∈ C(T), where C(T) is the Banach space of continuous functions on T.

Theorem. Suppose that ϕ ∈ TMS. Then for every f ∈ C(T), σn,ϕ ( f ) converges to f uniformly on T as n → ∞ if and only if ∞ 2 1 − |ak| ∑ 2 = ∞, for every t ∈ T. k=0 |1 −tak|

Acknowledgements. This work was supported by the Kyrgyz-Turkish Manas University (Bishkek / Kyrgyz Republic), project No. KTMÜ -BAP-2019.FBE.06.

14 Mathematical Analysis, Differential Equation & Applications - MADEA 9 Kyrgyz-Turkish Manas University, Bishkek, Kyrgyz Republic, June 21-25, 2021 ONSOLVABILITYOFTHESYNTHESISPROBLEMOFOPTIMAL CONTROLOFSYSTEMSWITHDISTRIBUTEDPARAMETERS Elmira Abdyldaeva Kyrgyz-Turkish Manas University, Bishkek, Kyrgyz Republic [email protected]

In the paper consider the synthesis problem of optimal control in the optimization of oscillation processes described by boundary value problem contains the Fredholm integral differential equation. The investigation is conducted according to the methodology of Professor A.I. Egorov developed by him on the basis of the Bellman scheme. Using the concepts of a generalized solution of the boundary value problem and the concept of the Fréchet differential for the Bellman functional, we obtain the integro-differential equation in partial derivatives. The solution procedure of this equation is found. Consider a controlled oscillation process described by the boundary value problem

Z T vtt − Av = λ K(t,τ)v(τ,x)dτ + f [t,x,u(t)], x ∈ Q, 0 < t < T, (1) 0

v(0,x) = ψ1(x), vt(0,x) = ψ2(x), x ∈ Q, (2) n

Γv(t,x) ≡ ∑ aik(x)vxk (t,x)cos(ν,xi) + a(x)v(t,x) = p[t,x,ϑ(t)],x ∈ γ,0 < t < T, (3) i,k=1 where A is an elliptic operator

Av(t,x) = ∑ (aikvxk (t,x))xi − c(x)v(t,x), (4) i,k=1

n Q is area of space R bounded by piecewise smooth curve γ ; f [t,x,u(t)] ∈ H(QT ) , ∀ control u(t) ∈ H(0,T), p[t,x,ϑ(t)] ∈ H(γT ), ∀ of the boundary control ϑ(t) ∈ H(0,T); H(Y) is a Hilbert space of square-summable functions ; λ is parameter; T is ﬁxed point in time; with respect to the function of external and boundary actions, we will assume that

fu[t,x,u(t)] 6= 0, ∀(t,x) ∈ QT ; pϑ [t,x,ϑ(t)] 6= 0, ∀(t,x) ∈ γT , (5) i.e., monotone with respect to the functional variable. In the synthesis problem, it is required to ﬁnd such controls u0(t) ∈ H(0,T) and ϑ 0(t) ∈ H(0,T) which minimizes the integral quadratic functional.

Z Z T 2 2 J[u(t),ϑ(t)] = {[v(T,x)−ξ1(x)] +[vt(T,x)−ξ2(x)] }dx+ {α |u(t)|+β |ϑ(t)|}dt, α,β > 0, Q 0 deﬁned on the set of generalized solutions of the boundary value problem (1)–(5). In this case, the desired controls u0(t) and ϑ 0(t) deﬁned as a function (functional) of the state of the controlled process, i.e. as

0 u (t) = u[t,v(t,x),vt(t,x)], (t,x) ∈ QT , QT = Q × (0,T),

0 ϑ (t) = ϑ[t,v(t,x),vt(t,x)], (t,x) ∈ γT , γT = γ × (0,T).

15 MATHEMATICAL ANALYSIS,DIFFERENTIAL EQUATION &APPLICATIONS - MADEA 9 KYRGYZ-TURKISH MANAS UNIVERSITY,BISHKEK,KYRGYZ REPUBLIC,JUNE 21-25, 2021

[1] Bellman R. The theory of dynamic programming, Bull. Amer. Math. Soc., 60 (6), 503-515 (1954). [2] Lyusternik L. A., Sobolev V.I. Elements of functional analysis, Nauka, Moscow, 1965. [3] Egorov A. Stabilization of the distributed parameter systems optimal stabilization of the distributed parameter systems, Lecture Notes in Computer Science, 27, 167-172 (1975). [4] Kerimbekov A., Abdyldaeva E., Duishenalieva U., and Seidakmat kyzy E. On solvability of optimization problem for elastic oscillations with multipoint sources of control, International Conference Functional analysis in interdisciplinary applications (FAIA2017), AIP Conference Proceedings 1880, 060009 (2017).

ABSOLUTECONTINUITYOFQUASISYMMETRICMAPPINGSIN METRIC SPACES Elena Afanas’eva Institute of Applied Mathematics and Mechanics of NAS of Ukraine, Sloviansk, Ukraine [email protected]

Let (X,d, µ) and (Y,d0, µ0) be two α-regular by Ahlfors metric spaces (α > 1) with locally 0 finite Borel measures µ and µ , respectively. In this talk we define the class ACSE of absolutely continuous functions on almost all compact subsets E ⊂ X and provide a sufficient condition for a mapping f : X → Y to be ACSE. Let η : [0,∞) → [0,∞) be a homeomorphism. A homeomorphism f : X → Y is called an η- quasisymmetric, if d ( f (x), f (y)) d (x,y) Y ≤ η X dY ( f (x), f (z)) dX (x,z) for every triple x,y,z ∈ X, x 6= z; see, e.g. [2]. n Recall that a finite real-valued function F is called absolutely continuous on the set E ⊂ R , if for any ε > 0 there exists δ > 0 such that for every sequence of disjoint segments {[ak,bk]}, whose ends belong to E, the inequality Σk(bk − ak) < δ yields Σk(F(bk) − F(ak)) < ε. The main result of talk is following Theorem. Let (X,d, µ) and (Y,d0, µ0) be two α-regular by Ahlfors metric spaces (α > 1) and f : X → Y be an η-quasisymmetric homeomorphism. Then f belongs to the class of functions absolutely continuous on almost all compact sets E ⊂ X. The talk is based on a joint work with V. Bilet; see [1].

[1] Afanas’eva E. S., Bilet V.V. Some properties of quasisymmetries in metric spaces, Ukr. Mat. Visn., 16 (1), 2–9 (2019) (Russian); translation in J. Math. Sci. (N.Y.) 242 (6), 754–759 (2019). [2] Väisälä J., Vuorinen M., Wallin H. Thick sets and quasisymmetric maps, Nagoya Math. J., 135, 121–148 (1994).

16 Mathematical Analysis, Differential Equation & Applications - MADEA 9 Kyrgyz-Turkish Manas University, Bishkek, Kyrgyz Republic, June 21-25, 2021 ORGANIZATION OF VIRTUAL LABORATORY WORK OF STUDENTS INPHYSICSINTHEDISTANCELEARNINGSYSTEM Azat Akmatbekova, Gulshat Muhametjanova Kyrgyz-Turkish Manas University, Bishkek, Kyrgyz Republic [email protected], [email protected]

Recently, virtual physics laboratories, virtual experiment, virtual laboratory work in physics are a promising area in physical education, which naturally attracts the attention of students and teachers. The relevance of the introduction of virtual laboratories into educational practice is due to the lack of support in the existing distance learning systems for the creation and use of virtual laboratory works in the disciplines of natural science, in particular in physics. However, in the scientific literature, there is not enough research that confirms the effectiveness of the use of virtual physics programs in the formation of the development of professional skills of students. In this article, we tried to analyze the possibilities of using virtual laboratory work in physics, as one of the constituent elements of self-preparation of students to perform real laboratory work. A methodology for performing virtual laboratory work is developed, the preparation and conduct of experimental training is described, and an analysis of its quantitative and qualitative results is given. The study showed that the use of virtual laboratory work in physics allowed students to show the effectiveness of work performance, the speed and high accuracy of the results obtained, which confirms the effectiveness of the proposed methodology. Thus, students can independently form practical skills and abilities at a convenient time for them, without limiting themselves to time and territorial remoteness from the educational organization. It is important to note that learning based on virtual laboratory work creates conditions for the effective manifestation of fundamental patterns of thinking, contributes to the development and activation of students’ creative abilities and optimizes the cognitive process of teaching students. In addition, students have the opportunity to develop skills for independent work, increased interest in the study of physics.

ON φ−WEAK PATA CONTRACTIONS IN METRIC SPACES Merve Aktay, Murat Özdemir Atatürk University, Erzurum, Turkey [email protected], [email protected]

In this paper, we give a φ−weak Pata contractions and establish some ﬁxed point results for such contractions. Our results generalizes some Pata type contractions and Banach contractions. Consequently, the obtained results encompass several results in the literature.

[1] Alber Ya. I., Guerre-Delabriere S. Principle of weakly contractive maps in Hilbert spaces. in: I. Gohberg, Yu. Lyubich (Eds.). New Results in Theory Operator Theory, in: Advances and Appl. Birkhauser, Basel 98, 7–22 (1997).

[2] Banach S. Sur les opérationes dans les ensembles abstraits et leur application aux équation intégrales, Fundam. Math., 3, 133–181 (1922).

[3] Chakraborty M., Samanta S. K. On a ﬁxed point theorem for a cyclical Kannan-type mapping, Facta Univ. Ser. Math. Inform., 28, 179–188 (2013).

[4] Kadelburg Z., Radenovic S. Fixed point theorems under Pata-type conditions in metric spaces, J. Egypt. Math. Soc., 24, 77–82 (2016).

17 MATHEMATICAL ANALYSIS,DIFFERENTIAL EQUATION &APPLICATIONS - MADEA 9 KYRGYZ-TURKISH MANAS UNIVERSITY,BISHKEK,KYRGYZ REPUBLIC,JUNE 21-25, 2021

[5] Nashine H. K., Kadelbur Z. Weaker cyclic (ϕ,φ)−contractive mappings with an application to integro- differential equations, Nonlinear Anal. Model. Control, 18, 427 (2013).

[6] Pata V. A ﬁxed point theorem in metric spaces, J. Fixed Point Theory Appl., 10, 299–305 (2011).

[7] Radenovic S., Kadelburg Z., Jandrlic D., Jandrlic, A. Some results on weakly contractive maps, Bull. Iranian Math. Soc., 38(3), 625 (2012).

[8] Rhoades B. H. Some theorems on weakly contractive maps, Nonlinear Analysis, 47, 2683–2693 (2001).

[9] Zhang Q., Song Y. Fixed point theory for generalized ϕ−weak contractions, Applied Mathematics Letters, 22, 75–78 (2009).

FORECASTINGTHEDEVELOPMENTOFTOURISTREGIONAL INFRASTRUCTURES Mahabat Amanalieva Kyrgyz Economic University, Bishkek, Kyrgyz Republic [email protected]

The article discusses the classiﬁcation of tourist infrastructure by region using the multidimensional average methodology. In addition, an econometric model was obtained for the point forecasting of the development of tourism infrastructure by region.

NEWCONCEPTSOF R1 SEPARATION IN FUZZY BITOPOLOGICAL SPACES IN QUASI-COINCIDENCESENSE Ruhul Amin1, Sahadat Hossain2 1 Begum Rokeya University, Rangpur, Bangladesh 2 University of Rajshahi, Rajshahi, Bangladesh [email protected], [email protected]

In this paper, we have deﬁned some new notions of R1-separation in fuzzy bitopological spaces using quasi-coincidence sense. We have discuss the relations among our and other such notions. We have observed that all these notions satisfy good extension property. We have shown that these notions are preserved under the bijective and FP-continuous mapping. Moreover, we have obtained some special properties of these concepts. Initial and ﬁnal topologies are also studied here.

18 Mathematical Analysis, Differential Equation & Applications - MADEA 9 Kyrgyz-Turkish Manas University, Bishkek, Kyrgyz Republic, June 21-25, 2021 APPROXIMATESOLUTIONOFTHESYSTEMOF VOLTERRA-STIELTJES LINEAR INTEGRAL EQUATIONS OF THE SECONDKINDWITHTHEGENERALIZEDTRAPEZOIDRULE Avyt Asanov, Kalyskan Matanova Kyrgyz-Turkish Manas University, Bishkek, Kyrgyz Republic [email protected], [email protected]

In this paper we investigate the system of linear Volterra-Stieltjes integral equations of the second kind Z x u(x) = K(x,s)u(s)dϕ(s) + f (x), x ∈ [a,b], (1) a where K(x,s) is a given m × m matrix-valued continuous function on G = {(x,s) : a ≤ s ≤ x ≤ b}, f (x) is a given continuous vector-function on [a,b], ϕ(x) is a given strictly increasing continuous function on [a,b] and u(x) is the sought vector-function on [a,b]. The numerical solution of the system of linear Volterra-Stieltjesintegral equations of the second kind (1) is established and investigated by using the generalized trapezoid rule. Also, the conditions on estimation of the error are determined and proved. Methods for the numerical solution of various integral equations were studied by many authors, for example, see [5, 6, 7]. In [2], using the concept of a derivative with respect to an increasing function, were considered systems of Volterra-Stilties integral equations of the ﬁrst and second kind. The notion of derivative of a function by means of a strictly increasing function was given by Asanov in [1]: 0 Deﬁnition. The derivative of a function f (x) with respect to ϕ(x) is the function fϕ (x) , whose value at x ∈ (a,b) is the number

0 f (x + ∆) − f (x) fϕ (x) = lim , ∆→0 ϕ(x + ∆) − ϕ(x)

where ϕ(x) is a given strictly increasing continuous function in (a,b). In the study [3], the generalized trapezoid rule was proposed to evaluate the Stieltjes integral approximately by employing the notion of derivative of a function by means of a strictly increasing function. In [4] the numerical solution of linear Volterra-Stieltjes integral equations of the second kind was obtained by using the generalized trapezoid rule.

[1] Asanov A. The derivative of a function by means of an increasing function, Manas Journal of Engineering, 3 (1), 18–64 (2001) (in Russian). [2] Asanov A. The system of Volterra-Stieltjes integral equations, Manas Journal of Engineering, 1 (4) 65–78 (2003) (in Russian). [3] Asanov A., Chelik M. H. and Chalish A. N. Approximating the Stieltjes integral by using the generalized trapezoid rule, Le Mathematiche, 66 (2), 13–21 (2011). [4] Asanov A., Hazar E., Eroz M., Matanova K and Abdyldaeva E. Approximate solution of Volterra–Stieltjes linear integral equations of the second kind with the generalized trapezoid rule, Advances in Mathematical Physics, 1–6 (2016). [5] Diogo T., Ford N. J., Lima P., Valtchev S. Numerical methods for a Volterra integral equation with non-smooth solutions, Journal of Computational and Applied Mathematics, 189 (1-2), 412–423 (2006). [6] Isaacson S.A. and Kirby R.M. Numerical solution of linear Volterra integral equations of the second kind with sharp gradients, Journal of Computational and Applied Mathematics, 235 (14), 4283–4301 (2011).

19 MATHEMATICAL ANALYSIS,DIFFERENTIAL EQUATION &APPLICATIONS - MADEA 9 KYRGYZ-TURKISH MANAS UNIVERSITY,BISHKEK,KYRGYZ REPUBLIC,JUNE 21-25, 2021

[7] Zhang S., Lin Y., Rao M. Numerical solutions for second-kind Voltera integral equations by Galerkin methods, Applications of Mathematics, 45 (1), 19–39 (2000).

ON ELEMENTARY SOFT COMPACT SPACES Ismet˙ Altınta¸s1,2, Arzıgul Imankulova˙ 1 1 Kyrgyz-Turkish Manas University, Bishkek, Kyrgyzstan 2 Sakarya University, Sakarya, Turkey [email protected], [email protected]

In this paper, we are working on the elementary soft (e-soft) compact spaces. e-soft topological space has been deﬁned in [1,2] and some topological properties have been proved in [1-4]. Today, many studies are carried out on e-soft topological spaces. The concept of compactness in e-soft topological spaces was introduced in [3]. Here we prove the following two properties of e-soft compact spaces.

1. Let (X˜1,τ1,P) and (X˜2,τ2,P) be two e-soft topological spaces. Then (X˜1xX˜2,τ1xτ2,PxP) is e-soft compact if and only if (X˜1,τ1,P) and (X˜2,τ2,P) are e-soft compact. 2. Let (X˜ ,τ,P) be an e-soft topological space. Then a soft set F ∈ S(X˜ ) is e-soft compact if and only if F(λ) is compact in (X,τλ ,P) for all λ ∈ P.

[1] Chiney M., Samanta S. K. Soft topology redeﬁned, arXiv preprint, arXiv: 1701.00466. [2] Ta¸sköprüK., Altınta¸s I.˙ A new approach for soft topology and soft function via soft element, Math. Method. Appl. Sci., Early Access, 2020. [3] Bousselsal M., Saadi A. Soft elemaentary compact in soft elementary topology, arXiv preprint, arXiv:1803.11443. [4] Altınta¸s I,˙ Ta¸sköprüK., Selvi B. Countable and separable soft topological space, Math. Method. Appl. Sci., Early Access, 2020.

ANINVARIANTOFREGULARISOTOPYFORDISORIENTED DIAGRAMS Ismet˙ Altınta¸s1,2, Hatice Parlatıcı2 1 Kyrgyz-Turkish Manas University, Bishkek, Kyrgyzstan 2 Sakarya University, Sakarya, Turkey [email protected], [email protected]

In this work, we deﬁne a two-variable Laurent polynomial invariant of regular isotopy for disoriented knots and links. This polynomial invariant is denoted DK for a disoriented link K, and it satisﬁes the axioms: 1. Regularly isotopic links take on the same polynomial.

2. D◦ = 1.

−1 3. DI+ = aDI0 , DI− = a DI0 . −1 4. D 0 = a D 0 , D 0 = aD 0 . I+ I0 I− I0

20 Mathematical Analysis, Differential Equation & Applications - MADEA 9 Kyrgyz-Turkish Manas University, Bishkek, Kyrgyz Republic, June 21-25, 2021

5. DK+ + DK− = z(DK0 + DK∞ ). where ◦ is the unknot, the other diagrams that are drawn in Figure 1 are small pieces of link diagrams. Regularly isotopy is the equivalence relation generated by the disoriented moves [1] of type II and type III. The existence of the polynomial DK will be proved by examining its behavior under the disoriented moves of type II and III. We prove also that the polynomial EK obtained by normalizing the polynomial DK with the help of the complete writhe [2] is an ambient isotopy invariant for the disoriented link K. It can easily be seen that the polynomial E is an extension of the Kauffman [3] polynomial F to disoriented link diagrams.

Figure 1: Diagrams, crossings and smoothings.

[1] Altınta¸s I.,˙ Parlatıcı H. Redeﬁnig disoriented knots and diagrammatic methods, Math. Method. Appl. Sci. (under review). [2] Altınta¸s I.˙ Introduction to disoriented knot theory, Open Math., 16 (1), 346–357 (2018). [3] Kauffman L. H. An invariant of regular isotopy, Trans. Amer. Math. Soc., 318, 417–471 (1990).

POINCARÉMETHODIN KAM-THEORY Alexandr Belyaev Kyiv, Ukraine [email protected]

Theorem 1. Let the perturbed Hamiltonian system be given by the Hamiltonian H(I,ϕ) = H0(I)+εH1(I,ϕ), where the functions H0,H1 are analytic in torus neighborhood, I = I(I1,...Im),ϕ = (ϕ1,...,ϕm) are the "action-angle" coordinates. Let the following relation be held on the torus I = I0

β m ∂H0 (α,k) > , k ∈ Z ,β,γ > 0, α = |I=I0 , |k|γ ∂I

2 and also the operator ∂ H0 is nondegenerate, then for that is small enough there are power series ∂I2 ε by ε : m n m n In(t)ε ϕn(t)ε Iε (t) = I0 + ∑ , ϕε (t) = ϕ0 + αt + ∑ , (1) n=1 n! n=1 n! determining the solution of the perturbed Hamiltonian system sgradH and converging uniformly in time. At the same time all the functions In(t),ϕn(t) are quasiperiodic and their explicit form can be represented in quadrature.

21 MATHEMATICAL ANALYSIS,DIFFERENTIAL EQUATION &APPLICATIONS - MADEA 9 KYRGYZ-TURKISH MANAS UNIVERSITY,BISHKEK,KYRGYZ REPUBLIC,JUNE 21-25, 2021

Remark. The existence of the series (1) in the theorem 1 is proved in [1]. Theorem 2. Let the solution of the perturbed Hamiltonian system H(I,ϕ) = H0(I)+εH1(I,ϕ), is given by converging series (1), but at the same time there exists an initial condition (I0,ψ0), for which corresponding series

m n m n Jn(t)ε ψn(t)ε Jε (t) = I0 + ∑ , ψε (t) = ψ0 + αt + ∑ n=1 n! n=1 n! are divergent. Then the functions Iε (t),ϕε (t) are Levitan’s functions without being quasiperiodic. Theorem 3. The invariant tori preserved under the perturbation of the Euler case of the 1 classical problem of the motion of a heavy body with a Hamiltonian H = 2 hAp, pi+ε hγ,ri, are determined by solutions having singular points t∗ ∈ C, with the asymptotics

0 −1 2 i 4 p(t) = p˜ t + α1u1 + ∑0 ψit ln t + ∑2 αivit + o(t), −1 0 0 2 i γ(t) = α1v1t + κ1 p˜ lnt + κ0v1 + α4 p˜ + α5v−1t +t ∑0 χi ln t + o(t).

Here arg(t) = const, α1,....α5 are free parameters and all coefﬁcients can be calculated explic- itly. Theorem 4. The fourth integral for perturbed invariant torus of the Euler problem of a motion of a heavy rigid body exists. It is single-valued and analytical in the vicinity of a torus it deﬁnes. This integral cannot be polynomial.

[1] Belyaev A.V. On the direct proof of the Poincaré theorem on invariant tori, Journal of Mathematical Sciences, 181, 18–27 (2012).

ONTHE FREDHOLMNESSOFTHE DIRICHLETPROBLEMFORA SECOND-ORDERELLIPTICEQUATIONINGRAND-SOBOLEV SPACES Bilal Bilalov Institute of Mathematics and Mechanics of NAS of Azerbaijan, Baku, Azerbaijan [email protected]

It is considered Banach function spaces (briefly BFS) in the sense of W.A.Luxembourg and some concrete well known non-standard cases of them. Some properties of these spaces are given. Sobolev spaces of differentiable functions, associated by these BFS are defined and elliptic equations are considered in these Sobolev spaces. Different types of solution of elliptic equation are given. Grand-Lebesgue and the corresponding grand-Sobolev spaces are considered. Some properties of convolution and singular operators in grand-Sobolev spaces are established. A second 2 order elliptic equation with nonsmooth coefficients is considered in grand-Sobolev classes Wq) (Ω) on a bounded n-dimensional domain Ω ⊂ Rn with a sufficiently smooth boundary ∂Ω, generated by the norm of the grand-Lebesgue space Lq) (Ω) is considered. These spaces are non-separable and therefore the definition of a reasonable solution in them faces certain difficulties. For this purpose, 2 a subspace Nq) (Ω) is distinguished in which infinitely differentiable and finite functions are dense. 2 2 2 The strict inclusion Wq (Ω) ⊂ Nq) (Ω) holds, where Wq (Ω) is the classical Sobolev space. This 2 raises specific questions dictated by the theory of spaces Wq (Ω), for example, the characterization

22 Mathematical Analysis, Differential Equation & Applications - MADEA 9 Kyrgyz-Turkish Manas University, Bishkek, Kyrgyz Republic, June 21-25, 2021

1 of the space of traces of functions from Nq) (Ω) cannot be characterized following the classical case. The corresponding theorems concerning traces, extensions, and compactness of a family of k functions from Nq) (Ω) are proved. These results are applied to obtain a Schauder-type estimate up to the boundary. Schauder-type estimates make it possible to establish the fredholmness of the 2 Dirichlet problem for the considered equation in spaces Nq) (Ω) with data from grand-Lebesgue type spaces that are different from Lebesgue spaces. Therefore, the results of this work cannot be directly obtained from the results of the Lp-theory. Acknowledgements. This work is supported by the Scientiﬁc and Technological Research Council of Turkey (TUBITAK) with Azerbaijan National Academy of Sciences (ANAS), Project Number: 19042020 and by the Science Development Foundation under the President of the Republic of Azerbaijan – Grant No. EIF-BGM-4-RFTF1/2017- 21/02/1-M-19.

METRIC SPACES WITH FINITE ASYMPTOTIC CLUSTERS OF PRETANGENT SPACES Viktoriia Bilet, Oleksiy Dovgoshey Institute of Applied Mathematics and Mechanics of NAS of Ukraine, Sloviansk, Ukraine [email protected], [email protected]

Let (X,d) be an unbounded metric space and letr ˜ = (rn)n∈N be a scaling sequence, i.e. a X sequence of positive real numbers tending to infinity. A pretangent space Ω∞,r˜ to (X,d) at infinity is a limit of the rescaling sequence X, 1 d . The set of all pretangent spaces ΩX is called an rn ∞,r˜ asymptotic cluster of pretangent spaces. Such a cluster can be considered as a weighted graph X (GX,r˜,ρX ) whose maximal cliques coincide with Ω∞,r˜ and the weight ρX is defined by metrics on X Ω∞,r˜. We describe the structure of metric spaces having finite asymptotic clusters of pretangent spaces. Let (X,d) be an unbounded metric space and let p be a point of X. Denote by Xn the set of all n n-tuples x = (x1,...,xn) with xk ∈ X for k = 1,...,n, n ≥ 2 and define the function Fn : X → R as  0, if (x1,...,xn) = (p,..., p)   min d(xk, p) ∏ d(xk,xl)  1≤k≤n 1≤k

Denote also n r o A(p,r,k) := x ∈ X : ≤ d(x, p) ≤ rk k and S(p,r) := {x ∈ X : d(x, p) = r} for r > 0 and k ≥ 1. The set S(p,r) is the sphere in (X,d) with the radius r and the center p. Ana- logically, we can consider A(p,r,k) as an annulus in (X,d) “bounded” by the concentric spheres r S(p,rk) and S(p, k ). In particular, the annulus A(p,r,1) coincides with the sphere S(p,r). Theorem. [1] Let (X,d) be an unbounded metric space, p ∈ X, and let n ≥ 2 be an integer number. Then the vertex set |V(GX,r˜)| ≤ n

23 MATHEMATICAL ANALYSIS,DIFFERENTIAL EQUATION &APPLICATIONS - MADEA 9 KYRGYZ-TURKISH MANAS UNIVERSITY,BISHKEK,KYRGYZ REPUBLIC,JUNE 21-25, 2021 holds for every scaling sequence r˜ if and only if

lim Fn(x1,...,xn) = 0 x1,...,xn→∞ and diam(A(p,r,k)) diam(S(p,r)) lim lim = lim = 0, k→1 r→∞ r r→∞ r n where r ∈ (0,∞) and k ∈ [1,∞), and the function Fn : X → R is deﬁned in (1). Remark. The annulus A(p,r,k) can be void. We use the convention diam(∅) = 0. [1] Bilet V., Dovgoshey O. Finite asymptotic clusters of metric spaces, Theory and Applications of Graphs, 5 (2), 33 pp. (2018).

MAXIMALCONVERGENCEANDINTERPOLATIONON UNCONNECTEDSETS Hans-Peter Blatt Catholic University Eichstaett-Ingolstadt, Eichstaett, Germany [email protected]

A theorem of Grothmann states that interpolating polynomials to a holomorphic function on a compact set E are maximally convergent to f only if a subsequence of the interpolation points converge to the equilibrium distribution of E in the weak* sense. Grothmann‘s proof applies only for connected sets E. The objective of this paper is to provide a new necessary condition for maximal convergence which is the crucial tool to prove Grothmann‘s theorem for unconnected sets E.

MINIMIZINGOFTHEFUNCTIONALSON HOPFIELD NETWORKS Oleksandr Boichuk, Oleksandr Pokutnyi, Viktor Feruk, Dmytro Bihun Institute of Mathematics of NAS of Ukraine, Kyiv, Ukraine [email protected], [email protected], [email protected], [email protected]

We consider a continuous Hopﬁeld model with a weak interaction of the network’s neurons, the evolution in time of which is described by a system of n non-linear differential equations (see [1, p. 690]) ! x (t) n a x t 0 j ˆ i i( ) x j(t) = − + ε Ij(t) + ∑ wi j tanh + Ij(t), j = 1,n, (1) R j i=1 2

1 where x j(t) ∈ C [0,T] is the potential of jth neuron; real parameters a j — gain coefﬁcients of the jth neuron, and wi j are elements of symmetric matrix W:   0 w12 ... w1n w 0 ... w   12 2n W =  . . .. . ,  . . . .  w1n w2n ... 0

24 Mathematical Analysis, Differential Equation & Applications - MADEA 9 Kyrgyz-Turkish Manas University, Bishkek, Kyrgyz Republic, June 21-25, 2021

which consists of synaptic weights of the connection of the ith neuron with the jth neuron, R j — 1 1 leakage resistance, Iˆj(t) ∈ C [0,T], Ij(t) ∈ C [0,T] are external signals (toque), ε << 1 is a small parameter characterizing the strength of the interaction of network’s neurons. Using the theory of pseudo-inverse matrices (see [2]), we investigate the questions of finding conditions for the existence and effective construction of solutions of the equation (1) with m boundary conditions lx(·,w,ε) = α, 1 m 1 l = col l1, l2,..., lm : C [0,T] → R is bounded linear vector functional, lν : C [0,T] → m R, ν = 1,m, α = col α1, α2,..., αm ∈ R , which for ε = 0 turns into the solution of the generating problem x0(t) = Ax(t) + I(t), lx(·) = α. An iterative algorithm for finding solutions with a quadratic rate of convergence has been constructed. The problem of finding the extremum of the target functions on the given problem’s solution is considered. To minimize a functional, an accelerated method of conjugate gradients is used. Results are illustrated with examples for the case of three neurons. Acknowledgements. Authors are grateful the financial support of the National Research Founda- tion of Ukraine (Project number 2020.02/0089).

[1] Haykin S. Neural networks and learning machines.3nd ed., New York: Pearson Education, 2009. [2] Boichuk A. A., Samoilenko A. M. Generalized inverse operators and Fredholm boundary value problems. 2nd ed., Berlin, Boston: Walter de Gruyter, 2016.

ANABSTRACTAPPROACHFORTHESTUDYOFABOUNDARY VALUEPROBLEMFORTHEBIHARMONICEQUATIONSETINA IINGULARDOMAIN Belkacem Chaouchi1, Lakhdar Benaissa2 1 Khemis Miliana University, Khemis Miliana, Algeria 2 University of Algiers Benyoucef Benkhedda, Algiers, Algeria [email protected], [email protected]

In this work, we will investigate a boundary value problem for biharmonic equation set in a ingular domain Ω containing a cuspidal point. Existence and maximal regularity results are obtained for the classical solutions by using the fractional powers of linear operators.

[1] Azzam A., Kreyszig E. On solutions of Elliptic Equations Satisfying Mixed Boundary Conditions, SIAM J. Math. Anal. 13, 254–262 (1982). [2] Berroug T. Problèmes aux Limites pour une Equation Différentielle Abstraite du Second Ordre, Thèse d’état. Université de Havre, 2006. [3] Chaouchi B., Labbas R., Sadallah B. K. Laplace Equation on a Domain With a Cuspidal Point in Little Hölder Spaces, Mediterranean Journal of Mathematics 10 (1), 157–175 (2013).

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ON NEW RESULTS IN CATEGORY OF UNIFORM SPACES Asylbek Chekeev Balasagyn Kyrgyz National University, Bishkek, Kyrgyz Republic [email protected]

From the beginning of works of J.Tukey [1], D.Kurepa [2], Yu.M.Smirnov [3], J.Isbell [4] and Czech Mathematics School the theory of uniform spaces took a central places in topology and has an application in topological algebra and functional analysis. The valuable contribution to the development of the theory of uniform spaces was made by V.V.Fedorcukˇ [5,6,7], E.V. Shchepin [8], A.A. Borubaev [9], K.L.Kozlov [10] and others. A.A.Borubaev solved the number of fundamental problems in the the theory of uniform spaces. He constructed the absolutes of uniform spaces and also uniform embeddings into n-dimensional Euclid spaces, varuous extensions of uniform spaces and etc. [9]. In this talk the new results of the theory of uniform spaces will be presented, generalizations of known results will be shown [11]. Namely, a functor from the category of uniform spaces to the category of topological groups will be constructed , and a number of properties of this functor will be proved [12].

[1] Tukey J. W. Convergence and uniformity in topology. Ann. of Math Studies, Princeton University Press, 1940. [2] Kurepa D. Sur les classes (E) et (D), Bul. Math. Belgrade, 5, 124–136 (1936). [3] Smirnov Yu.M. On proximity spaces, Mat. sbornik, 31, 543–574 (1952). [4] Isbel J. R. Uniform spaces, Providence, 1964. [5] Fedorcukˇ V.V. Some questions in the theory of ordered spaces, Sib. Mat. journ. 10, 172–187 (1969). [6] Fedorcukˇ V.V. Uniform spaces and perfect irreducible mappings of topological spaces, Sov. Math. Dokl., 193, 1228–1230 (1970). [7] Fedorcukˇ V.V., Kunzi H.-P. Uniformly open mappings and uniform embeddings of function spaces, Topol. Appl. 61, 61–84 (1995). [8] Shchepin E. V. On a problem of Isbell, Sov. Math. Dokl. 16, 685–687 (1975). [9] Borubaev A. A. Uniform spaces and uniformly continuous mappings, Frunze: Ilim, 1991, 171 pp. (in Russian). [10] Kozlov K. L. Rectangularity of products and completions of their subsets, Topol. Appl., 157, 698–707 (2010). [11] Chekeev A. A., Kasymova T. J. Ultraﬁlter-completeness on zero-sets of uniformly continuous functions, Topol.Appl., 252, 27–41 (2019). [12] Chekeev A. A., Kasymova T. J., Kasymova A. B. The Dieudonné τ-complete spaces and free topological groups of uniform spaces, Topol. Appl., 281, 107–210 (2020).

REALCOMPACTIFICATIONS OF A-SPACES Asylbek Chekeev, Azim Abdiev Balasagyn Kyrgyz National University, Bishkek, Kyrgyz Republic [email protected], [email protected]

Deﬁnition 1 [1]. A cozero-ﬁeld on the set X is a family A of subsets of X satisfying

(a) /0, X ∈ A .

(b) A is closed under ﬁnite intersection and countable union.

26 Mathematical Analysis, Differential Equation & Applications - MADEA 9 Kyrgyz-Turkish Manas University, Bishkek, Kyrgyz Republic, June 21-25, 2021

(c) If A,B ∈ A with (X \ A) ∩ (X \ B) = /0,then there are disjoint A1,B1 ∈ A with X \ A ⊂ A1 and X \ B ⊂ B1. T (d) If A ∈ A , then there are A1,A2,... ∈ A with X \ A = n∈N An. An Alexandroff space, or A-space, is a pair (X,A ), where A is a cozero-field on X. The sets in A are called the cozero-sets of (X,A ), and the complements are called zero-sets. An A-map (or coz-map) f : (X,A ) → (Y,B) between A-spaces is a map with f −1(B) ⊂ A . We will consider a category, whose objects are A-spaces and morphisms are A-maps. The set of morphisms we denote by A(X,Y), and A(X,R), where R is real line, is denoted by A(X). A(X) is uniformly closed inversion-closed algebra [1]. Definition 2. A-spaces (X,A ) and (Y,B) are said to be A-homeomorphic, if there exists a bijective A-map f : (X,A ) → (Y,B) such that a converse map f −1 : (Y,B) → (X,A ) is A-map too. All cozero-sets of uniformly continuous functions with ordinary uniformity Rτ form a cozero- field [2]. Definition 3. A-space (X,A ) is said to be real compact A-space or A-real compact, if it is τ A-homeomorphic to the closed subspace R for some cardinal τ ≥ ℵ0. It takes place the following generalization of the Hewitt Theorem [3]. Theorem. For any A-space X there exists the unique (up to A-homeomorphism) A-realcompact space νX having the following properties:

(i) there exists a A-homeomorphic embedding υ : X → υX such that υ(X) = υX.

(ii) For any A-function f : X → R ( f ∈ A(X)) there exists A-continuous function f˜ : υX → R such that f˜◦ υ = f .

A-realcompact space υX is called a realcompactiﬁcation of A-space X.

[1] Hager A. W. Real-valued functions on Alexandroff (zero-set) spaces, Comment. Math. Univ. Carol., 16 (4), 755–769 (1975). [2] Chekeev A. A. Uniformities for Wallman compactiﬁcations and realcompactiﬁcations, Topol. Appl., 201, 145– 156 (2016). [3] Hewitt E. Rings of real-valued continuous functions I, Trans. Amer. Math. Soc. 64, 45–99 (1948).

ON DIEUDONNE τ-COMPLETENESSOF A SPACE OF COMPACT SUBSETS Asylbek Chekeev, Tumar Kasymova Balasagyn Kyrgyz National University, Bishkek, Kyrgyz Republic [email protected], [email protected]

τ We denote as Mτ a set of all metrics spaces of a weight ≤ τ, and by uX a uniformity generated on the Tychonoff space X by all continuous mappings into spaces of Mτ [6], µτ X denotes a τ τ completion of X with respect to the uniformity uX [3]. In works [1, 2, 5, 6] the uniformity uX has been described by another ways. These results allow to obtain the spectral characterization for Dieudonne τ-complete spaces. Theorem 1. For Tychonoff space X the following are equivalent:

27 MATHEMATICAL ANALYSIS,DIFFERENTIAL EQUATION &APPLICATIONS - MADEA 9 KYRGYZ-TURKISH MANAS UNIVERSITY,BISHKEK,KYRGYZ REPUBLIC,JUNE 21-25, 2021

(1) X = µτ X, i.e. X is Dieudonne τ-complete.

β β (2) X = limS, where S = {Xα , fα ,β ≥ α} is an inverse system consisting of spaces of Mτ and ←− β fα are continuous "short" projections.

For Tychonoff space X by expcX we denote set of all compact subsets from X, equipped by the Vietoris topology [7]. For continuous mapping f : X → Y between Tychonoff spaces X and Y, assuming f˜(A) = f (A) for every A ∈ expcX, we obtain a continuous mapping f˜ : expcX → expcY [4]. α α α If S = {Xα ,πβ ,Σ} is an inverse system, then expcS = {expcXα ,π˜β ,Σ}, where π˜β : expcXα → expcX deﬁned as above, is an inverse system too, and limS˜ is homeomorphic to exp(limS) [8]. β ←− ←− By using all above we obtain the following Theorem 2. Tychonoff space X is Dieudonne τ-complete iff expcX is Dieudonne τ-complete. [1] Alo A. R., Shapiro H. L. Normal Topological Spaces, Cambridge University Press, 1974. [2] Chekeev A. A., Kasymova T. J. A Note on Dieudonne Complete Spaces, Filomat, 32 (14), . 5131–5136 (2018). [3] Chekeev A. A., Kasymova T. J., Kasymova A. B. The Dieudonné τ-complete spaces and free topological groups of uniform spaces, Topol. Appl., 281, 107210 (2020). [4] Choban M. Note sur la topologie exponentielle, Fund. Math., 71, 27–41 (1971). [5] Di Concilio A. Uniform properties and hyperspace topologies for ℵ-uniformities, Topol.Appl., 44, 115–123 (1992). [6] Kozlov K. L. Rectangularity of products and completions of their subsets, Topol. Appl., 157, 698–707 (2010). [7] Michael E. Topologies on spaces of subsets, Trans. Amer. Math. Soc., 71, 152–182 (1951). [8] Zenor P. On the completeness of the space of compact subsets, Proc. Amer. Math. Soc., 26, 190–192 (1970).

ON m-REFLECTIVE UNIFORM SPACES ANDTHEIREMBEDDINGSINTOPOLOGICALGROUPS Asylbek Chekeev, Ainaz Kasymova, Elnura Zhusupbekova Balasagyn Kyrgyz National University, Bishkek, Kyrgyz Republic [email protected], [email protected], [email protected]

For any uniform space uX all finite and countable uniform coverings form uniformity [2]. In work [3] it is proved that under the assumption of the generalized continuum hypothesis (i.e. under the assumption that for every cardinals n,m such that n < m the next inequality 2n ≤ m holds) all uniform coverings of a cardinality < m form the uniformity. In work [6] it is proved that the question does not depend on ZFC-axioms. So, it takes place Definition. Let m ≥ ℵ0 be an arbitrary cardinal. Uniform space uX is called m-reflective, if all uniform coverings from u of a cardinality < m form the uniformity um. It is clear, that every uniform space uX is ℵ0 and ℵ1-reflective [2] and if m = |exp(expX)|, then every uniform space uX is m-reflective. Let uX be m-reflective uniform space. We denote by smX a completion of X with respect to the uniformity um. It is evident, that sℵ0 X = suX is the

Samuel compactiﬁcation, and sℵ1 X = υuX is the Hewitt completion of the uniform space uX [2]. Theorem 1. A uniform space uX is m-reﬂective iff smX coincides with the completion of X with respect to the uniformity u.

28 Mathematical Analysis, Differential Equation & Applications - MADEA 9 Kyrgyz-Turkish Manas University, Bishkek, Kyrgyz Republic, June 21-25, 2021

For a uniform space uX we denote by F(uX) the free topological group [4, 5, 1]. Theorem 2. A uniform space uX is m-reﬂective iff the group F(uX) is topologically embedded into the group F(smX). Corollary 1 [4]. A uniform space uX is precompact iff the group F(uX) is topologically embedded into the group F(suX). Corollary 2 [4]. A uniform space uX is ℵ0-bounded iff the group F(uX) is topologically embedded into the group F(υuX).

[1] Chekeev A. A., Kasymova T. J., Kasymova A. B. The Dieudonné τ-complete spaces and free topological groups of uniform spaces, Topol. Appl., 281, 107210 (2020). [2] Isbell J. R. Uniform spaces: Mathematical Survey, Providence, 1964. [3] Kucia A. On coverings of a uniformity, Coll. Math., 27, 73–74 (1973). [4] Nakayama T. Note on free topological groups, Proc. Imp. Acad. Sci., 19, 471–475 (1943). [5] Nummela E. C. Uniform free topological groups and Samuel compactiﬁcation, Topol. Appl., 13, 77–83 (1982). [6] Pelant J. , Cardinal reﬂections and point character of uniformities, Seminar Uniform Spaces 1973–74, directed by Z. Frolík, MÚ CSAVˇ Praha, 149–158(1975).

NEWNOTIONIN R1 SUPRAFUZZYTOPOLOGICALSPACE Md. Lalin Chowdhury, Ruhul Amin Begum Rokeya University, Rangpur, Bangladesh [email protected], [email protected]

Sometime we need to minimize the conditions of topology for different reasons such as obtaining more convenient structures to describe some real-life problems, or constructing some counter examples which shows the interrelations between certain topological concepts, or preserving some properties under fewer conditions of those on topology. To contribute this research area, in this paper, we establish some notions of R1 separation axioms in supra fuzzy topological space in quasi-coincidence sense. Also we investigate some of its properties and establish certain relationship among them and other such concepts. Moreover some of their basic properties are examined. The concept of separation axioms is one of the most important parts in fuzzy mathematics, mainly modern topological mathematics, which plays an important role in modern networking system.

BOUNDARY VALUE PROBLEM FOR A SYSTEM OF MATRIX DIFFERENTIAL-ALGEBRAICEQUATIONSWITHPULSE PERTURBATION Sergii Chuiko1, Maryna Dziuba2 1 Donbas State Pedagogical University, Sloviansk, Ukraine, 2 Donbas State Engineering Academy, Kramatorsk, Ukraine, [email protected]

We construct necessary and sufﬁcient conditions for the existence of solutions [1]

1 1 α×β Z(t) ∈ Cα×β {[a,b] \{τi}I} := C {[a,b] \{τi}I} ⊗ R , i = 1, 2,..., p

29 MATHEMATICAL ANALYSIS,DIFFERENTIAL EQUATION &APPLICATIONS - MADEA 9 KYRGYZ-TURKISH MANAS UNIVERSITY,BISHKEK,KYRGYZ REPUBLIC,JUNE 21-25, 2021 of linear matrix boundary value problem for a system of matrix differential-algebraic equations with pulse perturbation

0 µ×ν A Z (t) = BZ(t) + F(t), t 6= τi, L Z(·) = A, A ∈ R , (1) where [2] 0 1 A Z (t) : Cα×β {[a,b] \{τi}I} → Cγ×δ {[a,b] \{τi}I} matrix differential-algebraic operator and [2]

p 1 µ×ν L Z(·) := ∑ LiZ(·) : Cα×β {[a,b] \{τi}I} → R , i = 1, 2,..., p., i=0

1 µ×ν LiZ(·) : Cα×β [τi,τi+1[→ R , i = 0,..., p − 1, τ0 := a, 1 µ×ν LpZ(·) : Cα×β [τp,b] → R . We also construct a generalized Green’s operator of the linear boundary conditions for a system of matrix differential-algebraic equations with pulse perturbation. To solve the matrix differential- algebraic boundary problem with pulse action used original solvability conditions and the structure of the general solution of the linear matrix equation. In the case of nonsolvability, the difference- algebraic boundary-value problem for a system of matrix differential-algebraic equations with pulse perturbation can be regularized analogously [4,5]

[1] Boichuk A. A., Samoilenko A. M. Generalized Inverse Operators and Fredholm Boundary-value Problems, 2-nd edition, Walter de Gruyter GmbH & Co KG, 2016. [2] Chuiko S. M., Dzyuba M. V. Matrix boundary-value problem with pulsed action, Nonlinear Oscillations (N.Y.), 238 (3), 333–343 (2019). [3] Chuiko S. M. On a reduction of the order in a differential-algebraic system, Journal of Mathematical Sciences, 235 (1), 2–14 (2018). [4] Tikhonov A. N., Arsenin V.Ya. Solution of Ill-Posed Problems, Winston, Washington, DC, 1977. [5] Chuiko S. M. On the regularization of a linear Fredholm boundary-value problem by a degenerate pulsed action, Journal of Mathematical Sciences, 197, 138–150 (2014).

LINEAR NOETHERIAN BOUNDARY-VALUE PROBLEM FOR A SYSTEM OF LINEAR DIFFERENCE-ALGEBRAIC EQUATIONS Sergii Chuiko, Yaroslav Kalinichenko Donbas State Pedagogical University, Sloviansk, Ukraine, [email protected]

We investigate the problem of ﬁnding bounded solutions [1–3]

n z(k) ∈ R , k ∈ Ω := {0, 1, 2, ... , ω} of linear Noetherian (n 6= υ) boundary-value problem for a system of linear difference-algebraic equations υ A(k)z(k + 1) = B(k)z(k) + f (k), `z(·) = α, α ∈ R ; (1)

30 Mathematical Analysis, Differential Equation & Applications - MADEA 9 Kyrgyz-Turkish Manas University, Bishkek, Kyrgyz Republic, June 21-25, 2021

m×n here A(k), B(k) ∈ R are bounded matrices and f (k) are real bounded column vectors,

n υ `z(·) : R → R is a linear bounded vector functional deﬁned on a space of bounded functions. We assume that the matrix A(k) is, generally speaking, rectangular: m = n. It can be square, but singular. The problem of ﬁnding bounded solutions z(k) of a boundary-value problem for a linear non-degenerate [2]

detB(k) 6= 0, k ∈ Ω system of ﬁrst-order difference equations

υ z(k + 1) = B(k)z(k) + f (k), `z(·) = α ∈ R was solved by A.A. Boichuk [2]. Thus, the boundary-value problem (1) is a generalization of the problem solved by A.A. Boichuk. We investigate the problem of ﬁnding bounded solutions linear Noetherian boundary-value problem for a system of linear difference-algebraic equations (1) in case 1 ≤ rank A(k) = σ0, k ∈ Ω. We construct necessary and sufﬁcient conditions for the existence of solution of linear boundary-value problem for a system of difference-algebraic equations in the critical and noncritical case [3]. In the case of nonsolvability, the difference-algebraic boundary- value problems can be regularized analogously [4,5].

[1] Boichuk A. A., Samoilenko A. M. Generalized Inverse Operators and Fredholm Boundary-value Problems, 2-nd edition, Walter de Gruyter GmbH & Co KG, 2016. [2] Boichu A. A. Boundary-value problems for systems of difference equations, Ukrainian Math. J., 49 (6), 930– 934 (1997). [3] Chuiko S. M., Chuiko E. V., Kalinichenko Y.V. Boundary-value problems for systems of linear difference- algebraic equations, Journal of Mathematical Sciences, 254 (2), 318–333 (2021). [4] Tikhonov A. N., Arsenin V.Ya. Solution of Ill-Posed Problems, Winston, Washington, DC, 1977. [5] Chuiko S. M. On the regularization of a linear Fredholm boundary-value problem by a degenerate pulsed action, Journal of Mathematical Sciences, 197, 138–150 (2014).

BOUNDARY-VALUEPROBLEMSFORSYSTEMSOF INTEGRAL-DIFFERENTIAL EQUATIONS OF FREDHOLMTYPE Sergii Chuiko, Vlada Kuzmina Donbas State Pedagogical University, Sloviansk, Ukraine [email protected]

We investigate the problem of ﬁnding solutions [1,2]

2 0 2 y(t) ∈ D [a;b], y (t) ∈ L [a;b] of linear Noetherian (n 6= υ) boundary-value problem for a system of linear integral-differential equations of Fredholm type with degenerate kernel

Z b 0 0 p A(t)y (t) = B(t)y(t) + Φ(t) F(y(s),y (s),s)ds + f (t), `y(·) = α, α ∈ R . (1) a

31 MATHEMATICAL ANALYSIS,DIFFERENTIAL EQUATION &APPLICATIONS - MADEA 9 KYRGYZ-TURKISH MANAS UNIVERSITY,BISHKEK,KYRGYZ REPUBLIC,JUNE 21-25, 2021

We seek a solution of the Noetherian boundary-value problem (1) in a small neighborhood of the solution 2 0 2 y0(t) ∈ D [a;b], y0(t) ∈ L [a;b] of the generating problem 0 A(t)y0(t) = B(t)y0(t) + f (t), `y0(·) = α. (2) Here

2 2 m×n 2 2 A(t),B(t) ∈ Lm×n[a;b] := L [a;b] ⊗ R , Φ(t) ∈ Lm×q[a;b], f (t) ∈ L [a;b]. We assume that the matrix A(t) is, generally speaking, rectangular: m 6= n. It can be square, but singular. Assume that the function F(y(t),y0(t),t) is linear with respect to unknown y(t) in a small neighborhood of the generating solutions and with respect to the derivative y0(t) in a small 0 0 neighborhood of the function y0(t). In addition, we assume that the function F(y(t),y (t),t) is con- p tinuous in the independent variable t on the segment [a,b]; `y(·) : D2[a;b] → R is a linear bounded vector functional defined on a space D2[a;b]. The problem of finding solutions of a boundary-value problem (1) in case A(t) = In was solved by A.M. Samoilenko and A.A. Boichuk [1,2]. Thus, the boundary-value problem (1) is a generalization of the problem solved by A.M. Samoilenko and A.A. Boichuk. We investigate the problem of finding solutions linear Noetherian boundary-value problem (2) in the paper [3]. We found the conditions of the existence and constructive scheme for finding the solutions of the linear integro-differential boundary-value problem (1).

[1] Boichuk A. A., Samoilenko A. M. Generalized Inverse Operators and Fredholm Boundary-value Problems, 2-nd edition, Walter de Gruyter GmbH & Co KG, 2016. [2] Boichuk A. A., Samoilenko A. M., Krivosheya S.A. Boundary value problems for systems of integro-differential equations with degenerate kernel, Ukrainian Math. J., 48 (11), 1785–1789 (1996). [3] Chuiko S. M. On a reduction of the order in a differential-algebraic system, Journal of Mathematical Sciences, 235 (1), 2–14 (2018).

NONLINEAR BOUNDARY-VALUE PROBLEMS FOR DEGENERATE DIFFERENTIAL-ALGEBRAICSYSTEMS Sergii Chuiko1, Olga Nesmelova2 1 Donbas State Pedagogical University, Sloviansk, Ukraine, 2Institute of Applied Mathematics and Mechanics of NAS of Ukraine, Sloviansk, Ukraine [email protected], [email protected]

We will study the problem of construction of solutions

1 z(t,ε) : z(·,ε) ∈ C [a,b], z(t,·) ∈ C[0,ε0] of the seminonlinear differential-algebraic boundary-value problem [1,2] A(t)z0(t,ε) = B(t)z(t,ε) + f (t) + ε Z(z,t,ε), `z(·,ε) = α + ε J(z(·,ε),ε). (1) We will seek the solutions of the boundary-value problem (1) in a small neighborhood of a solution 1 z0(t) ∈ C [a,b] of the generating Noetherian (n = k) boundary-value problem [3] 0 A(t)z0(t) = B(t)z0(t) + f (t), `z0(·) = α. (2)

32 Mathematical Analysis, Differential Equation & Applications - MADEA 9 Kyrgyz-Turkish Manas University, Bishkek, Kyrgyz Republic, June 21-25, 2021

Here, A(t), B(t) ∈ Cm×n[a,b] are continuous matrices, f (t) ∈ C[a,b] is a continuous vector; Z(z,t,ε) is a nonlinear function which is continuously differentiable with respect to the unknown z(t,ε) in a small neighborhood of a solution of the generating problem, continuous in t ∈ [a,b], and continuous in a small parameter; `z(·,ε) and J(z(·,ε),ε) are, respectively, a linear and nonlinear vector functionals, k `z(·,ε), J(z(·,ε),ε) : C[a,b] → R . Moreover, the second functional is continuously differentiable with respect to the unknown z(t,ε) and continuous in the small parametere ε in a small neighborhood of a solution of the generating problem (2) and on the segment [0,ε0]. The seminonlinear differential-algebraic boundary-value problem (1) is a generalization of nu- merous statements of nonlinear boundary-value problems [1]. We will study the case of degeneration [3] of the generating boundary-value problem (2), namely: PA∗ (t) 6= 0; here, PA∗ (t) is the m ∗ orthoprojector [1]: PA∗ (t) : R → N(A (t)). Generally speaking, the degenerate system (2) is not solvable relative to the derivative. The necessary and sufﬁcient conditions of solvability of seminonlinear differential-algebraic boundary-value problem (1) and a convergent iterative scheme of construction of approximations to their solutions are found. An improved classiﬁcation of the solutions of such problems is given.

[1] Boichuk A. A., Samoilenko A. M. Generalized Inverse Operators and Fredholm Boundary-value Problems, 2-nd edition, Walter de Gruyter GmbH & Co KG, 2016. [2] Chuiko S. M., Nesmelova O. V. Nonlinear boundary-value problems for degenerate differential-algebraic systems, Journal of Mathematical Sciences, 252 (4), 799–803 (2021). [3] Chuiko S. M. A generalized Green operator for a linear Noetherian differential-algebraic boundary value problem, Siberian Advances in Mathematics, 30, 177–191 (2020).

ENTROPYNUMBERSAND MARCINKIEWICZ-TYPE DISCRETIZATION Feng Dai University of Alberta, Edmonton, Canada [email protected]

This is a joint work with A. Prymak, A. Shadrin, V.N. Temlyakov, and S. Tikhonov. In this talk I will discuss the behavior of the entropy numbers of classes of functions with bounded integral norms from a given ﬁnite dimensional linear subspace. Upper bounds of these entropy numbers in the uniform norm are obtained and applied to establish a Marcinkiewicz-type discretization result for these classes.

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APPROXIMATIONOFFUNCTIONOFSEVERALVARIABLESBY MULTIDIMENSIONAL A-FRACTIONSWITHINDEPENDENT VARIABLES Roman Dmytryshyn Vasyl Stefanyk Precarpathian National University, Ivano-Frankivsk, Ukraine [email protected]

Let N be a ﬁxed natural number and Ik = i(k) : i(k) = (i1,i2,...,ik), 1 ≤ ip ≤ ip−1, 1 ≤ p ≤ k, i0 = N , k ≥ 1, be the sets of multiindices. We consider the problem of approximation of functions of several variables by multidimensional A-fractions with independent variables

N i δi ,i i δi ,i pi( )zi 1 (−1) 1 2 pi( )zi zi 2 (−1) 2 3 pi( )zi zi ∑ 1 1 ∑ 2 1 2 ∑ 3 2 3 ··· , 1 + q zi + 1 + q zi + 1 + q zi + i1=1 i(1) 1 i2=1 i(2) 2 i3=1 i(3) 3 where the pi(k) ∈ C\{0} and qi(k) ∈ C for i(k) ∈ Ik, k ≥ 1, and δi, j is a Kronecker delta, z = N (z1,z2,...,zN) ∈ C . An algorithm for the expansion of the given formal multiple power series into the corresponding multidimensional A-fraction with independent variables was constructed in [3]. Some questions of convergence of the above-mentioned branched continued fractions were considered in the works [1, 2, 4, 5]. For some functions of several variables, we have constructed multidimensional A-fractions with independent variables representations and investigated their domains of convergence.

N ai(1) [1] Antonova T. M., Dmytryshyn R. I. Truncation error bounds for branched continued fraction ∑i =1 1 + a a 1 i1 i(2) i2 i(3) ..., Ukr. Math. J., 71 (7), 1018–1029 (2020). ∑i2=1 1 + ∑i3=1 1 + [2] Antonova T. M., Dmytryshyn R. I. Truncation error bounds for branched continued fraction whose partial de- nominators are equal to unity, Mat. Stud., 54 (1), 3–14 (2020). [3] Bodnar D. I., Dmytryshyn R. I. Multidimensional associated fractions with independent variables and multiple power series, Ukr. Math. J., 71 (3), 370–386 (2019). [4] Dmytryshyn R. I. Convergence of multidimensional A- and J-fractions with independent variables, Comput. Methods Funct. Theory, 2021. doi:10.1007/s40315-021-00377-6 [5] Dmytryshyn R. I. Convergence of some branched continued fractions with independent variables, Mat. Stud., 47 (2), 150–159 (2017).

NONLINEAREVOLUTIONINCLUSIONSWITHTIMELAG Tzanko Donchev1, Alina Lazu2 1 University of Architecture, Civil Engineering and Geodesy, Soﬁa, Bulgaria 2 "Gh. Asachi" Technical University, Ia¸si, Romania [email protected], [email protected]

The theory of causal differential equations has the powerful quality of unifying ordinary differential equations, integro-differential equations, differential equations with ﬁnite or inﬁnite delay, Volterra integral equations and neutral equations.

34 Mathematical Analysis, Differential Equation & Applications - MADEA 9 Kyrgyz-Turkish Manas University, Bishkek, Kyrgyz Republic, June 21-25, 2021

In this paper we study multivalued perturbations of m-dissipative evolution equations with multivalued term depending on causal operators. Namely we study the following problem ( x˙(t) ∈ Ax(t) + fx(t), t ∈ (t0,T) fx(t) ∈ F(t,Q(x)(t))

with initial condition x(s) = ϕ(s), s ∈ [−τ,0] - local initial condition. x(s) = g(x(·))(s), s ∈ [−τ,0] - non local initial condition. Here A is m-dissipative operator, F is a multimap and Q is casual operator. Existence of solution and some qualitative properties of the solution set are considered under dissipative type conditions. Illustrative example is then provided. Acknowledgements The work was supported by the Bulgarian National Science Fund under Project KP- 06-N32/7.

INTRODUCTIONTOORDERING DUALNUMBERS Olgun Durmaz1, Bu¸sraAkta¸s,2, Halit Gündogan˘ 2 1 Atatürk University, Erzurum, Turkey 2 Kırıkkale University, Kırıkkale, Turkey [email protected], [email protected], [email protected]

A dual number has the form a = a + εa∗, where a and a∗ are real numbers and ε is called dual unit that satisfies the condition ε2 = 0. Inequalities are basic tools used in various fields of mathematics including calculus, algebra and geometry. That’s why, it is clear that the order relation is needed so as to carry out mathematical studies. In this paper, using the dictionary order relation defined on R2, we investigate the properties of order relations in the dual numbers system.

[1] Beckenbach F. E., Bellman R. Inequalities. Berlin: Springer-Verlag, 1961. [2] Dimentberg F. M. The Screw Calculus and its Applications in Mechanics. Ohio: Foreign Technology Division, Wright-Patterson Air Force Base, 1965. [3] Hardy G. H., Littlewood J. E., Polya G. Inequalities. Cambridge: Cambridge Univ. Press., 1934. [4] Halmaghi E., Liljedahl P. Inequalities in the History of Mathematics: From Peculiarities to a Hard Discipline, GeoGebra International Journal of Romania, 4 (2), 43–56 (2015). [5] Frydryszak M. A. Dual Numbers and Supersymmetric Mechanics, Czech. J. Phys., 55 (11), 1409–1414 (2005). [6] Herman J., Kucera R., Simsa J. Equations and Inequalities: Elementary Problems and Theorems in Algebra and Number Theory. Berlin-New York: Springer-Verlag, 2000. [7] Kotelnikov A. P. Screw Calculus and Some of its Applications in Geometry and Mechanics. Kazan: Annals of the Imperial University, 1895. [8] Mitrinovic S. D. Analytic Inequalities. Berlin-New York: Springer-Verlag, 1970. [9] Munkres J. R. Topology. The Second Edition, ISBN 0-13-181629-2, 2000. [10] Study E. Geometrie der Dynamen. Leipzig: Druck und Verlag von B.G. Teubner, 1903. [11] Wald R. M. A new Type of Gauge Invariance for a Collection of Massles Spin-2 Fields, II. Geometrical Inter- pretation, Class. Quant. Gav., (5) 4, 1279–1316 (1987).

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[12] Veldkamp G. R. On the Use of Dual Numbers, Vectors and Matrices in Instantaneous Spatial Kinematics, Mechanism and Machine Theory, 11, 141–156 (1976).

SINGULARLY NONSYMMETRIC FINITE RANK PERTURBATIONS H−2-CLASSOFSELF-ADJOINT OPERATORS Olga Dyuzhenkova, Mykola Dudkin National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Kyiv, Ukraine [email protected], [email protected]

We generalize results of [1,2] on the case of nonsymmetric H−2-class perturbations of finite rank. Namely, we consider the formal expression n ˜ A = A + ∑ α jh·,ω jiδ j, j=1 where A is unperturbed self-adjoint operator in the separable Hilbert space H , α j ∈ C, 0 < |αi| < ∞ and ω j, δ j, j = 1,2,...,n < ∞ vectors from the negative space H−2, constructed by A. The action of A on vectors from H−2 is underspending as the extension be continuity. n n Definition. For sets of linearly independent vectors {ω j} j=1 ⊂ H−2, {δ j} j=1 ⊂ H−2, n < ∞, n n such that Ω∩H = {0}, ∆∩H = {0}, where Ω := span{ω j} j=1, ∆ := span{δ j} j=1, the operator ˜ ˜ n A is called singularly rank n perturbed H−2-class with respect to A (and denoted A ∈ P−2(A)), if for a fixed z ∈ ρ(A) its domain has a form: n ˜ −1 D(A) = {ϑ = φ − ∑ αibi, j(z)hφ,ωii(A − z) δ j | φ ∈ D(A)}, i, j=1 where b (z) are elements of the matrix inverse to G(z) = ( 1 θ + h(1 + zA)(1 + A2)−1δ ,(A − i, j αi i j j −1 n z¯) ωii)i, j=1, θi j – Kronecker symbol, under the condition det G(z) 6= 0; and ˜ ˙ −1 n D(A) =DH2 +span{(A − z) δ j} j=1, −1 DH2 = φ ∈ D(A) | ((A − z)φ,(A − z¯) ω j) = 0, j = 1,2,...,n , under the condition detG(z) = 0; and the action on vectors from D(A˜) is given by the rule (A˜ − z)ϑ = (A − z)φ. Theorem. The resolvent Rz, z ∈ ρ(A) of the unperturbed operator A and R˜z, z ∈ ρ(A˜) – the ˜ n perturbed one A ∈ P−2(A) in H are connected by like M.Krein formula: n ˜ ˜ Rz = Rz + ∑ bi, j(z)(·,ni(z¯))m j(z), z,ξ ∈ ρ(A) ∩ ρ(A) i, j=1 with vector-valued functions −1 −1 n j(z) = (A − ξ)(A − z) n j(ξ), m j(z) = (A − ξ)(A − z) m j(ξ), j = 1,2,...,n, −1 n where ni(z) = Rzδi, m j(z) = Rzω j, and the matrix-valued function G(z) = {bi, j(z)}i, j=1 is such that G(z) − G(ξ) = (z − ξ)Γ(ni(ξ),m j(z¯)), where Γ( ·· ) is the Gramm-matrix of correspondence vectors.

36 Mathematical Analysis, Differential Equation & Applications - MADEA 9 Kyrgyz-Turkish Manas University, Bishkek, Kyrgyz Republic, June 21-25, 2021

[1] Albeverio S., Kurasov P. Singular perturbations of differential operators; solvable Schrödinger type operators. Cambridge, Univ. Press, 2000. [2] Dudkin M. E., Vdovenko T. I. Dual pair of eigenvalues in rank one singular perturbations, Mat. Stud., 48 (2), 156–164 (2017).

DEGREESOFCOCONVEXAPPROXIMATIONOF PERIODICFUNCTIONS German Dzyubenko1, Victoria Voloshyna2 1Institute of Mathematics of NAS of Ukraine, Kyiv, Ukraine 2Taras Shevchenko National University of Kyiv, Ukraine; University of Toulon, La Garde, France [email protected], [email protected]

2s In [1] we prove, that for each r ∈ N, n ∈ N and s ∈ N there are a collection {yi}i=1 of points (∞) y2s < y2s−1 < ··· < y1 < y2s + 2π =: y0 and a 2π - periodic function f ∈ C (R), such that

2s 00 f (t)∏(t − yi) ≥ 0, t ∈ [y2s,y0], (1) i=1 and for each trigonometric polynomial Tn of degree ≤ n (of order ≤ 2n + 1), satisfying

2s 00 Tn (t)∏(t − yi) ≥ 0, t ∈ [y2s,y0], (2) i=1 the inequality r−1 (r) n k f − TnkC(R) ≥ crk f kC(R) holds, where cr > 0 is a constant, depending only on r. 2s Moreover, we prove, that for each r = 0,1,2 and any such collection {yi}i=1 there is a 2π - (r) i−1 periodic function f ∈ C (R), such that (−1) f is convex on [yi,yi−1], 1 ≤ i ≤ 2s, and, for each ∞ sequence {Tn}n=0 of trigonometric polynomials Tn, satisfying (2), we have r n k f − TnkC( ) limsup R = + , (r) ∞ n→∞ ω4( f ,1/n) where ω4 is the fourth modulus of continuity. Note, that in [2] the corresponding positive result is proved, i.e. for any 2π - periodic function (r) i−1 f ∈ C (R), such that (−1) f is convex on [yi,yi−1], 1 ≤ i ≤ 2s, there exists a trigonometric polynomial Tn, satisfying (2), for which

k f − TnkC(R) ≤ csω3( f ,1/n), n ≥ N, where cs is a constant, depending only on s, N is a number, depending only on Ys.

[1] Dzyubenko G, Voloshyna V., Yushchenko L. Negative results in coconvex approximation of periodic functions, Journal of Approx. Theory, 267 (7), 2021. [2] Zalizko V. Coconvex approximation of periodic functions, Ukr. Math. J., 59 (1), 28–44 (2007); translated from Ukr. Mat. Zh., 59 (1), 29–43 (2007).

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AN APPROXIMATION PROBLEM INTHEWEIGHTED ORLICZ SPACE Idris˙ Ellik, Ugur˘ Deger˘ Mersin university, Mersin, Turkey [email protected], [email protected]

Recently, the investigation of approximation problems for spaces that are more general than Lebesgue space has been one of the curious topics. In [3], the author has considered approximation to functions belonging to the Lipschitz class by Woronoi-Nörlund mean and Riesz mean in weighted Lebesgue space with variable exponent. What we will do here is to address the speed of approach by the more general methods including the trigonometric polynomials given in [3] for the extended Lipschitz classes in the weighted Orlicz space [1-2]. We see that the speed depends on the used methods and the properties of the function class under consideration.

[1] Israﬁlov D. M., Guven A. Approximation by trigonometric polynomials in weighted Orlicz spaces, Studia Math., 174 (2), 147–168 (2006). [2] Guven A., Israﬁlov D. On approximation in weighted Orlicz spaces, Mathematica Slovaca, 62(1), 77–86 (2012). [3] Testici A. Approximation by Nörlund and Riesz means in weighted Lebesgue space with variable exponent, Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 68(2), 2014– 2025 (2019).

APPROXIMATIVECHARACTERISTICSOFTHE NIKOL’SKII-BESOV-TYPECLASSESOFPERIODICFUNCTIONSOF ONEANDSEVERALVARIABLES Oksana Fedunyk-Yaremchuk, Svitlana Hembars’ka Lesya Ukrainka Volyn National University, Lutsk, Ukraine [email protected], [email protected]

Ω We study the classes Bp,θ of periodic functions of one and several variables [1] with Ω(t) = d ω ∏ j=1 t j , where ω is a given function (of one variable) of the type of a modulus of continuity α of order l that satisﬁes the conditions (S ) and (Sl), which are called the Bari-Stechkin conditions Ω [2]. For a certain choice of function Ω, the classes Bp,θ coincide with analogs of the well-known r Nikol’skii-Besov classes Bp,θ [3]. d Let L∞(πd),πd = ∏ j=1[0;2π), be the space of essentially bounded functions f (x) = f (x1,...,xd), which are 2π-periodic in each variable, with the norm k f k∞ = esssup| f (x)|. x∈πd B Ω We obtain exact order estimates of similar to the Fourier-widths quantities dM(Bp,θ ,B∞,1) in Ω the space B∞,1, which norm is stronger than the L∞-norm. For the functional classes Bp,θ ⊂ B∞,1, these quantities are deﬁned as follows:

dB (BΩ ,B ) = inf sup k f − G f k . M p,θ ∞,1 B∞,1 G∈LM(B)∞ Ω f ∈Bp,θ ∩D(G)

Here by LM(B)∞ we denote the set of linear operators satisfying the conditions:

38 Mathematical Analysis, Differential Equation & Applications - MADEA 9 Kyrgyz-Turkish Manas University, Bishkek, Kyrgyz Republic, June 21-25, 2021

a) the domain of deﬁnition D(G) of these operators contains all trigonometric polynomials, and their domain of values is contained in a subspace with dimension M of the space B∞,1; b) there exists a number B ≥ 1 such that, for all vectors k = (k1,...,kd), k j ∈ Z, j = 1,d, the

inequality Gei(k,·) ≤ B holds. 2 The following theorems hold true. d α Theorem 1. Let d ≥ 2, 1 ≤ θ ≤ ∞, Ω(t) = ω(∏ j=1 t j , where ω satisﬁes condition (S ) with n d−1 some α > 1 and condition (Sl). Then for any M,n ∈ N, such that M 2 n , the relation

B Ω −n n (d−1)(1−1/θ) dM(B1,θ ,B∞,1) ω(2 )2 n

holds. Theorem 2. Let d = 1, 1 ≤ θ ≤ ∞, and ω satisﬁes condition (Sα ) with some α > 1 and condition (Sl). Then the relation

B Ω −1 dM(B1,θ ,B∞,1) ω(M )M holds. Remark. In the one-dimensional case, in contrast to the multidimensional one, the obtained estimates are independent of the parameter θ.

[1] Yongsheng S., Heping W. Representation and approximation of multivariate periodic functions with bounded mixed moduli of smoothness, Tr. Mat. Inst. Steklova, 219, 356–377 (1997). [2] Bari N.K., Stechkin S.B. The best approximations and differential properties of two conjugate function, Trans. Moscow Math. Soc., 5, 483–522 (1997). [3] Lizorkin P.I., Nikol’skii S.M. Spaces of functions with mixed smoothness from the decomposition point of view, Tr. Mat. Inst. Steklova, 187, 143–161 (1989).

ELLIPTIC BOUNDARY VALUE PROBLEMS WHERE A PARAMETER AFFECTS BOTH THE EQUATION AND THE BOUNDARY CONDITIONS Nalin Fonseka1, Ratnasingham Shivaji2, Byungjae Son3, Keri Spetzer4 1 Carolina University, Winston-Salem, USA 2 University of North Carolina at Greensboro, Greensboro, USA 3 University of Maine, Orono, USA 4 University of North Carolina at Greensboro, Greensboro, USA [email protected], [email protected], [email protected], [email protected]

We study positive solutions to steady state reaction diffusion equations of the form: ( −∆u = λ f (u); Ω, ∂u ∂η + µ(λ)u = 0; ∂Ω,

N ∂u where λ > 0, Ω is a bounded domain in R ; N ≥ 1 with smooth boundary ∂Ω, ∂η is the outward 2 normal derivative of u, µ ∈ C([0,∞)) is strictly increasing such that µ(0) ≥ 0 and f ∈ C ([0,r0)) 2 with 0 < r0 ≤ ∞. If r0 < ∞ we assume f ∈ C ([0,r0]) with f (r0) = 0 and f (s) ≤ 0 for s ∈ (r0,∞),

39 MATHEMATICAL ANALYSIS,DIFFERENTIAL EQUATION &APPLICATIONS - MADEA 9 KYRGYZ-TURKISH MANAS UNIVERSITY,BISHKEK,KYRGYZ REPUBLIC,JUNE 21-25, 2021

f (s) and if r0 = ∞ we assume lim f (s) > 0 and lim = 0 (sublinear at ∞). Note here that the pa- s→∞ s→∞ s rameter λ inﬂuences both the equation and the boundary condition. We discuss existence, nonex- istence, multiplicity and uniqueness results for the cases when (A) f (0) = 0, (B) f (0) < 0, and (C) f (0) > 0. We obtain existence and multiplicity results by the method of sub-super solutions and uniqueness results by comparison principles and a priori estimates.

LIMIT CYCLES OF MULTI-PARAMETER POLYNOMIAL DYNAMICALSYSTEMS Valery Gaiko United Institute of Informatics Problems, National Academy of Sciences of Belarus, Minsk, Belarus [email protected]

We carry out a global qualitative analysis of multi-parameter polynomial dynamical systems. To control all their limit cycle bifurcations, especially, bifurcations of multiple limit cycles, it is necessary to know the properties and combine the effects of all their rotation parameters. It can be done by means of the development of new bifurcational geometric methods based on Perko’s planar termination principle [1]. This principle is a consequence of the principle of natural termination which was applied by A. Wintner for studying one-parameter families of periodic orbits of the restricted three-body problem to show that in the analytic case any one-parameter family of periodic orbits can be uniquely continued through any bifurcation except a period-doubling bifurcation. Such a bifurcation can happen, e. g., in a three-dimensional Lorenz system. But this cannot happen for planar systems. That is why the Wintner–Perko termination principle is applied for studying multiple limit cycle bifurcations of planar polynomial dynamical systems [1]. If we do not know the cyclicity of the termination points, then, applying canonical systems with ﬁeld rotation parameters, we use geometric properties of the spirals ﬁlling the interior and exterior domains of limit cycles. Applying this approach, we have solved, e. g., Hilbert’s Sixteenth Problem on the maximum number and distribution of limit cycles for the general Liénard polynomial system with an arbitrary number of singular points [2], the Kukles cubic-linear system [3], the Euler–Lagrange–Liénard polynomial mechanical system [4], Leslie–Gower systems which model the population dynamics in real ecological or biomedical systems [5] and a reduced planar quartic Topp system which models the dynamics of diabetes [6]. Finally, applying a similar approach, we have considered various applications of three-dimensional polynomial dynamical systems and, in particular, completed the strange attractor bifurcation scenario in Lorenz type systems globally connecting the homoclinic, period-doubling, Andronov–Shilnikov, and period-halving bifurcations of their limit cycles [7].

[1] Gaiko V.A. Global Bifurcation Theory and Hilbert’s Sixteenth Problem. Boston: Kluwer, 2003. [2] Gaiko V.A. Maximum number and distribution of limit cycles in the general Liénard polynomial system, Adv. Dyn. Syst. Appl., 10 (2), 177–188 (2015). [3] Gaiko V.A. Global bifurcation analysis of the Kukles cubic system. Int. J. Dyn. Syst. Differ. Equ., 8 (4), 326–336 (2018). [4] Gaiko V.A., Savin S. I., Klimchik A. S. Global limit cycle bifurcations of a polynomial Euler–Lagrange– Liénard system. Comput. Res. Model., 12 (4), 693–705 (2020). [5] Gaiko V.A., Vuik C. Global dynamics in the Leslie–Gower model with the Allee effect. Int. J. Bifurcation Chaos, 28 (12), 1850151 (2018).

40 Mathematical Analysis, Differential Equation & Applications - MADEA 9 Kyrgyz-Turkish Manas University, Bishkek, Kyrgyz Republic, June 21-25, 2021

[6] Gaiko V.A., Broer H. W., Sterk A. E. Global bifurcation analysis of Topp system. Cyber. Phys., 8 (4), 244–250 (2019). [7] Gaiko V.A. Global bifurcation analysis of the Lorenz system. J. Nonlinear Sci. Appl., 7 (6), 429–434 (2014).

APPLICATION OF NATURALDECOMPOSITIONMETHODTO SOLVE NONLINEAR VOLTERRA FUZZY INTEGRO-DIFFERENTIAL EQUATIONS Atanaska Georgieva, Mira Spasova University of Plovdiv, Plovdiv, Bulgaria aﬁ[email protected], [email protected]

In this paper, we consider the following nonlinear Volterra fuzzy integro-differential equation

x Z u(n)(x) = g(x) ⊕ k(x − s) G(u(s))ds, x ∈ [o,b], 0

(i) u (0) = pi where k : [0,b] → R is continuous function, G : E1 → E1 is continuous function on E1, g, u : 1 1 1 [0,b] → E are continuous fuzzy functions and pi ∈ E are constant. The set E is the set of all fuzzy numbers. We develop a method to obtain approximate solutions for this equations with the help of Natural decomposition method (NDM). The technique is based on the application of Natural transform to nonlinear Volterra fuzzy integro-differential equations. The nonlinear term can easily be handled with the help of Adomian polynomials. We illustrate this technique with the help of example.

Acknowledgements. This research was funded by the Bulgarian National Science Fund under Project KP-06- N32/7.

SOME PROPERTIES OF MULTIPLICATIVE HERMITE EQUATION Sertac Gokta¸s1, Emrah Yılmaz2 and Ayse Cigdem Yar3 1Mersin University, Mersin, Turkey 2,3Firat University, Elazıg,˘ Turkey [email protected], [email protected], [email protected]

In 1960’s, Grossman and Katz [5] constructed a comprehensive family of calculus that includes classical calculus as well as an inﬁnite sub-branches of non-Newtonian calculus. Non-Newtonian calculus is divided into many sub-branches as geometric, anageometric, bio- geometric, quadratic and harmonic calculus, etc. Geometric calculus, which is one of these, is also deﬁned as multiplicative calculus. Changes of arguments and values of a function are measured by differences and ratios in multiplicative calculus, respectively, while they are measured by differences in the classical case. Multiplicative calculus is especially useful in situations where products and ratios provide the natural methods of combining and comparing magnitudes. Multiplicative calculus has been addressed in many studies by some authors [1-9].

41 MATHEMATICAL ANALYSIS,DIFFERENTIAL EQUATION &APPLICATIONS - MADEA 9 KYRGYZ-TURKISH MANAS UNIVERSITY,BISHKEK,KYRGYZ REPUBLIC,JUNE 21-25, 2021

In this study, we consider the multiplicative Hermite equation:

y∗∗(y∗)−2ty2n = 1 where, n is a real number [7]. We reconstruct this equation from multiplicative Sturm-Liouville equation. Eigenfunctions of the constructed problem are obtained by the power series solution technique. While making these solutions, multiplicative Hermite polynomials were used strongly. We get a generator for multiplicative Hermite polynomials and construct integration representations for these polynomials. Finally, some spectral properties of the multiplicative Hermite problem are examined in detail.

[1] Bashirov A. E., Kurpınar E. M., Özyapıcı A. Multiplicative calculus and its applications, Journal of Mathemat- ical Analysis and Applications, 337, 36–48 (2008). [2] Bashirov A. E., MısırlıE.,˜ Tandogdu Y., Özyapıcı A. On modeling with multiplicative differential equations, Applied Mathematics-A Journal of Chinese Universities, 26 (4), 425–438 (2011). [3] Boruah K., Hazarika B. G-Calculus, TWMS Journal of Applied and Engineering Mathematics, 8 (1), 94–105 (2018). [4] Florack L., Assen Hv. Multiplicative calculus in biomedical image analysis, Journal of Mathematical Imaging and Vision, 42, 64–75 (2012). [5] Grossman M., Katz R. Non-Newtonian calculus. Pigeon Cove:Lee Press, 1972. [6] Stanley D. A multiplicative calculus, Primus, 9 (4), 310–326 (1999). [7] Yalçın N. The solutions of multiplicative Hermite differential equation and multiplicative Hermite polynomials. Rend. Circ. Mat. Palermo, II. Ser. 70, 9–21 (2021). [8] Yalçın N., Çelik E., Gokdogan A. Multiplicative Laplace transform and its applications, Optik, 127, 9984–9995 (2016). [9] Yalçın N., Çelik E. Solution of multiplicative homogeneous linear differential equations with constant expo- nentials, NTMSCI, 6 (2), 58–67 (2018).

MAPPINGTHEORYFROMMETRICPOINTOFVIEW Anatoly Golberg1, Elena Afanas’eva2 1 Holon Institute of Technology, Holon, Israel 2 Institute of Applied Mathematics and Mechanics of NAS of Ukraine, Sloviansk, Ukraine [email protected], [email protected]

In the talk we study the main relationships between various classes of mappings whose defi- nitions rely on metric approaches and techniques: finitely bi-Lipschitz mappings, quasisymmetric mappings, quasimöbius and quasiconformal mappings, mappings of finite metric distortion and mappings of finite area distortion. The latter is the cental object in our presentation. Although no analytic restrictions are assumed, some nice and important regularity properties like the Lusin (N)- condition, absolute continuity on almost all compacts, etc. are derived. We also involve classes of mappings which are called the ring, lower and hyper Q-homeomorphisms and are defined purely geometrically. The interplay between the above classes of mappings allows us to investigate the boundary correspondence problems related to the weakly flat and strongly accessible boundaries on Riemannian manifolds. Several illustrated examples are also presented.

42 Mathematical Analysis, Differential Equation & Applications - MADEA 9 Kyrgyz-Turkish Manas University, Bishkek, Kyrgyz Republic, June 21-25, 2021 BERNSTEIN-WALSH TYPE ESTIMATIONS FOR THE DERIVATIVES OF ARBITRARY ALGEBRAIC POLYNOMIALS Cevahir Doganay˘ Gün Gaziantep University, Gaziantep, Turkey [email protected]

Let C denote the complex plane; G ⊂ C be a bounded Jordan region with boundary L := ∂G, 0 ∈ G; Pn(z) algebraic polinom of degree at most n ∈ N; h(z) be a generalized Jacobi weight l function respect to the points z j j=1 ∈ L. For any p > 0 and Jordan region G, we introduce:

 1/p ZZ kP k : = kP k := h(z)|P (z)|p d ,0 < p < , n p n Ap(h,G)  n σz ∞ G

kPnk∞ : = kPnkA (1,G) := max|Pn(z)|, p = ∞, ∞ z∈G

and Ap(1,G) ≡ Ap(G), where σ be the two-dimensional Lebesgue measure. 0 In this work, we study the pointwise estimations for the derivative |Pn(z)| in unbounded region Ω with zero angles as the following type

0 Pn(z) ≤ c2ηn(G,h, p,d(z,L),|Φ(z)|)kPnkp , z ∈ Ω, where c1 = c1(G, p) > 0 is a constant independent of n,z and ηn(G,h, p,d(z,L),|Φ(z)|) → ∞, n → ∞, depending on the properties of the G, h and from the distance of point z ∈ Ω to the G.

A CONVEXITYPROBLEMFORASEMI-LINEAR PDE Layan El Hajj American University in Dubai, Dubai, UAE [email protected]

In this paper we prove convexity of super-level sets of a semi-linear PDE with a non-monotone right hand side, and with a free boundary

( n ∆u = χ{0

Here D is assumed to be convex, and n ≥ 2. The main difﬁculty of this problem is that the right hand side is non-monotone and no apriori regularity is known about the boundary ∂{u > 0}.

43 MATHEMATICAL ANALYSIS,DIFFERENTIAL EQUATION &APPLICATIONS - MADEA 9 KYRGYZ-TURKISH MANAS UNIVERSITY,BISHKEK,KYRGYZ REPUBLIC,JUNE 21-25, 2021

ULAM TYPE STABILITY FOR FRACTIONAL DIFFERENTIAL EQUATIONWITHGENERALIZEDPROPORTIONALFRACTIONAL DERIVATIVES Snezhana Hristova University of Plovdiv, Plovdiv, Bulgaria [email protected]

This paper is dedicated to studying the Ulam type stability of solutions to a system of fractional differential equations with the generalized proportional fractional derivatives of Riemann-Liouville type n R α,ρ (a D u)(t) = ψ1 (t,u(t),v(t)), t ∈ [a,∞), equipped with the initial value fractional integral conditions:

n 1−α,ρ (aI u)(a) = c,

R α,ρ where ρ ∈ (0,1], a D denotes the generalized proportional fractional derivatives of Riemann- 1−α,ρ Liouville type of order α ∈ (0,1], aI is the generalized proportional fractional integrals of order 1−α and ψ1,ψ2 : [a,b]×R×R → R are given continuous functions. An equivalent formula for the initial value problem given by the generalized proportional fractional integral is obtained. An expicit solution for the linear generalized proportional fractional differential equation is given. Sufﬁcint conditions for Ulam ype stability of the studied equations are obtained. An example is proposed to illustrate the applicability of the theoretical results.

Acknowledgments. This research was funded by the Bulgarian National Science Fund under Project KP-06- N32/7.

ON THE INFLUENCE OF INTEGRAL PERTURBATIONS OF THE VOLTERRA-STILTYESTYPETOTHEBOUNDEDNESSOF SOLUTIONSOFASECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS ON THE SEMI-AXIS Samandar Iskandarov1,a, Abdibait Baigesekov2,b 1 Institute of Mathemtics of NAS of Kyrgyz Republic, Bishkek, Kyrgyz Republic 2 Sulyukta Humanitarian and Economic Institute, Batken SU, Kyrgyz Republic [email protected], [email protected]

All functions and their derivatives are continuous and the relations take place at t ≥ t0, t ≥ τ ≥ t0. The Problem. Establish sufﬁcient conditions for the boundedness of all solutions and their ﬁrst derivatives on the half-interval I = [t0,∞) linear integro-differential Volterra-Stieltjes equation:

00 Z t x (t) + a(t)x(t) + K(t,τ)x´(τ)dg(τ) ≥= f (t), t ≥ t0 (1) t0 in the case, when the corresponding linear homogeneous and inhomogeneous differential equations: 00 x (t) + a(t)x(t) = 0, t ≥ t0

44 Mathematical Analysis, Differential Equation & Applications - MADEA 9 Kyrgyz-Turkish Manas University, Bishkek, Kyrgyz Republic, June 21-25, 2021

00 x (t) + a(t)x(t) = f (t), t ≥ t0 can have unbounded on a half-interval I solutions. Note that in (1) g(t) is increasing function and dg(t) is understood in the sense of the deﬁnition from [1]. To solve the problem under consideration, a method is developed based on the use of the ideas of the works [2,3].

[1] Asanov A. The derivative of a function with respect to an increasing function, Tabigiy ilimder journals. Bishkek: Kyrgyz-Turkish University "Manas", 18–64 (2001) (In Russian). [2] Iskandarov S., Baygesekov A. M. On the asymptotic properties of solutions of a weakly nonlinear integro- differential equation of the second order Volterra-Stieltjes on the semiaxis, Izvestiya Vuzov Kyrgyzstan, 9, 3–8 (2016) (In Russian). [3] Iskandarov S. On lemmas with a Stieltjes integral and their application to the study of the asymptotic properties of solutions of Volterra integro-differential and integral equations, Investigations on integro-different. equations. Bishkek: Ilim, 28, 64–74 (1999) (In Russian).

ONTHEMETHODOF LYAPUNOVFUNCTIONALSFORLINEAR VOLTERRA INTEGRO-DIFFERENTIAL EQUATION OF FIRST ORDER WITH A DELAY ON THE HALF-AXIS Samandar Iskandarov, Atahan Khalilov Institute of Mathemtics of NAS of Kyrgyz Republic, Bishkek, Kyrgyz Republic [email protected], [email protected]

All functions and their derivatives are continuous and the relations take place at t ≥ t0, t ≥ τ ≥ t0. The Problem. Establish sufﬁcient conditions for estimating, boundedness, and power absolute integrability on a semi-interval J, striving to zero at t → ∞ all solutions of a ﬁrst-order linear IDE of Volterra type

0 Z t x (t) + a(t)x(t) + b(t)x(λ(t)) + [K(t,τ) +C(t,τ)]x(τ)dτ = f (t), t ≥ t0 (1) t0

with argument delay λ(t) : t0 ≤ λ(t) ≤ t, initial set Et0 consists of a single point t0. To solve this problem, a new generalized Lyapunov functional is constructed, based on the ideas of [1–6], and in combination with other methods, a class of IDE of the form (1) is found, for which we solve the above problem.

[1] Krasovsky N. N. Some problems of the theory of stability of motion. Moscow: Fizmatgiz, 1959 [2] Levin J. J. The asymptotic behavior of the solution of a Volterra equation, Proc. Amer. math. soc., 14, 534–541 (1963). [3] Vinokurov V.R. Asymptotic behavior of solutions of one class of Volterra integro-differential equations, Dif- ferent. equations, 3 (10), 1732–1744 (1967) (In Russian). [4] Levin J. J. Nonlinear Volterra equation not of convolution type, Different. equations, 4, 176–186 (1968). [5] Burton T. A. Volterra integral and differential equations. New York a.o.: Acad.press, 1983. [6] Iskandarov S. Method of weighting and cutting functions and asymptotic properties of solutions of integro- differential and integral equations of Volterra type. Bishkek: Ilim, 2002.

45 MATHEMATICAL ANALYSIS,DIFFERENTIAL EQUATION &APPLICATIONS - MADEA 9 KYRGYZ-TURKISH MANAS UNIVERSITY,BISHKEK,KYRGYZ REPUBLIC,JUNE 21-25, 2021

ON T2 SEPARATION AXIOMS IN FUZZY SOFT TOPOLOGICAL SPACES Raihanul Islam, Ruhul Amin, Sudipto Kumar Shaha Begum Rokeya University, Rangpur, Bangladesh [email protected], [email protected], [email protected]

Fuzzy set was introduced by Zadeh in his classical paper of 1965. Three years later, Chang gave the deﬁnition of fuzzy topology, which is a family of fuzzy sets satisfying the three classical axioms. In 1999, the Russian researcher Molodtsov introduced the concept of a soft set, and started to develop the basics of the corresponding theory as a new approach for modeling uncertainties. In this paper, we have introduced and studied some new notions of T2 separation axiom in fuzzy soft topological spaces by using quasi-coincident relation for fuzzy soft points. We have observed that all these notions satisfy good extension property. We have shown that these notions are preserved under the bijective and FSP continuous mapping. Moreover, we have obtained some other properties of this new concept. Finally, we have studied initial and ﬁnal soft topologies.

HISTORICALDEVELOPMENTOFUNIVALENTFUNCTIONS SUMMARY Ekrem Kadıoglu Atatürk University, Erzurum, Turkey [email protected]

In this study, some studies on univalent functions and their subclasses will be given in a historical process. The fundamental points of the research for Bieberbach’s Conjecture and its proof will be introduced. As a result, it is aimed to arouse interest in this subject.

BELONGTOASUBCLASSOFANALYTICFUNCTIONS FEKETE-SZEGÖINEQUALITY Muhammet Kamali Kyrgyz-Turkish Manas University, Bishkek, Kyrgyz Republic [email protected]

In this study, a subclass of analytic functions that provide the subordination condition associated with polynomials are deﬁned. For the functions belonging to this class, |a2| and |a3| coefﬁcient estimates are given and the Fekete-Szegö problem is examined.

46 Mathematical Analysis, Differential Equation & Applications - MADEA 9 Kyrgyz-Turkish Manas University, Bishkek, Kyrgyz Republic, June 21-25, 2021 NEWASPECTSOFTHEPHYSICALWORLDVIEWINTHECOURSE CONCEPTSOFMODERNNATURALSCIENCE Tamara Karasheva Kyrgyz-Turkish Manas University, Bishkek, Kyrgyz Republic [email protected]

One of the important disciplines, that form the natural-scientific understanding of the world among students of higher educational institutions, is the course “Concepts of modern natural science”. The article discusses new aspects of the physical worldview, which must be reflected in the content of the course. The author discussed in the article [1], the experience of distance teaching of this course at the Kyrgyz-Turkish Manas University, which made it possible to rationally use both the material- technical and human resources of the educational institution. This experience was useful in a pandemic, because it turned out to be the most acceptable way of sharing knowledge. The core of a unified natural-scientific view of the world is the physical worldview, since physics is the foundation of the modern world outlook. The development of physics has led to the creation of an integral natural-scientific view of the world, constantly supplementing and expand- ing our knowledge about it. The matter is in the center of the physical worldview. The evolution of the physical worldview is associated precisely with a change in our understanding of the matter. By replacing the one physical worldview to another complements, it expands and deepens our knowledge of matter. The quantum-mechanical worldview is the most modern worldview and in order to understand it, it requires certain knowledge from us. As the scientists suggests, the current generation of students in 15-20 years will witness the transition from semiconductor computers to quantum ones, which are truly the brainchild of modern quantum physics. Quantum computers will change our lives in the same way that semiconductor computers have changed it over the last half century [2, 3]. This means that our ideas about the world must change. And this will primarily affect the youth, whom we must prepare for the coming future.

[1] Karasheva T. T. New approaches in teaching Concepts of modern natural science, Osh State University Bul- letin, 3 150-153 (2014) [2] Fedichkin L. Quantum Computers, Science and Life, 4 (2021): https://www.nkj.ru/archive/articles/5309/ [3] Khmeleva A. How quantum computers can change the world, Computerra. Legendary magazine about modern technologies: https://www.computerra.ru/261820/kak-kvantovye-kompyutery-mogut-izmenit-mir/

47 MATHEMATICAL ANALYSIS,DIFFERENTIAL EQUATION &APPLICATIONS - MADEA 9 KYRGYZ-TURKISH MANAS UNIVERSITY,BISHKEK,KYRGYZ REPUBLIC,JUNE 21-25, 2021

ABOUTTHERELATIONBETWEENTHEEXISTENCEOFBOUNDED SOLUTIONS OF THE DIFFERENTIAL EQUATIONS AND EQUATIONS ONTIMESCALES Olha Karpenko1, Victoria Mogylova2, Tetiana Dobrodzii3 1 Zhejiang Normal University, Jinhua, China 2 National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Kyiv, Ukraine 3 Chuiko Institute of Surface Chemistry of NAS of Ukraine, Kyiv, Ukraine [email protected], [email protected], [email protected]

It is obtained the conditions, under which the existence of bounded solution of differential equation implies the existence of bounded solution of this equation, deﬁned on time scales, and vice versa [1].

Consider the following system of differential equations dx = X (t,x), (1) dt

d t ∈ R,x ∈ D, D be a domain in R . Consider the set of time scales Tλ and the system (1) deﬁned on Tλ

∆ xλ (t) = X(t,xλ ), (2)

d ∆ ∆ where t ∈ Tλ , xλ : Tλ → R , xλ (t) be delta derivative x for a function x(t) deﬁned on Tλ , infTλ = −∞, supTλ = ∞, λ ∈ Λ ⊂ R, and λ = 0 is a limit point of Λ. We assume that the function X(t,x) continuously differentiable and bounded together with its partial derivatives, i.e. ∃C > 0 such that

∂X(t,x) ∂X(t,x) |X(t,x)| + + ≤ C (3) ∂t ∂x

∂X for t ∈ R,x ∈ D, where ∂x is the corresponding Jacobi matrix. The following theorem holds. Theorem 1. Let the system (1) has a bounded on R , asymptotically stable uniformly in t0 ∈ R solution x(t), which lies in the domain D with some ρ - neighborhood. Then there exists λ0 > 0 such that for all λ < λ0 the system (2) has a bounded on Tλ solution xλ (t). Theorem 2. If there exists λ0 > 0 such that for all λ < λ0 the system (2) has an asymptotically stable uniformly in t0 ∈ Tλ and λ bounded on axis solution xλ (t), which lies in the domain D with some ρ-neighborhood, then the system (1) has a bounded on axis solution.

[1] Karpenko O., Stanzhytskyi O., Dobrodzii T. The relation between the existence of bounded global solutions of the differential equations and equations on time scales, Turk. J. Math. 44, 2099–2112 (2020).

48 Mathematical Analysis, Differential Equation & Applications - MADEA 9 Kyrgyz-Turkish Manas University, Bishkek, Kyrgyz Republic, June 21-25, 2021 SOME RESULTS ON FOURIER–BOAS-LIKE WAVELETS Leena Kathuria1, Shashank Goel1, Nikhil Khanna2 1 Amity University, Noida, India 2 University of Delhi, Delhi, India [email protected], [email protected], [email protected]

In this paper, we introduce Fourier-Boas-Like wavelets with the motive to characterize the wavelets whose Fourier transform vanishing a.e. on ]−1,1[ and to reinforce the incompetency of wavelets to study both the symmetries of an asymmetric signal. Further, we derive sufﬁcient conditions for their higher vanishing moments. Some properties of Fourier–Boas-Like wavelets connected to Riesz projectors are also given. Finally, we formulate a variation diminishing wavelet allied with a Fourier–Boas-Like wavelet.

[1] Khanna N., Kumar V., Kaushik S. K. Vanishing moments of wavelet packets and wavelets associated with Riesz projectors. Proceedings of the 12th International Conference on Sampling Theory and Applications (SampTA), IEEE, Tallinn, Estonia, 2017, 222–226. [2] Soares L. R., de Oliveira H. M., Cintra R. J. The Fourier-Like and Hartley-Like wavelet analysis based on Hilbert transforms. https://arxiv.org/abs/1502.02049, 2015. [3] Khanna N., Kathuria L. On convolution of Boas transform of wavelets. Poincare Journal of Analysis and Applications, 6 (1), 53–65 (2019). [4] Khanna N., Kaushik S. K., Jarrah A. M. Some remarks on Boas transforms of wavelets. Integral Transforms and Special Functions, 31 (2), 107–117 (2020). [5] Walnut D. F. An Introduction to Wavelet Analysis. Birkhäuser, Boston, Inc., Boston, MA, 2004.

r APPROXIMATIONOFFUNCTIONFROMTHECLASS Wβ,∞ BY INTEGRALSOF ABEL-POISSONTYPE Yurii Kharkevych, Inna Kal’chuk Lesya Ukrainka Volyn National University, Lutsk, Ukraine [email protected], [email protected]

Let L be a space of 2π-periodic summable on a period functions f equipped with the norm π R k f kL = | f (t)|dt; C be a space of 2π-periodic continuous functions f in which the norm is set −π by means of the equality k f kC = max| f (t)|; L∞ be a space of 2π-periodic measurable essentially t bounded functions f with the norm k f k∞ = esssup| f (t)|. t ∞ a0 Assume, that f ∈ L and S[ f ] = 2 + ∑ (ak coskx + bk sinkx) is the corresponding Fourier se- k=1 ries. Let, further, r > 0 and β ∈ R. If the series

∞ r βπ βπ ∑ k ak cos kx + + bk sin kx + k=1 2 2

is the Fourier series of a summable function ϕ, then we call the function ϕ a (r,β)-derivative of f r in the Weyl–Nagy sense and denote it by fβ . A set of functions for which this condition is satisﬁed

49 MATHEMATICAL ANALYSIS,DIFFERENTIAL EQUATION &APPLICATIONS - MADEA 9 KYRGYZ-TURKISH MANAS UNIVERSITY,BISHKEK,KYRGYZ REPUBLIC,JUNE 21-25, 2021

r r r r is denoted by Wβ . If f ∈ Wβ and, besides, fβ (·) ∞ ≤ 1, then f belongs to the class Wβ,∞ (see e.g. [1]). Let f ∈ L, δ > 0. The quantity

∞ γ a0 − k Aδ,γ ( f ,x) = + ∑ e δ (ak coskx + bk sinkx), 0 < γ ≤ 2, 2 k=1 is called the integral of Abel-Poisson type of the function f . The paper is devoted to investigation of asymptotic behavior as δ → ∞ of the quantity

r E (W , ;Aδ,γ )C = sup k f (·) − Aδ,γ ( f ,·)kC. β ∞ r f ∈Wβ,∞

[1] Stepanets A. I. Methods of Approximation Theory. VSP: Leiden, Boston, 2005.

COMBINATORICSOFDOUBLESEQUENCES Ljubiša D.R. Kocinacˇ 1, Dragan Djurciˇ c´2 1 University of Niš, Niš, Serbia 2 University of Kragujevac, Caˇ cak,ˇ Serbia [email protected], [email protected]

Our talk concerns selective properties of real double sequences convergent in the Pringsheim sense. We present recent results on this topic. Also, we will discuss another interesting direction of further investigation related to the Primgsheim limit points of double sequences. The talk is based on a chapter in the forthcoming book [1].

[1] Kocinacˇ LJ. D. R., Djurciˇ c´ D. Some applications of double sequences. In: B. Hazarika, S. Acharjee, H.M. Srivastava (eds.), Advances in Mathematical Analysis and its Multi-Disciplinary Applications, CRC Press, in preparation.

POLYNOMIAL APPROXIMATION WITH JACOBIANDOTHER DOUBLINGWEIGHTS Kirill Kopotun University of Manitoba, Manitoba, Canada [email protected]

I’ll discuss matching direct and inverse results for univariate weighted approximation by algebraic polynomials in the Lp, 0 < p ≤ ∞, (quasi)norm weighted by (i) certain “averages” of doubling weights, or (ii) doubling weights having ﬁnitely many zeros and singularities and not “too rapidly changing” away from those points (classical Jacobi weights is one example of such weights).

50 Mathematical Analysis, Differential Equation & Applications - MADEA 9 Kyrgyz-Turkish Manas University, Bishkek, Kyrgyz Republic, June 21-25, 2021 STRONG RECTANGULAR FEJÉRMEANSON SCHURCLASSIN POLYDISK Yulia Kozachenko1, Viktor Savchuk2 1 Donbass State Pedagogical University, Sloviansk, Ukraine 2 Institute of Mathematics of NAS of Ukraine, Kyiv, Ukraine [email protected], [email protected]

d d Let d be natural, D := {z := (z1,...,zd) ∈ C : max1≤ j≤m |z j| < 1} be the unit polydisk and d d let H (D ) be the set of holomorphic functions in D . d d k k1 kd For a multiindex k := (k1,...,kd) ∈ Z and z ∈ C , we denote z := z1 ···zd . d Let a function f ∈ H (D ) and let

k d f (z) = ∑ ckz , z ∈ D , d k∈Z+

d be its power series expansion in D . The rectangular partial sums Sn( f ) and Fejér means σn( f ) of d order n ∈ N are deﬁned by k Sn( f )(z) := ∑ ckz 0≤k≤n−1 and 1 σn( f )(z) := ∑ Sk( f )(z), n1 ···nd 0≤k≤n−1 respectively. Here the notion denote the multiple sum n1−1 ... nd−1. ∑0≤k≤n−1 ∑k1=0 ∑kd=0 d d Consider the Schur class B(D ) which consist of functions f ∈ H (D ) with

k f k := sup | f (z)| ≤ 1. z∈Dd

d f ∈ B( ) d k ( f )k ≤ . It is well known [1, p.68] that D if and only if supn∈N σn 1 In the present talk, we supplement this assertion as follows

d d Theorem. Let d ∈ N and f ∈ H (D ). Then f ∈ B(D ) if and only if ! 1 sup kS ( f )k ≤ 1. n ···n ∑ k n∈Nd 1 d 0≤k≤n−1

[1] Schur I. On power series which are bounded in the interior of the unit circle II. Operator Theory: Advances and Applications, 18, 61–88 (1986).

51 MATHEMATICAL ANALYSIS,DIFFERENTIAL EQUATION &APPLICATIONS - MADEA 9 KYRGYZ-TURKISH MANAS UNIVERSITY,BISHKEK,KYRGYZ REPUBLIC,JUNE 21-25, 2021

NONLOCALPROBLEMWITHINTEGRALCONDITIONSFOR HOMOGENEOUS SYSTEM OF PARTIAL DIFFERENTIAL EQUATIONSOFSECONDORDER Grzegorz Kuduk1, Michael Symotyuk2 1 Graduate of University of Rzeszow, Rzeszow, Poland 2 Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, Lviv, Ukraine [email protected], [email protected]

2 Let Π(T) = {(t,x) ∈ R : t ∈ [0,T],x ∈ R}, T > 0. Let us denote Eα,β ,α > 0,β > 0., to the space of functions ϕ ∈ L2(R), with the ﬁnite norm [1] v u +∞ u Z 1 2 2α kϕkE = u |ϕˆ(ξ)| (1 + |ξ|) exp(2β|ξ|)dξ α,β t2π −∞

where ϕˆ(ξ) is the Fourier transform of the function ϕ(x) In the strip Π(T) we consider nonlocal-integral problem

∂ ∂ ∂ nu(t.x) n ∂ nu(t,x) L , u(t,x) ≡ n + ∑ a j n− j j = 0, a j ∈ C, (t,x) ∈ Π(T), (1) ∂t ∂x ∂t j=1 ∂t ∂x

T k  k  Z ∂ U  ∂ U  k  −  + t U(t,x)dt = ϕk(x), k = 0,...,n − 2, (2) ∂t t=0 ∂t t=T 0 T T Z 1 Z n−1 n−1 t U(t,x)dt + t U(t,x)dt = ϕn−1(x)(3)

0 T2 n n−1 where a1,a2 ∈ C. Assuming that the real parts of roots of polynomial λ + a1λ + ... + an are nonzero and different, the correctioness of the problem (1) - (3) in the space of functions 2 C ([0,T],Eα,β ) is established. Obtained results continue the research of the work [2].

[1] Dubinskii Yu.A. The algebra of pseudodifferential operators with analytic symbols and its applications to mathematical physicals, Russian Mathematical Surveys, 37 (5), 109–153 (1982). [2] Kalenyuk P.I, Nytrebych Z.M, Symotyuk M.M, Kuduk G. Integral problem for partial differential equation of second order in unboudud layer, Bukovinian Mathematical Journal, 4 (3–4), 69–74 (2016).

52 Mathematical Analysis, Differential Equation & Applications - MADEA 9 Kyrgyz-Turkish Manas University, Bishkek, Kyrgyz Republic, June 21-25, 2021 SOMEPROPERTIESOFFRACTIONAL BOASTRANSFORMSOF WAVELETS Dilip Kumar University of Delhi, Delhi, India [email protected],

In this presentation, we introduce fractional Boas transforms and discuss some of their properties. We also introduce the notion of wavelets associated with fractional Boas transforms and give some results related to their vanishing moments. Finally, a comparative study of Hilbert transforms and fractional Boas transforms is done.

Joint work with Dr. Nikhil Khanna, Dr. A. Zothansanga and Dr. Shiv Kumar Kaushik.

DITOPOLOGIESASSOCIATEDWITHTEXTUREGRAPHS Tugçe˘ Kunduracı1, Ceren Sultan Elmalı2, Tamer Ugur˘ 1 1Atatürk University, Erzurum, Turkey 2Erzurum Technical University, Erzurum, Turkey [email protected], [email protected], [email protected]

In this study, it is shown how a ditopology can be obtained on the given set and texture with the help of this graph. Some relations between obtained that these ditopological texture spaces and graphs are investigated.

[1] Brown L. M. Ditopological fuzzy structures 1, Fuzzy Systems A. I M. 3 (1), 12–17 (1993).

[2] Brown L. M. Ditopological fuzzy structures 2, Fuzzy Systems A. I M. 3(2), 5–14 (1993).

[3] Brown L. M., Ertürk R., Dost S. Ditopological texture spaces and fuzzy topology, 1. Basic concepts, Fuzzy Sets and Systems, 147, 171–199 (2004).

[4] Evans J. W., Harary F., Lynn M. S. On Computer enumeration of ﬁnite topologies, Comm. Assoc. Comp. Mach. 10, 295–298 (1967).

[5] Bhargav T. N., Ahlborn T. J. On topological spaces associated with digraphs. Acta Math. Acad. Sc. Hungar., 19, 47–52 (1968).

[6] Kannan K., A note on some generalized closed sets in bitopological spaces associated to digraphs, Journal of App. Math. 2012, 1–5 (2012).

[7] Sampathkumar E. Transitive Digraphs and topologies on a set, Journal of the Karnatak UnivScience, 266–273 (1972).

[8] Özçag˘ S., Brown L. M. Di-uniform Texture Space, Applied General Topology, 4 (1), 157–192 (2003).

[9] Diestel R. Graph Theory. Springer-Verlag, 2005.

[10] Bang-Jensen J., Gutin G. Digraphs: Theory, Algorithms and Applications. Springer–Verlag, 2002.

[11] Rosenfeld A. Fuzzy graphs, In: L.A.Zadeh, K.S.Fu and M.Shimura, Eds, Fuzzy Sets and their Applications, Academic Press, New York, 77–95 (1975).

53 MATHEMATICAL ANALYSIS,DIFFERENTIAL EQUATION &APPLICATIONS - MADEA 9 KYRGYZ-TURKISH MANAS UNIVERSITY,BISHKEK,KYRGYZ REPUBLIC,JUNE 21-25, 2021

[12] Girija B., Pilakkat R. Bitopological spaces associated with digraphs, South Asian Journal of Mathematics, 3 (1), 56–65 (2013). USING FUZZY-ROUGH SUBSET EVALUATION FOR FEATURE SELECTIONANDNAIVE BAYES TO CLASSIFY THE PARKINSON’S DISEASE Naiyer Mohammadi Lanbaran, Ercan Celik Atatürk University, Erzurum, Turkey [email protected], [email protected]

Abstract: Feature selection is one of the issues in machine learning as well as statistical pat- tern recognition. This is important in many fields (such as classification) because there are many features in these areas, many of which are either unused or have little information load. Not elim- inating these features does not make a problem in terms of information, but it does increase the computational burden for the intended application. Besides, it causes to store of so much useless information along with useful data. A problem for machine learning research occurs when there are many possible features with few attributes of training data. One way is to first specify the best attributes for prediction and then to classify features based on a measure of their dependence. In this study, the Fuzzy- Rough subset evaluation has been used to take features in core of similar features. Fuzzy-rough set-based feature selection (FS) has been demonstrated to be extremely advantageous at reducing dataset size but has various problems that yield it unproductive for big datasets. Fuzzy- Rough subset evaluation algorithm indicates that the techniques greatly decrease dimensionality while keeping classification accuracy. This paper considers classifying attributes by using fuzzy set similarity measures as well as the dependency degree as a relatedness measure. Here we use Artificial Neural Network, Naïve Bayes as classifiers, and the performance of these techniques are compared by accuracy, precision, recall, and F-measure metrics. [1] Feng Q., Li R. Discernibility Matrix Based Attribute Reduction in Intuitionistic Fuzzy Decision Systems. Paper presented at the RSFDGrC, 2013. [2] Tiwari A. K., Shreevastava Sh., Som T., Shukla K. K. Tolerance-based intuitionistic fuzzy-rough set approach for attribute reduction. Expert System, 101 (C July), 205–212 (2018). [3] Terano T., Asai K., Sugeno M. Applied Fuzzy Systems. Academic Press Professional, Boston, 1994. [4] Heisters D. Parkinson’s: symptoms treatments and research. British Journal of Nursing, 1 (20), 548–554 (2011). [5] Jensen R., Shen Q. New approaches to fuzzy-rough feature selection. IEEE Transactions on fuzzy systems, 17 (4), 824–838 (2008). MULTI-DELAYED PERTURBATION OF MITTAG-LEFFLER TYPE FUNCTIONS Nazim Mahmudov Eastern Mdeiterranean University, Famagusta, T. R. Northen Cyprus [email protected]

We propose a multi-delayed perturbation of Mittag-Leffler type matrix function. It is an extension of the classical Mittag-Leffler type matrix function and delayed Mittag-Leffler type matrix function. With the help of the multi-delayed perturbation of Mittag-Leffler type matrix function, we give an explicit analytical solution to linear nonhomogeneous Riemann-Liouville fractional multi-delay differential equations of order l − 1 < α ≤ l. Our results are new even for the case

54 Mathematical Analysis, Differential Equation & Applications - MADEA 9 Kyrgyz-Turkish Manas University, Bishkek, Kyrgyz Republic, June 21-25, 2021

1 < α < 2 and for the classical (α = 2) second order single delay and multi-delay linear (fractional) differential equations.

[1] Diblík J, Khusainov DYa. Representation of solutions of discrete delayed system x(k + 1) = Ax(k) + Bx(k − m) + f (k) with commutative matrices, J. Math. Anal. Appl., 318 (1), 63–73 (2006).

[2] Mahmudov NI. Delayed linear difference equations: the method of Z-transform. Electron J Qual Theory Differ Equ., 53, 1–12 (2020).

[3] Mahmudov NI. Representation of solutions of discrete linear delay systems with non permutable matrices. Appl Math Lett., 85 (1), 8–14 (2018).

[4] Medved’ M, Pospíšil M. Sufﬁcient conditions for the asymptotic stability of nonlinear multidelay differential equations with linear parts deﬁned by pairwise permutable matrices. Nonlinear Anal., 75 (11), 3348–3363 (2012).

[5] Pospíšil M. Representation and stability of solutions of systems of functional differential equations with multiple delays. Electronic Journal of Qualitative Theory of Differential Equations, 53–54, 1–15 (2012).

[6] Pospíšil M. Representation of Solutions of Systems of Linear Differential Equations with Multiple Delays and Nonpermutable Variable Coefﬁcients. Mathematical Modelling and Analysis, 25 (2), 303–322 (2020).

ANADDENDUMTO JACKSONINEQUALITY Oksana Motorna, Igor Shevchuk Taras Shevchenko National University of Kyiv, Kyiv, Ukraine [email protected], [email protected]

We are going to discuss recent results on the constrained approximation of periodical continuous and differentiable functions. To this end some results in unconstrained approximation are needed. Among them, say, the following clariﬁcation of Bernstein-Privalov inequality, recently obtained by D. Leviatan and the authors: For each b ∈ (0,π] and every trigonometric polynomial Tn of degree ≤ n there holds the inequality n πn kT 0k ≤ kT k ≤ kT k , n [−b/2,b/2] b n [−b,b] b n [−b,b] sin 2 where k · k[a,b] := k · kC([a,b]).

55 MATHEMATICAL ANALYSIS,DIFFERENTIAL EQUATION &APPLICATIONS - MADEA 9 KYRGYZ-TURKISH MANAS UNIVERSITY,BISHKEK,KYRGYZ REPUBLIC,JUNE 21-25, 2021

AN ESTIMATE OF THE REMAINDER OF THE INTERPOLATION CONTINUED C–FRACTION Julia Myslo1, Mykhaylo Pahirya1,2 1 Uzhhorod National University, Uzhhorod, Ukraine 2 Mukachevo State University, Mukachevo, Ukraine [email protected], [email protected]

The problem of the interpolation of functions of a real variable by interpolating continued C– fraction is investigated. The relationship between the continued fraction and the continuant was used. The properties of the continuant are established. Proven theorem. (n+ ) Theorem. Let R ⊂ R be a compact, f ∈ C 1 (R) and the interpolating continued C– fraction (C–ICF) of the form

Pn(x) n Dn(x) = = a0 + K (ak(x − xk−1)/1), ak ∈ R, k = 0,n, Qn(x) k=1 be constructed by the values the function f at nodes

X = {xi : xi ∈ R,xi 6= xi,i 6= j,i, j = 0,n}.

If the partial numerators of C–ICF satisfy the condition of the Paydon–Wall type, that is

0 < a∗diamR ≤ p, then n ∗ f ∏ |x − xk| r n+1−2k k=0 n+1 ∗ k | f (x) − Dn(x)| ≤ κn+1(p) + ∑ k (a ) ∑ κi1 (p)× (n + 1)!Ωn(t) k=1 i1=1

n+3−3k n−3 n−1 × ∑ κi2−i1−1(p)··· ∑ κik−1−ik−2−1(p) ∑ κik−ik−1−1(p)κn−ik (p) , i2=i1+2 ik−1=ik−2+2 ik=ik−1+2 √ n √ n ∗ n+1−m (1 + 1 + 4p) − (1 − 1 + 4p) ∗ where f = max max| f (x)|, κn(p) = √ , a = max |ai|, 0≤m≤r x∈R 2n 1 + 4p 2≤i≤n ∗ 1 ρ = a diamR, p = t(1 −t),t ∈ (0, 2 ],r = [n/2],

 n+1 1 − 4(1 −t)t  1  n , if 0 < t < 2 , Ωn(t) = 2 (1 − 4(1 −t)t) n + 1  , if t = 1 . 2n 2

[1] Pahirya M. M. A continuant and an estimate of the remainder of the interpolating continued C–fraction. Matem- atychni Studii, 54, 32–45 (2020). [2] Pahirya M. M. Estimation of the remainder for the interpolation continued C-fraction. Ukr. Math. J., 66, (6), 905–915 (2014).

56 Mathematical Analysis, Differential Equation & Applications - MADEA 9 Kyrgyz-Turkish Manas University, Bishkek, Kyrgyz Republic, June 21-25, 2021 DYNAMICAL BEHAVIOR OF RATIONAL DIFFERENCE EQUATION xn−4 xn+1 = ±1±xnxn−1xn−2xn−3xn−4 Burak Ogul˘ 1, Dagıstan˘ ¸Sim¸sek2 1 Istanbul Aydin University, Istanbul, Turkey 2 Konya Technical University, Konya, Turkey [email protected], [email protected]

The study and solution of nonlinear rational recursive sequence of high order is quite challeng- ing and rewarding. In the recent times, nonlinear difference equations have a critical role in the ﬁelds of pyhsics, economy, ecology and computational science engineering, etc. Many researchers have investigated the behavior of the solution of nonlinear difference equations. So, recently there has been an increasing interest in the study of qualitative analysis of rational difference equations. Elsayed [1], investigated the solution of difference equations,

xn−5 xn+1 = . −1 + xn−2xn−5 Elsayed [2, 3], obtained the solutions of the following difference equations,

xn−7 xn−9 xn+1 = , xn+1 = . ±1 ± xn−1xn−3xn−5xn−7 ±1 ± xn−4xn−9 Simsek et al. [4], gave the solution of difference equation,

xn−(4k+3) xn+1 = 2 1 + ∏t=0 xn−(k+1)t−k where initial conditions are positive real numbers. Our aim in this paper is to investigate the behavior of the solution of the following nonlinear recursive sequences xn−4 xn+1 = , n ∈ N0 (1) ±1 ± xnxn−1xn−2xn−3xn−4 where the initial conditions are arbitrary nonzero positive real numbers. Also, we get explicit forms of the solutions.

xn−5 [1] Elsayed E.M. On The Difference Equation xn+1 = , Inter. J. Contemp. Math. Sci., 33 (3), 1657– −1+xn−2xn−5 1664 (2008). [2] Elsayed E.M. Behavior of a rational recursive sequences. Studia Universitatis Babes-Bolyai Mathematica, 56 (1), 27–42 (2011). [3] Elsayed E.M. Solution of a Recursive Sequence of Order Ten. General Mathematics, 19 (1), 145–162 (2011).

xn−(4k+3) [4] Simsek D., Abdullayev F. On the recursive sequence xn+1 = 2 . Journal of Mathematical Sci- 1+∏t=0 xn−(k+1)t−k ences, 222 (6), 762–771 (2017).

57 MATHEMATICAL ANALYSIS,DIFFERENTIAL EQUATION &APPLICATIONS - MADEA 9 KYRGYZ-TURKISH MANAS UNIVERSITY,BISHKEK,KYRGYZ REPUBLIC,JUNE 21-25, 2021

ASYMPTOTICSOFTHESOLUTIONOFPARABOLICPROBLEMS WITHNONSMOOTHBOUNDARYFUNCTIONS Asan Omuraliev, Ella Abylaeva Kyrgyz-Turkish Manas University, Bishkek, Kyrgyzstan [email protected], [email protected]

Singularly perturbed problems with nonsmooth regular or boundary layer functions were studied in [1] - [5]. To construct the asymptotics of the solution of such problems [1], the method of matching asymptotic expansions was used, and in [2] - [5], an asymptotic expansion of the zero and ﬁrst orders was constructed by the smoothing procedure. Using the method [5] in the papers [6] , a method is proposed for constructing an asymptotics of an arbitrary order, without using the procedure for matching the solution. Papers [5], [6] are devoted to the study of problems with nonsmooth degeneracy for a singularly perturbed problem. We have constructed, using the regularization method for singularly perturbed problems, the asymptotics of the solution of the following problem:

2 2 √ Lε u ≡ −∂tu + ε a(x)∂x u + εb(x)∂xu − c(x,t)u = f (x,t), (x,t) ∈ Ω. (1)

u|t=0 = u|x=0 = 0, which is studied [6], [7], where ε > 0- a small parameter, a(x),b(x),c(x,t), f (x,t)-endlessly differentiable, a(x) > 0,b(x) > 0,Ω = (0 < x < ∞) × (0 < t ≤ T). To regularize the problem, we introduce independent regularizing variables

ξl = ϕl(x,t),ηl = ψ(x,ε)(2) and extended functionu ˜(M,ε),M = (x,t,ξ,η,ζ,τ) such that:

u˜(M,ε)|χ=χ(x,t,ε)≡u(x,t,ε). (3)

By regularizing the problem, we will look for solutions to the extended function.

[1] Ilin A. Matching asymptotic expansions of boundary value problems. Moscow, Nauka, 1989. [2] Butuzov B., Neterov A. On the asymptotics of the solution of a parabolic equation with a small parameter in the highest derivatives, Journal of Computational Mathematics and Mathematical Physics, 22 (4), 865–870 (1982). [3] Butuzov B. Asymptotics of the solution of the bisingular problem for systems of linear parabolic equations, Journal of Modeling and Analysis of Information Systems, 20 (1), 5–17 (2013). [4] Butuzov B., Buchnev V. On the asymptotics of the solution of a singularly perturbed problem in the two- dimensional case. Journal of Differential Equations, 25 (3), 453–461 (1989). [5] Bulcheva O., Sushko V. Construction of an approximate solution for a singularly perturbed parabolic problem with nonsmooth degenerations. Journal of Fundamental and Applied Mathematics, 1 (4), 881–905 (1995). [6] Lomov S. Introduction to the general theory of singular perturbations. Moscow, Nauka, 1981.

58 Mathematical Analysis, Differential Equation & Applications - MADEA 9 Kyrgyz-Turkish Manas University, Bishkek, Kyrgyz Republic, June 21-25, 2021 ASYMPTOTICSOFTHESOLUTIONOFAPARABOLICSYSTEM WITHANIRREVERSIBLELIMITOPERATOR Asan Omuraliev, Peyil Esengul kyzy Kyrgyz-Turkish Manas University, Bishkek, Kyrgyz Republic [email protected], [email protected]

In this paper, we consider a boundary value problem and construct the asymptotics of the solution:

2 2 ε∂tu = ε A(x)∂x u +C(t)u + f (x,t), (x,t) ∈ Ω = (0 < x < 1) × (0 < t ≤ T)

u(x,t,ε)|t=0 = 0, u(x,t,ε)|x=0 = u(x,t,ε)|x=1 = 0, (1) where ε > 0 is small parametre, u(x,t,ε) = col(u1(x,t,ε),u2(x,t,ε),...,un(x,t,ε)).

Let the following conditions be satisﬁed:

∞ n×n 1. the matrix of a simple structure C(t) ∈ C ([0,T],C ) has a p multiple eigenvalue λi(t) = 0, i = 1, p, and for nonzero eigenvalues λk(t),k = 1 + p,2 + p,...,n the relations Reλk(t) ≤ 0,λk(t) 6= λ j(t), ∀k 6= j, t ∈ [0,T] hold; ∞ n×n 2. the matrix A(x) ∈ C ([0,1],C ) has n distinct eigenvalues βi(x), i = 1,n such that βi(x) > 0,∀x ∈ [0,1];

3. the given functions f (x,t),h(x) are smooth, the initial and boundary conditions are consistent h(0) = h(1) = 0. Problem (1) was studied in [1], [2], when A(x) is a scalar function. The work [1] is devoted to the construction of the asymptotics of the boundary layer type [3], and the regularized asymptotics [4] is constructed in [2]. The asymptotics contains 2n corner boundary layer functions, half of which describe the boundary layer in the vicinity of the corner points (0,0), and the other half in the vicinity of (0,1). The construction of the asymptotics is based on the method of S.A. Lomov [4] and for regularization in this problem the technique of [5] is used, where the eigenvalues of the matrix A(x) are used to regularize the problem, in contrast to [5], to regularize the problem, additionally, the eigenvalues of the matrix C(t). To regularize the problem, we introduce regularizing variables

ϕli(x) ψk(t) t ξli = √ ,i = 1,n,l = 1,2, τk = ,k = p + 1, p + 2,...,n,η = (2) ε3 ε ε2 and extended function

u˜(M,ε)|η=q(x,t,ε) ≡ u(x,t,ε),M = (x,t,ξ,τ), (3) We will be looking for solutions to the advanced function. [1] Lomov S. A. Introduction to the general theory of singular perturbations. Moscow, Nauka, 1981. [2] Lightill M. J. A technique for rendering approximate solutions to physical problems uniformly valid, Phil. Mag., 40 (7), 1179–1201 (1949).

59 MATHEMATICAL ANALYSIS,DIFFERENTIAL EQUATION &APPLICATIONS - MADEA 9 KYRGYZ-TURKISH MANAS UNIVERSITY,BISHKEK,KYRGYZ REPUBLIC,JUNE 21-25, 2021

[3] Naife A. Perturbation methods. Moscow, Mir, 1968. [4] Qian Xue-Sen. Poincaré-Lighthill-Guo Method, Problems of Mechanics, 2, 7–62 (1959). [5] Pontryagin L. S. Selected Works, T.2. Moscow, 1988.

SOLUTIONESTIMATESFORONECLASSOFELLIPTICAND PARABOLIC NONLINEAR EQUATIONS Mukhtarbai Otelbaev, Nurbek Kakharman Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan [email protected], [email protected]

The separation of differential expressions and related fundamental results have been obtained by Everitt and Giertz [1,2]. Significant development of the theory of separation and a number of results concerning a property referred to separation of differential expressions have been discussed by Otelbaev [3] and Boimatov [4,5]. The interesting fact that the separation problem in non- reflective spaces is solved more successfully than in reflexive spaces was first noticed [6].

n n In the space L1,m(R ) of m dimensional vector functions with components belong to L1(R ) we will consider an operator

Lu :≡ −∆u(x) + l (x,u(x))A(x,u(x)) · u(x), where l(x,u) and A(x,u) are continuous scalar function and matrix-function, respectively. n Deﬁnition. In the space L1,m(R ) an operator L is called separated, if the following a priori estimate k−∆uk + kl(x,u(x))A(x,u(x)) · u(x)k ≤ ckLuk holds for some constant c independent of u. Theorem 1. If conditions

∗ −1 l(x,u) ≥ δ0 > 0, A(x,u) = A (x,u) ≥ E, A (x,u) + kA(x,u)k ≤ c0 < ∞ (1) are satisﬁed, then the operator L is separated. Theorem 2. Let

Du :≡ ut + (−∆)u + l(t,x,u(t,x))A(t,x,u(t,x)) · u(t,x).

If conditions u(t,x)|t=0 = 0 and (1) are satisﬁed, and if

T Z kDu(t,x)k (n) dt < ∞, L1,m(R ) 0 then for any T > 0 the operator D is separated.

[1] Everitt W. N., Giertz M. Some properties of the domains of certain differential operators, Proc. London Math. Soc., 3 (23), 301–324 (1971). [2] Everitt W. N., Giertz M. Some inequalities associated with certain ordinary differential operators, Math Ztachr., 126, 308-326 (1972). [3] Otelbaev M. On the separation of elliptic operators. Dokl. Acad. Nauk SSSR, 234, 540–543 (1977).

60 Mathematical Analysis, Differential Equation & Applications - MADEA 9 Kyrgyz-Turkish Manas University, Bishkek, Kyrgyz Republic, June 21-25, 2021

[4] Boimatov K. Kh. Coercive estimates and separation for second order elliptic differential equations. Soviet Math. Dokl., 38, 157–160 (1989). [5] Boimatov K. Kh., Sharefov A. Coercive properties of non-linear operators Schrodinger and Dirac, Dokl. Acad. Nauk SSSR, 326, 393–398 (1992). [6] Oinarov R. On seperation of the Schrodinger’s operator in the space of integrable functions, Dokl. Akad. Nauk SSSR, 285 (5), 1062–1064 (1985).

ON SOME APPROXIMATE PROPERTIES OF THE p-BIEBERBACH POLYNOMIALSINCLOSEDREGIONS Pelin Özkartepe, Cevahir Doganay˘ Gün Gaziantep University, Gaziantep, Turkey

Let C be a complex plane; G ⊂ C, 0 ∈ G, be a ﬁnite Jordan region and L := ∂G. Let us denote by A(G) the class of analytic in G and continuous on G functions g(z) with the norm kgkC(G) := max |g(z)|, z ∈ G . For any p > 0, we deﬁne: ZZ 1 0 0 p Ap(G) := { f ∈ A(G), f (0) = 0, f (0) = 1, f (z) dσz < ∞} G and 1   p ZZ 0 p k f k := k f k 1 := f (z) dσ , p Ap(G)  z G where σ be the two-dimensional Lebesgue measure. Let ℘n denotes the class of all algebraic polynomials Pn(z) of degree at most n ∈ N and sat- 0 isfying the conditions: Pn(0) = 0 and Pn(0) = 1. Let the function ϕ maps G conformally and univalently onto {w : |w| < 1} which is normalized by ϕ(0) = 0 and ϕ0(0) = 1. We consider the following extremal problem: n o kϕ − Pnkp , Pn ∈℘n → inf, p > 0. (1)

∗ According to [1, p.137], it can be seen that, for any p > 0, there exists a polynomial Pn,p ∈℘n ∗ solving the problem (1), and if p > 1, this polynomial Pn,p(z) is unique [1, p.142]. This unique ∗ solution Pn,p(z) was denoted by Bn,p(z) and called p−Bieberbach polynomial for the pair (G,0). In this we study the possibility of convergence of p−Bieberbach polynomials in the closed regions G, when these regions having both interior and exterior zero angles.

[1] Davis P.J. Interpolation and Approximation. Blaisdell Publishing Company, 1963.

61 MATHEMATICAL ANALYSIS,DIFFERENTIAL EQUATION &APPLICATIONS - MADEA 9 KYRGYZ-TURKISH MANAS UNIVERSITY,BISHKEK,KYRGYZ REPUBLIC,JUNE 21-25, 2021

THEBESTNON-SYMMETRICAPPROXIMATIONOFCLASSESOF DIFFERENTIABLE FUNCTIONS BY SPLINES Nataliia Parﬁnovych Oles Honchar Dnipro National University, Dnipro, Ukraine [email protected]

Let Lp, 1 ≤ p ≤ ∞, be the spaces of measurable functions f : R → R with norms k·kp, M ⊂ Lp be a class. The quantity E(M,H)p = sup inf k f − hkp f ∈M h∈H we will call the best approximation of class M by set H in Lp. Let Cr (r = 0,1,...) be the spaces of r times continuously differentiable (continuous for r = 0) r (r−1) (0) functions. By Wp denote the class of functions f ∈ Lp, such that f ( f = f ) is locally (r) absolutely continuous and k f kp ≤ 1. m−1 For h > 0 and m = 0,1,2,..., by Sh,m denote the collection of functions s ∈ C (m ≥ 1) such (m) that s |( jh,( j+1)h) = c j, j ∈ Z. The set Sh,m we will call the space of polynomial splines of order m defect 1 with knots at the points jh, j ∈ Z. In [1] G.G. Magaril-Il’yaev proved, that for all h > 0, n,r ∈ N, m ∈ Z, m ≥ r − 1: K E(W r,S ∩ L ) = r hr, (1) 1 h,m 1 1 πr where Kr are Favard constants. We obtained non-symmetric analogs of the equality (1).

[1] Magaril-Il’yaev G. G. On the best approximation of functional classes by splines on the real line. Trudy MIAN, 194, 148–159 (1992).

ABOUTGENERALIZATIONOFPOINTWISEINTERPOLATION ESTIMATESOFAMONOTONEAPPROXIMATIONOFFUNCTIONS HAVING A FRACTIONAL DERIVATIVE OF ARBITRARY ORDER Tamara Petrova, Irina Petrova Taras Shevchenko National University of Kyiv, Kyiv, Ukraine [email protected], [email protected]

r Let W ,r ∈ N Sobolev‘s space on [0,1]. Telyakovskiy for r = 1 and Gopengauz for r ∈ N strengthened the direct theorem Nikolskogo-Timmana proving, that each function f ∈ W r can be approximated by an algebraic polynomial pn ∈ Πn degree < n so r px(1 − x) | f (x) − p (x)| ≤ c(r)( ) ,n > r (1) n n where c - absolute constant. DeVore and Yu prove, that at r = 1,2 estimation (1) is also valid for the approximation of a monotone function by a monotone polynomial. Gonska, Leviatan, Shevchuk, Wenz proved that for a natural r > 2 the estimate (1) is, in general, false. For r ∈ R introduce a r r−1 r class of functions W [0,1], such that D0+ f absolutely continuous and D0+ f ≤ 1 a.e. on [0,1]

62 Mathematical Analysis, Differential Equation & Applications - MADEA 9 Kyrgyz-Turkish Manas University, Bishkek, Kyrgyz Republic, June 21-25, 2021

r−1 1 (here D0+ f - left fractional derivative). Let’s denote ∆ the set of monotonically nondecreasing on [0,1] function. The main result is a theorem that summarizes the result of the works [1],[2] in classes W r[0,1]∩ ∆1 with r ∈ (2,3),r ∈ R. r 1 Theorem. Let r ∈ (2,3). Then for everyone n exists function F = Fr,n ∈ W [0,1]∩∆ such that 1 for each polynomial pn ∈ Πn ∩∆ and for any positive on (0;1) function ψ, such that lim ψ(x) = 0 x→0 and lim ψ(x) = 0 one of such properties is valid: x→1 or |F(x) − p (x)| lim n = +∞ x→0 ϕ2(x)ψ(x) or |F(x) − p (x)| lim n = +∞. x→0 ϕ2(x)ψ(x) [1] Kopotun K. A., Leviatan D., Shevchuk I. A. Interpolatory estimates for convex piecewise polynomial approximation. J. Math. Anal. Appl., 474 (1), 467–479 (2019). [2] Petrova T. A. On pointwise interpolation estimates of monotonic approximation of functions with fractional derivative. Visnyk KNU. Mathematics-Mechanics, 9-10, 125–127 (2003).

SAMPLING RECOVERY INTHEREPRODUCINGKERNEL HILBERT SPACE SETTING Kateryna Pozharska Institute of Mathematics of NAS of Ukraine, Kyiv, Ukraine [email protected] In the talk, we present the results of our joint work with Tino Ullrich [1]. We investigate the d recovery of complex-valued multivariate functions on a compact domain D ⊂ R from function samples at a discrete set of nodes X := {x1,...,xn} ⊂ D. The error is measured in the uniform norm `∞, i.e.,

k f k`∞(D) := sup| f (x)|. x∈D The functions are modeled as elements from some reproducing kernel Hilbert space H(K) with bounded kernel K : D × D → C, i.e., p kKk∞ := K(x,x) < ∞ The nodes X are drawn independently according to a tailored probability measure depending on the spectral properties of the embedding of H(K) in L2(D,ρD). This “random information” determines a weighted least squares recovery operator which is used for the whole class of functions and our main focus is on worst-case errors with high probability. In one scenario the measure ρD is at our disposal and represents a certain degree of freedom. In another scenario the measure ρD is a ﬁxed measure, namely the one where the data is coming from. The main feature of this approach is that the nodes are drawn once for the whole class. This is usually termed as “random information” (in contrast to Monte Carlo algorithms). Surprisingly, this approach, which is actually tailored for the L2(D,ρD) setting, also leads to near-optimal results for the `∞-setting expressed in bounds on gn(Id: H(K) → `∞(D)):

gn (Id) := inf inf sup k f − R( f (X))k`∞(D), X={x1,...,xn−1} R∈ ( n−1,` (D)) L C ∞ k f kH(K)≤1

63 MATHEMATICAL ANALYSIS,DIFFERENTIAL EQUATION &APPLICATIONS - MADEA 9 KYRGYZ-TURKISH MANAS UNIVERSITY,BISHKEK,KYRGYZ REPUBLIC,JUNE 21-25, 2021 where by L (F,`∞(D)) we denote the set of linear continuous operators A: F → `∞(D). The comparison to the corresponding benchmark approximation numbers

an (Id: H(K) → `∞(D)) := inf sup k f − A f k`∞(D) A∈L (H(K),`∞(D)) k f k ≤1 rankA

[1] Pozharska K., Ullrich T. A note on sampling recovery of multivariate functions in the uniform norm. arXiv: math/2103.11124, 2021.

APPROXIMATION BY MODIFIED LUPA ¸S-STANCU OPERATORS BASEDON (p,q)-INTEGERS Mohd Qasim Baba Ghulam Shah Badshah University, Rajouri India [email protected]

The purpose of this paper is to construct a new class of Lupa¸soperators in the frame of post quantum setting. We obtaine Korovkin type approximation theorem, study the rate of convergence of these operators by using the concept of K-functional and modulus of continuity, also give a convergence theorem for the Lipschitz continuous functions.

GENERALIZED JAIN-BETA OPERATORS AND THEIR APPROXIMATION PROPERTIES Mohd Qasim Baba Ghulam Shah Badshah University, Rajouri, India [email protected]

The main intent of this paper is to innovate a new construction of generalized Jain operators with weights of some Beta basis functions whose construction depends on ρ such that ρ(0) = 0 0 and inf ρ (y) ≥ 1. For these operators we compute rate of convergence by weighted modulus of y∈[0,∞) continuity, pointwise convergence by proving Voronovskaya type theorem. Finally, we ﬁgure out quantitative estimates for the local approximation.

64 Mathematical Analysis, Differential Equation & Applications - MADEA 9 Kyrgyz-Turkish Manas University, Bishkek, Kyrgyz Republic, June 21-25, 2021 ACCELERATED EXPLICIT TSENG’SEXTRAGRADIENTMETHODS FOR SOLVING DIFFERENT CLASSES OF VARIATIONAL INEQUALITIES Habib ur Rehman1, Murat Özdemir2, Poom Kumam1 1 King Mongkuts Universityof Technology Thonburi, Bangkok, Thailand. 2 Atatürk University, Erzurum, Turkey. [email protected], [email protected], [email protected] A plethora of applications from mathematical programming such as minimax and numerical programming, saddle point problems, fixed point and variational inequalities are formulated as an equilibrium problem. Mainly the procedures for solving such problems require iterative techniques, so, in this study, we construct three novel extragradient-type iterative schemes to solve variational inequalities in real Hilbert spaces. The proposed iterative schemes are equivalent to the extragradient method used to solve variational inequalities in the setting of a real Hilbert space. The main advantage of these iterative schemes is a simple format of step size rule depends on the information of an operator evaluation instead of the Lipschitz constant or some other line search method. The strong convergences are well proved, corresponding to the proposed methods and imposing certain control parameters conditions. Finally, to confirm the efficacy and supremacy of the iterative methods, we present some numerical experiments.

Acknowledgements. The authors acknowledge the ﬁnancial support provided by the Center of Excel- lence in Theoretical and Computational Science (TaCS-CoE), KMUTT. Moreover, this research project is supported by Thailand Science Research and Innovation (TSRI) Basic Research Fund: Fiscal year 2021 under project number 64A306000005.

[1] Korpelevich G. The extragradient method for ﬁnding saddle points and other problems, Matecon, 12, 747–756 (1976). [2] Stampacchia G. Formes bilineaires coercitives sur les ensembles convexes. Comptes Rendus Hebdomadaires Des Seances De L Academie Des Sciences, 258 (18), 4413–4416. (1964).

ENTROPYNUMBERSOFTHE NIKOL’SKII–BESOVCLASSESIN THE SPACE OF QUASI-CONTINUOUSFUNCTIONS Anatolii Romanyuk, Sergii Yanchenko Institute of Mathematics of NAS of Ukraine, Kyiv, Ukraine [email protected], [email protected]

r We consider the approximations of the Nikol’skii–Besov classes Bp,θ (see, for example, [1]) of periodic functions of many variables and establish the estimates for the entropy numbers of these classes in the metric of the QC-space of quasi-continuous functions [2]. d Let R , d ≥ 1, be the d-dimensional Euclidean space with the elements x = (x1,...,xd) and d (x,y) = x1y1 + ... + xdyd. Denote by Lp(πd), πd = ∏ j=1[0,2π], 1 ≤ p ≤ ∞, the space of functions f (x) that are 2π-periodic in each variable, with a standard ﬁnite norm. Let X be a Banach space, and let BX (y,r) = x ∈ X : kx − yk ≤ r be a ball centered at the point y with the radius r. For the compact set A ⊂ X and for ε > 0 the quantity

2k n 1 2k [ j o εk(A,X ) = inf ε : ∃y ,...,y ∈ X : A ⊆ BX (y ,ε) j=1

65 MATHEMATICAL ANALYSIS,DIFFERENTIAL EQUATION &APPLICATIONS - MADEA 9 KYRGYZ-TURKISH MANAS UNIVERSITY,BISHKEK,KYRGYZ REPUBLIC,JUNE 21-25, 2021 is called the entropy number of the set A with respect to the space X , where k ∈ N. Further, let µ be the normed Lebesgue measure on the unit circle. Given a function f ∈ L1(dµ) with the Fourier series ∞ Z 2π ikx f ∼ ∑ δs( f ,x), δ0( f ,x) = f dµ, δs( f ,x) = ∑ fb(k)e , s = 1,2,..., s=0 0 2s−1≤|k|<2s let us introduce the value ∞ Z 1 k f kQC ≡ ∑ rs(ω)δs( f ,x) dω, 0 s=0 L∞(dµ) ∞ where rs(ω) s=0 is the Rademacher system. By the space of quasi-continuous functions (denote QC) we mean the closure of the set of trigonometric polynomials with respect to this norm. Spaces of quasi-continuous functions can also be deﬁned in the multidimensional case (d ≥ 2):

1 k f kQC ≡ k f (·,x )kQC ∞,

1 where, for x = (x1,...,xd) ∈ πd, we set by deﬁnition x = (x2,...,xd) ∈ πd−1. In other words, in this quantity, the QC-norm is considered with respect to the variable x1 and the sup-norm is considered with respect to the remaining variables. r d We assume that the vector r in the deﬁnition of classes Bp,θ has the form r = (r1,...,r1) ∈ R+. 1 Theorem. Let 2 ≤ p ≤ ∞, 2 ≤ θ < ∞, r1 > 2 . Then for d ≥ 2 the following relation holds:

1 1 r −r1 d−1 r1+ 2 − p εM Bp,θ ,QC M (log M) θ logM.

[1] Lizorkin P.I., Nikol’skii S. M. Function spaces of mixed smoothness from the decomposition point of view, Proc. Steklov Inst. Math., 187, 163–184 (1990). [2] Kashin B. S., Temlyakov V.N. On a norm and approximation characteristics of classes of functions of several variables. Metric theory of functions and related problems in analysis, Izd. Nauchno-Issled. Aktuarno-Finans. Tsentra (AFTs), Moscow, 1999, 69 – 99. (in Russian)

ANEXTREMALPROBLEMFOR CESÀROMEANSONSOMECLASS OFHOLOMORPHICFUNCTIONS Olga Rovenska1, Viktor Savchuk2 1 Donbas State Engineering Academy, Kramatorsk, Ukraine 2 Institute of Mathematics of NAS of Ukraine, Kyiv, Ukraine [email protected], [email protected]

Let f be a function holomorphic in the unit disk D := {z ∈ C : |z| < 1} and let

∞ ( j) j f (0) f (z) = ∑ fbjz , fbj := j=0 j! be its Taylor expansion.

66 Mathematical Analysis, Differential Equation & Applications - MADEA 9 Kyrgyz-Turkish Manas University, Bishkek, Kyrgyz Republic, June 21-25, 2021

α The Cesàro means σn ( f ), n = 1,2,..., α > −1, for f are deﬁned by

n−1 α 1 α j α Γ(k + α + 1) k + α σn ( f )(z) = α ∑ An− j−1 fbjz , Ak := = An−1 j=0 Γ(k + 1)Γ(α + 1) k where Γ(n) — Gamma function. Let n be natural, ρ ∈ [0,1] and α > −1. Denote

α 1 C (n,ρ,α) := sup | f (ρ) − σn ( f )(ρ)| : f ∈ B ,

B1 f f 0 z where is the class of holomorphic functions with supz∈D | ( )| ≤ 1. In this talk, we consider the extremal√ problem about the exact value of C (n,ρ,α). Theorem 1. For any ρ ∈ [0,2 − 2] and for any natural n, we have ρ C (n,ρ,1) = . n

Theorem 2. For any ρ ∈ [0,1] and for any natural n, we have

2ρ C (n,ρ,2) = . n + 1

[1] Savchuk V.V., Chaichenko S. O., Savchuk M. V. Approximation of holomorphic and harmonic functions by Fejer means. Ukr. Math. J. 71 (4), 589–618 (2019).

THEEXTENDEDNONSYMMETRICBLOCK LANCZOSMETHODS FOR SOLVING LARGE-SCALE DIFFERENTIAL LYAPUNOV EQUATIONS Lakhlifa Sadek, Hamad Talibi Alaoui Chouaib Doukkali University, El Jadida, Morocco [email protected], [email protected]

By using the extended nonsymmetric block Lanczos method we present a new approach for solving large-scale differential Lyapunov equations, which is based on a low-rank approximation of the solution and compare it to the extended block Arnoldi method technique. The numerical results demonstrate the high efﬁciency and accuracy of suggested algorithms.

[1] Abou-kandil H., Freiling G., Ionescu V., et al. Matrix Riccati equations in control and systems theory. Birkhäuser, 2012. [2] Barkouki H., Bentbib A. H., Heyouni M., et al. An extended nonsymmetric block Lanczos method for model reduction in large scale dynamical systems. Calcolo, 55 (1), 1–23 (2018). [3] Hached M., Jbilou K. Numerical solutions to large-scale differential Lyapunov matrix equations, Numerical Algorithms, 79 (3), 741–757 (2018).

67 MATHEMATICAL ANALYSIS,DIFFERENTIAL EQUATION &APPLICATIONS - MADEA 9 KYRGYZ-TURKISH MANAS UNIVERSITY,BISHKEK,KYRGYZ REPUBLIC,JUNE 21-25, 2021

ON RIEMANNPROBLEMINWEIGHTED SMIRNOVCLASSESWITH POWERWEIGHT Sabina Sadigova Institute of Mathematics and Mechanics of NAS of Azerbaijan, Baku, Azerbaijan Khazar University, Baku, Azerbaijan [email protected]

Weighted Smirnov classes with power weight in bounded and unbounded domains are deﬁned in this work. Nonhomogeneous Riemann problem with a measurable coefﬁcient whose argument is a piecewise continuous function is considered in these classes. A Muckenhoupt type condition is imposed on the power type weight function and the orthogonality condition is found for the solvability of nonhomogeneous problem in weighted Smirnov classes, and the formula for the index of the problem is derived. Some special cases with power type weight function are also considered, and conditions on degeneration order are found.

Acknowledgements. This work is supported by the Scientiﬁc and Technological Research Council of Turkey (TUBITAK) with Azerbaijan National Academy of Sciences (ANAS), Project Number: 19042020 and by the Science Development Foundation under the President of the Republic of Azerbaijan – Grant No. EIF-BGM-4-RFTF1/2017- 21/02/1-M-19.

LIFT PROBLEMS OF DIFFERENTIAL GEOMETRIC OBJECTS Arif Salimov, Tarana Sultanova Baku State University, Baku, Azerbaijan [email protected], [email protected]

The main purpose of this presentation is to investigate a holomorphic pure Riemannian metric with respect to the commutative algebraic structure. We show that the holomorphic manifold admitting this metric is a Kähler type manifold and such type structure exists on the bundle of 2-jets. 2 Let T (Vr) be the bundle of 2-jets, i.e. the tangent bundle of order 2 over the Riemannian man- 2 2 ifold (Vr,g), dimT (Vr) = 3r. We can prove that the bundle T (Vr) has a natural integrable struc- 2 2 3 = I, , I = id 2 R , = ture Π γ γ , T (Vr), which is an r-regular representation of the algebra ε ε 0 [1]. 2 If G ji, Hji are a symmetric tensor ﬁelds of types (0,2) in Vr, then we have an R ε −holomorphic 2 pure Riemannian metric g in T (Vr):

 s 1 t s s s  x ∂sg ji + 2 x x ∂t∂sg ji + x ∂sG ji + Hji x ∂sg ji + G ji g ji s ge=  x ∂sg ji + G ji g ji 0 . g ji 0 0 de f II 2 We denote ge by g and call it a deformed 2-nd lift of a Riemannian metric to T (Vr). We have 2 de f II Theorem. If Vr is a Riemannian manifold with metric g, then the triple T (Vr), Π, g is a Kähler type manifold.

[1] Salimov A., Cengiz N., Behboudi Asl. M. On holomorphic hypercomplex connections, Adv. Appl. Clifford Algebras, 23 (1), 179–207 (2013).

68 Mathematical Analysis, Differential Equation & Applications - MADEA 9 Kyrgyz-Turkish Manas University, Bishkek, Kyrgyz Republic, June 21-25, 2021 ONTOPOLOGICALSTRUCTURESOFTHESIMPLEUNDIRECTED GRAPHS Hatice Kübra Sarı, Abdullah Kopuzlu Atatürk University, Erzurum, Turkey [email protected], [email protected]

Graph theory is a branch of mathematics. The theory has also a very important place in our lives, since the graphs are used effectively in many fields. As a result of this, the topological structure of the graphs has been a interesting research subject. Many researchers has studied on this subject. In this paper, we investigate some topological notions such as accumulation point, interior point, relative topology, T0,T1 and Hausdorff space on the topological spaces generated the graphs. Firstly, it is examined that the states of being an accumulation point and an interior point of a point in these spaces. Relative topology on a subgraph of a graph is defined. It is shown that this topology is not equal to the topology generated by this subgraph. Finally, necessary and sufficient conditions for being T0−space, T1−space and Hausdorff space of the topological space generated from this graph are presented. This enables to examine whether the topological space is T0,T1 and Hausdorff without obtaining the topology generated from the graph .

[1] Abdu K. A., Kılıçman A., Topologies on the Edges Set of Directed Graphs, J. Math. Computer Sci., 18 2018, 232-241.

[2] Amiri S. M., Jafarzadeh A., Khatibzadeh H. An Alexandroff Topology on Graphs, Bulletin of the Iranian Mathematical Society. 39 (4), 647–662 (2013).

[3] Bondy J. A., Murty U. S. R. Graph Theory. Springer, Berlin, 2008.

[4] Euler L. Solutio problematis ad geometriam situs pertinentis. Commentarii Academiae Scientiarum Imperialis Petropolitanae, 8, 128–140 (1736).

[5] Lipschutz S. Schaum’s Outline of Theory and Problems of General Topology. Mcgraw-Hill Book Company, New York, St. Louis,San Francisco, Toronto,Sydney, 1965.

[6] Munkres J. R. Topology Second Edition. Prentice Hall, Upper Saddle River, New Jersey, 2000.

[7] Sarı H. K., Kopuzlu A., A Note on a Binary Relation corresponding to a Bipartite Graph, ITM Web of Confer- ences 22, 2018, 01039.

[8] Sarı H. K., Kopuzlu A., On topological spaces generated by simple undirected graphs. Aims Mathematics, 5 (6), 5541–5550 (2020).

INVERSELOADEDTWO-SPEEDTRANSFERPROBLEMSOF KATZ TYPEINANUNBOUNDEDDOMAINWITHACOLLISION INTEGRAL Zhyldyz Sarkelova, Taalaibek Omurov J. Balasagyn Kyrgyz National University, Bishkek, Kyrgyz Republic [email protected], [email protected]

We study a loaded two-velocity singularly perturbed inverse problem (SPIP) in this work. To solve this problem, an algorithm is taken into account that generalized method of boundary layer function [1], when the a priory information is given from L2(R2)[2].

69 MATHEMATICAL ANALYSIS,DIFFERENTIAL EQUATION &APPLICATIONS - MADEA 9 KYRGYZ-TURKISH MANAS UNIVERSITY,BISHKEK,KYRGYZ REPUBLIC,JUNE 21-25, 2021

We consider an inverse problem:

β ∂ 1,1,1 1,1,1 ε E Uε + λE Uε + h∗(x,y)Uε = ε [UεUεx(x0,y,t) + KUε ] + Zε (x,y) f (t), ∂t (a1,a2) (a1,a2)

( (i) (i) −1 −1 i x2+y2 (Uεt (t,x,y)|t=0 = Vt (0,x,y) + (2a1xε + 2a2yε ) exp − ε , (i) 2 Vt (0,x,y) = ϕi(x,y), (i = 0,1), ∀(x,y) ∈ R ,

( 1,1,1 (E Uε )|t=T = g0(x,y) + gε (x,y), (a1,a2) 1,1,1 2 (E V)|t=T = g0(x,y), ∀(x,y) ∈ R , (a1,a2)

where (Uε ,Zε ) are unknown functions. 0 < h∗(x,y) = h1 (x) + h2(y) + h(x,y), f (t), ϕi(x,y), 1 g0(x,y), gε (x,y),K(x,y,τ1,τ2),0 < ai, λ = const, (i = 0,1),0 < β < 2 are known to us, moreover Z (KUε ) ≡ K(x,y,τ1,τ2)h(τ1,τ2)Uε (t,τ1,τ2)dτ1,dτ2,(0 ≤ K). R2 Algorithm-based:

2 2 ( (x−a1t) +(y−a2t) Uε = V(t,x,y) + ξε (t,x,y) + exp − ε , Zε = Z˜(x,y) + ηε (x,y), we prove the uniqueness of the solution for SPIP, and deﬁne the closeness of the SPIP solution to 2 the solution of corresponding degenerate inverse problem in Wh (Ω0). [1] Imanaliev M. I. Asymptotic methods in the theory of singularly perturbed integro-differential systems. Frunze: Ilim, 1972. [2] Omurov T. D., Tuganbaev M. M. Direct and inverse problems of one-velocity transport theory. Bishkek: Ilim, 2010.

DEFERREDALMOSTSEQUENCESSPACES Ekrem Savas Usak University, Usak, Turkey [email protected]

In this paper we present the notion of deferred almost sequences spaces deﬁned by using a modulus. Considering this concept we introduce a series of inclusion theorems that characterize the relationship between a modulus function and regular summability methods and present theorems.

[1] Agnew R. P. On deferred Cesaro means, Ann. of Math., 33 (3), 413–421 (1932). [2] Bhardwaj V.K., Singh N. On some sequence spaces deﬁned by a modulus, Indian J. Pure Appl. Math. 30(8), 809–817 (1999). [2] Ruckle W. H. FK Spaces in which the sequences of coordinate vectors on bunded, Canad. J. Math., 25, 973– 978 (1973).

70 Mathematical Analysis, Differential Equation & Applications - MADEA 9 Kyrgyz-Turkish Manas University, Bishkek, Kyrgyz Republic, June 21-25, 2021 AN EXTREMAL PROBLEM FOR THE INVARIANT DIFFERENTIAL OPERATORS ON CLASS OF CAUCHYTYPEINTEGRALS Viktor Savchuk1, Maryna Savchuk2 1 Institute of Mathematics of NAS of Ukraine, Kyiv, Ukraine 2 National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Kyiv, Ukraine [email protected], [email protected]

2 0 2 The differential operators D1 ( f )(z) := (1 − |z| ) f (z) and D2 ( f )(z) := D1 ( f )(z) = (1 − |z|2)2 f 00(z) − 2z(1 − |z|2) f 0(z), deﬁned on the space of holomorphic functions in the unit disk D := {z ∈ C : |z| < 1}, are invariant with respect to compositions with fractionally linear functions. They arise naturally on studies of holomorphic functions from the Bloch class B := f is holomorphic in D : sup|D1( f )(z)| ≤ 1 , z∈D which plays an important role in the geometric theory of functions. It is known that the images of the operators |D j|, j = 1,2, on B are Lipschitz with respect to the pseudo-hyperbolic metric ρ (z,w) in D. Namely, it was proved in [1] that √ ||D1 ( f )(z)| − |D1 ( f )(w)|| 3 3 sup = , z,w ∈ D. f ∈B ρ (z,w) 2

In this talk, we give the solution of extremal problem about the exact value of the quantity

 |D ( f )(z) − D ( f )(w)| sup 1 2 , w 6= z, A(z,w) := f ρ (z,w) sup f |D2( f )(z)|, w = z,

when f runs a class of Cauchy type integrals K , which consist from the holomorphic functions f , such that Z ϕ(t) dt f (z) = , z ∈ D, T 1 −tz 2πit where ϕ is essentialy bounded function on T := {t ∈ C : |t| = 1} with dist ϕ,H∞(T) ≤ 1. The function for which the supremum is attained called the extremal for A(z,w).

Theorem. Let z,w ∈ D and argz = argw. Then we have A(z,w) = 2. The only functions extremal for A(z,w) are (t − z)(t − w) f (t) = λ , (1 −tz)(1 −tw) where λ is unimodular constant.

[1] Xiong C. On the Lipschitz continuity of the dilation of Bloch functions, Per. Math. Hung. 47 (1–2), 233–238 (2003).

71 MATHEMATICAL ANALYSIS,DIFFERENTIAL EQUATION &APPLICATIONS - MADEA 9 KYRGYZ-TURKISH MANAS UNIVERSITY,BISHKEK,KYRGYZ REPUBLIC,JUNE 21-25, 2021

ORDER ESTIMATES OF THE UNIFORM APPROXIMATIONS BY ZYGMUNDSUMSONTHECLASSESOFCONVOLUTIONSOF PERIODICFUNCTIONS Anatolii Serdyuk1, Ulyana Hrabova2 1 Institute of Mathematics of NAS of Ukraine, Kyiv, Ukraine 2 Lesya Ukrainka Volyn National University, Lutsk, Ukraine [email protected], grabova−[email protected] ψ Denote by Cβ,p the class of 2π-periodic functions f representable by the convolution f (x) = π a0 1 R 2 + π Ψβ (x−t)ϕ(t)dt, a0 ∈ R, kϕkp ≤ 1, 1 ≤ p ≤ ∞, with a fixed generating kernel Ψβ ∈ Lp0 , −π 1 1 ∞ βπ p + p0 = 1, which has the Fourier series of the form Ψβ (t) ∼ ∑k=1 ψ(k)cos(kt + 2 ), ψ(k) ≥ 0, ψ β ∈ R. It is clear that Cβ,p ⊂ C. Consider the problem of finding the exact orders for the quantities ψ s s E (C ;Zn−1)C := sup k f − Zn−1( f ;·)kC, β,p ψ f ∈Cβ,p s a0 n−1 k s where Zn−1( f ;t) := 2 + ∑k=1 1 − n ak( f )coskt + bk( f )sinkt , s > 0 are the Zygmund sums of the function f (ak( f ) and bk( f ) are the Fourier coefficients). We indicate the conditions under which the Zygmund sums realize the order of the quantities ψ En(Cβ,p)C := sup inf k f −tn−1kC. ψ tn−1∈T2n−1 f ∈Cβ,p of best uniform approximations by the subspace T2n−1 of trigonometric polynomials of order n−1. We assume that the sequences ψ(k) are traces of some positive continuous convex down- wards functions ψ(t), t ≥ 1, on the set such that lim ψ(t) = 0 and write ψ ∈ M. We put N t→∞ n ψ(t) o M0 = ψ ∈ M : ∃K > 0 ∀t ≥ 1 α(ψ;t) := t|ψ0(t)| ≥ K > 0 . + ∞ Denote by GM the set of sequences g = gk k=1 for which there exists a constant A ≥ 1 m−1 satisfying the following inequalities: gn1 + ∑ |gk − gk+1| ≤ Agm for all positive integers n1 ≤ n2, k=n1 + ∞ m = n1,n2. Denote by GA the set of sequences g = gk k=1 for which there exists a number −ε ε > 0 such that gkk almost increases i.e., there exists a constant K such that for all positive −ε −ε δ integers n1 ≤ n2 gn1 n1 ≤ Kgn2 n2 . For a fixed δ > 0, we also set gδ (t) := ψ(t)t , t ≥ 1. + + Theorem. Assume that s > 0, 1 ≤ p < ∞, g1/p ∈ M0, gs+1/p ∈ GM ∩GA , β ∈ R and n ∈ N. ∞ p0 p0−2 p0 1 1 i) If 1 < p < ∞, ∑k=n ψ (k)k < ∞ and inf α(g1/p;t) > , + 0 = 1, then the following order t≥1 2 p p estimates hold: ∞ 1/p0 ψ ψ s p0 p0−2 En Cβ,p E Cβ,p;Zn−1 ∑ ψ (k)k . C C k=n ∞ ii) If p = 1, ∑k=n ψ(k) < ∞ and inf α(g1;t) > 1, then the following order estimates take place: t≥1 ∞  βπ ψ ψ s  ∑ ψ(k), cos 2 6= 0; En Cβ,1 E Cβ,1;Zn−1 k=n C C  βπ ψ(n)n, cos 2 = 0.

72 Mathematical Analysis, Differential Equation & Applications - MADEA 9 Kyrgyz-Turkish Manas University, Bishkek, Kyrgyz Republic, June 21-25, 2021 JACKSONINEQUALITIES IN BESICOVITCH-STEPANETS SPACES Anatolii Serdyuk, Andrii Shidlich Institute of Mathematics of NAS of Ukraine, Kyiv, Ukraine [email protected], [email protected] For a ﬁxed 1 ≤ p < ∞ consider the spaces of Besicovitch almost periodic functions f of the order 1 (B-a.p. functions) with the ﬁnite norm 1/p k f k := k f k = |A ( f )|p , p S p ∑ λk k∈N

1 R T −iλkx where Aλ ( f ) = lim 0 f (x)e dx, k ∈ N, are the Fourier coefficients of the function f . We k T→∞ T denote these spaces by BS p and call the Besicovitch-Stepanets spaces. By definition the B-a.p. functions are assumed to be equivalent in the space BS p when their Fourier series are the same. In the talk, direct and inverse approximation theorems will be formulated in terms of best approximations and generalized moduli of smoothness for almost periodic functions from the spaces BS p, the sequences of Fourier exponents of which have a single limit point at infinity. For such functions f , we write the Fourier series in the symmetric form: 1 Z T iλkx −iλkx S[ f ](x) = ∑ Ake , where Ak = Ak( f ) = lim f (x)e dx, (1) T→∞ T 0 k∈Z λ0 := 0, λ−k = −λk, |Ak| + |A−k| > 0, λk+1 > λk > 0 for k > 0. Consider the set Φ of all continuous bounded nonnegative pair functions ϕ such that ϕ(0) = 0 and the Lebesgue measure of the set {t ∈ R : ϕ(t) = 0} is equal to zero. For a fixed function ϕ ∈ Φ, we define the generalized modulus of smoothness of a function f ∈ BS p by the equality: 1/p ( f , ) = p(kh)|A ( f )|p , ≥ . ωϕ δ p : sup ∑ ϕ λk δ 0 |h|≤δ k∈N

Let Gλn denotes the set of all B-a.p. functions whose Fourier exponents belong to the interval p (−λn,λn) and let Eλn ( f )p = inf k f − gkp be the best approximation of the function f ∈ BS . g∈Gλn Let also M (τ), τ > 0, be a set of bounded nondecreasing functions µ that differ from a constant on [0,τ]. Theorem. Assume that the function f ∈ BS p, 1 ≤ p < ∞, has the Fourier series of the form (1). Then for any τ > 0, n ∈ N and ϕ ∈ Φ, the following inequality holds: τ Eλn ( f )p ≤ Cn,ϕ,p(τ)ωϕ f , , (2) λn p where τ 1/p Z µ(τ) − µ(0) pλkt Cn,ϕ,p(τ) = inf , In,ϕ,p(τ, µ) = inf ϕ dµ(t). (3) µ∈M(τ) In,ϕ,p(τ, µ) k∈N:k≥n λn 0 There exists a function µ∗ ∈ M (τ) that realizes the greatest lower bound in (3). Inequality (2) is not improved on the set of all functions f ∈ BS p, f 6≡ const, with the Fourier series of the form (1) in the sense that for any ϕ ∈ Φ and n ∈ N the following equality holds: ( E ( f ) ) λn p p Cn, ,p(τ) = sup : f ∈ BS , f 6≡ const . ϕ ω ( f , τ ) ϕ λn p

73 MATHEMATICAL ANALYSIS,DIFFERENTIAL EQUATION &APPLICATIONS - MADEA 9 KYRGYZ-TURKISH MANAS UNIVERSITY,BISHKEK,KYRGYZ REPUBLIC,JUNE 21-25, 2021

ONASYMPTOTICEQUATIONSFORTHEWIDTHSOFCLASSESOF THEGENERALIZED POISSONINTEGRALS Anatolii Serdyuk, Igor Sokolenko Institute of Mathematics of NAS of Ukraine, Kyiv, Ukraine [email protected], [email protected]

Denote by Cα,r the set of all 2π-periodic functions f , representable as convolution β¯,2

π a0 1 Z f (x) = + ϕ(x −t)Pα,r,β (t)dt, a0 ∈ R, (1) 2 π −π

∞ −αkr βkπ the ﬁxed kernel Pα,r,β¯ (t) = ∑ e cos kt − , α > 0, r > 0, βk ∈ R, with functions k=1 2 0 ϕ ∈ B2 = {g ∈ L2 : kgkL2 ≤ 1, g ⊥ 1}. If β ≡ β,β ∈ , then the classes Cα,r are denoted by Cα,r which are well-known classes of k R β¯,2 β,2 the generalized Poisson integrals (see [1]). Further, let K be a convex centrally symmetric subset of C and let B be a unit ball of the space C. Let also FN be an arbitrary N-dimensional subspace of space C, N ∈ N, and L (C,FN) be a set of linear operators from C to FN. By P(C,FN) denote the subset of projection operators of the set L (C,FN), that is, the set of the operators A of linear projection onto the set FN such that A f = f when f ∈ FN. The quantities

bN(K,C) = sup sup{ε > 0 : εB ∩ FN+1 ⊂ K}, FN+1

dN(K,C) = inf sup inf k f − ukC , FN f ∈K u∈FN

λN(K,C) = inf inf sup k f − A f kC , FN A∈L (C,FN ) f ∈K

πN(K,C) = inf inf sup k f − A f kC , FN A∈P(C,FN ) f ∈K are called Bernstein, Kolmogorov, linear, and projection N-widths of the set K in the space C, respectively. ¯ ∞ r Theorem. Let β = {βk}k=1,βk ∈ R and α > 0, r > 1, n ∈ N be such that (n − 1) > 1/α. Then as n → ∞ the following asymptotic equalities hold

α,r ) P2n(C ¯ ,C) r 1 1 r−1 β,2 = e−αn √ + O(1) 1+ e−αr(n−1) , P (Cα,r,C) r(n− )r−1 2n−1 β¯,2 π α 1 where PN is any of the widths bN,dN,λN or πN, and O(1) are the quantities uniformly bounded in all parameters.

Acknowledgements. This work was partially supported by the Grant H2020-MSCA-RISE-2019, project number 873071 (SOMPATY: Spectral Optimization: From Mathematics to Physics and Advanced Technol- ogy).

[1] Stepanets A. I. Methods of Approximation Theory. Utrecht: VSP, 2005.

74 Mathematical Analysis, Differential Equation & Applications - MADEA 9 Kyrgyz-Turkish Manas University, Bishkek, Kyrgyz Republic, June 21-25, 2021 UNIFORMAPPROXIMATIONSBY FOURIERSUMSONCLASSESOF (ψ,β)–DIFFERENTIABLE FUNCTIONS Anatolii Serdyuk1, Tetiana Stepaniuk1,2 1 Institute of Mathematics of NAS of Ukraine, Kyiv, Ukraine 2 University of Luebeck, Luebeck, Germany [email protected], [email protected]

Let ψ(k) be an arbitrary ﬁxed sequence of real, nonnegative numbers and let β be a ﬁxed real ψ number. And let Cβ,1 be the set of all functions f , which are represented for all x as convolutions of the form π Z a0 1 0 f (x) = + ϕ(t)Ψβ (x −t)dt, a0 ∈ R, ϕ ∈ B , 2 π 1 −π where 0 B1 := {ϕ : ||ϕ||L1 ≤ 1, ϕ ⊥ 1}.

and Ψβ is a ﬁxed kernel of the form

∞ ∞ βπ Ψβ (t) = ∑ ψ(k)cos kt − , ψ(k) ≥ 0, ∑ ψ(k) < ∞, β ∈ R. k=1 2 k=1

ψ For the classes Cβ,1 we consider the quantities

ψ En(C )C = sup k f (·) − Sn−1( f ;·)kC, (1) β,1 ψ f ∈Cβ,1

where Sn−1( f ;·) are the partial Fourier sums of order n − 1 for a function f . We consider Kolmogorov–Nikolsky problem about ﬁnding of asymptotic equalities of the quantity (1) as n → ∞. The following statement holds. ∞ Theorem. Let ∑ kψ(k) < ∞, ψ(k) ≥ 0, k = 1,2,... and β ∈ R. Then as n → ∞ the following k=1 asymptotic equality holds

∞ ∞ ψ 1 O(1) En(Cβ,1)C = ∑ ψ(k) + ∑ kψ(k + n), (2) π k=n n k=1

where O(1) is a quantity uniformly bounded in all parameters. Formula (2) becomes an asymptotic equality for the sequence ψ(k), which decreases to zero r faster than arbitrary power function. In the case, when ψ(k) = e−αk , α > 0, r > 0, the equality (2) was established in [1] and [2].

[1] Serdyuk A. S. Approximation of classes of analytic functions by Fourier sums in uniform metric, Ukr. Math. J. 57 (8), 1275–1296 (2005). [2] Serdyuk A. S., Stepanyuk T. A., Uniform approximations by Fourier sums on classes of generalized Poisson integrals, Analysis Mathematica 45 (1), 201–236 (2019).

75 MATHEMATICAL ANALYSIS,DIFFERENTIAL EQUATION &APPLICATIONS - MADEA 9 KYRGYZ-TURKISH MANAS UNIVERSITY,BISHKEK,KYRGYZ REPUBLIC,JUNE 21-25, 2021

WOVEN FRAMES IN QUATERNIONIC HILBERT SPACES Nitin Sharma University of Delhi, Delhi, India [email protected]

In this presentation, we introduce and study woven frames in quaternionic Hilbert spaces. We also give some properties of woven frames and give some conditions on family of frames under which it is woven in quaternionic Hilbert spaces. Also a characterization of weaving frames in terms of a surjective bounded right linear operator is given.

Joint work with Dr. Sumit Kumar Sharma and Dr. Khole Timothy Poumai.

POINTWISE ESTIMATE OF DEVIATION OF KRIAKIN POLYNOMIAL FROMAFUNCTION, CONTINUOUSONASEGMENT Mykyta Shchehlov Taras Shevchenko National University of Kyiv, Kyiv, Ukraine [email protected]

Let Qk−1 be the polynomial of degree ≤ k − 1 of integral approximation of the continuous m function f on the segment I := [0;1] in the points xm := k , i.e., Z i/k ( f (t) − Qk−1( f ,t))dt = 0, i = 0,1...k. 0 From [1] it’s known that

|| f − Qk−1|| ≤ We (k)ωk( f ,1/k), (1) where We (k) = 2 when k ≤ 82000 and We (k) = 2 + exp(−2) when k > 82000. But this estimation is achieved near the ends of the segment I (namely on the segments [0;1/k] and [(k − 1)/k,1]). So here arises the question: if it is possible to improve this estimation "inside" the segment, i.e, obtain for x ∈ [1/k;(k − 1)/k] inequality

| f (x) − Qk−1(x)| ≤ p(x)ωk( f ,1/k), where p(x) is a function that depends on x (possibly constant), but whose values are less then 2, already known estimation. It can help to improve overall estimation on the segment. The main results relating to this matter are the following: Theorem. 4mlnk |g(x)| ≤ m , (2) Ck where x ∈ [m/k,(m + 1)/k],m < k/2 and g := f − Q, . Lemma. Let f is a continuous function on the segment [I and P is an approximating polynomial. Let also the minimum and the maximum of f are achieved on the segments [0,1/k] and [(k − 1)/k,1] and aren’t opposite. Then there exists such a polynomial Q := P + P1 that || f − Q|| < || f − P|| with deg P1 ≤ 1.

[1] Gilewicz J., Kryakin Yu.V., Shevchuk I. A. Boundedness by 3 of the Whitney Interpolation Constant, Journal of Approximation Theory, 119 (2), 271–290 (2002).

76 Mathematical Analysis, Differential Equation & Applications - MADEA 9 Kyrgyz-Turkish Manas University, Bishkek, Kyrgyz Republic, June 21-25, 2021 MATHEMATICAL MODELING AS A METHOD OF COGNITION Olga Shvai Lesya Ukrainka Volyn National University, Lutsk, Ukraine [email protected]

The purpose of our research is to consider historical aspects of the formation of the modeling method and to highlight its epistemological functions. The use of mathematical modelling method in educational process was investigated in the paper [1]. Modeling in the educational process acts simultaneously as:

• a method of scientiﬁc knowledge,

• a part of the content of educational material

• and an effective mean of its study.

The development of students’ ideas about the role of mathematical modeling in scientific knowledge and practice, the development of their ability to build mathematical models of life phenomena, are important task of modern school. In particular, we need to pay special attention to developing students’ skills to reformulate an applied problem into the language of mathematics and to create adequate mathematical models. It is important, that students concentrate correctly, highlighting the essential and non-essential properties of objects; abstracted from insignificant properties; correctly interpreted the relationships between objects of the problem. A teacher should form a particular attitude of students to the acquired knowledge through the disclosure of the essence of mathematical modeling. Despite the widespread use of the method of mathematical modeling, the development of rele- vant students’ skills is not systematic and is mostly done during mathematics lessons. This significantly reduces didactic effectiveness of the use of this method in learning process, in particular, in increasing mathematical literacy of students. To overcome this limitation, in our opinion, it is possible to use interdisciplinary connections more effectively. For example, when generalizing the basic properties of directly proportional and linear functions in mathematics lessons, it is advisable to use the knowledge of students that were already obtained in physics lessons when studying thermal phenomena. We offer them, as a homework, to build graphs of the amount of heat obtained during the combustion of this type of fuel, its mass, for example, for dry firewood, anthracite, gasoline. We analyze these graphs in math lessons. The development of correct ideas of students about the nature of the reflection of mathematical phenomena and processes of the real world, the role of mathematical modeling in scientific knowledge and in practice, is of great importance for the formation of their mathematical literacy.

[1] Shvai O. Solving exercises with physical content in the mathematics lessons as a part of formation of pupils’ universal learning activities, Physical and Mathematical Education, 2 (28), 83–88 (2021).

77 MATHEMATICAL ANALYSIS,DIFFERENTIAL EQUATION &APPLICATIONS - MADEA 9 KYRGYZ-TURKISH MANAS UNIVERSITY,BISHKEK,KYRGYZ REPUBLIC,JUNE 21-25, 2021

THELONGTIMEBEHAVIOROFNONLINEARSTOCHASTIC FUNCTIONAL-DIFFERENTIAL EQUATIONS OF NEUTRAL TYPE IN HILBERT SPACES WITH NON-LIPSCHITZNONLINEARITIES Andriy Stanzhytskyi National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Kyiv, Ukraine [email protected].

We study the asymptotic behaviour of the solutions of neutral type stochastic functional- differential equations of the form

d[u(t) + g(ut)] = [Au + f (ut)]dt + σ(ut)dW(t) for t > 0;u(t) = ϕ(t),t ∈ [−h,0), h > 0.

Here A is an iniﬁnitesimal generator of a strong continuous semigroup {S(t),t ≥ 0} of bounded linear operators in real separable Hilbert space H. The noise W(t) is a Q-Wiener process on a separable Hilbert space K. For any h > 0 denote Ch := C([−h,0],H) to be a space of continuous H-valued functions ϕ : [−h,0] → H, equipped with the norm

kϕkCh := sup kϕ(t)kH, t∈[−h,0] where k · kH stands for the norm in H. By k · kH will be denoted with k · k. The solution u(t) of this equation is sometimes referred as a state process. We also denote ut := u(t + θ), where 0 0 1/2 θ ∈ [−h,0). The functionals f and g map Ch to H, and σ : Ch → L2 , where L2 = L (Q K,H) is the space of Hilbert-Schmidt operators from Q1/2K to H. Finally, ϕ : [−h,0] × Ω → H is the initial condition, where (Ω,F ,P) is the probability space. The main goal of our research is to establish the existence of invariant measure for this equation using Krylov-Bogoliubov theory on the tightness of a family of measures. More precisely, we will implement the compactness approach of Da Prato and Zabczyk.

THE AVERAGING METHOD FOR BOUNDARY VALUE PROBLEMS FOR DIFFERENTIAL EQUATIONS WITH NON-FIXED IMPULSIVE MOMENTS Oleksandr Stanzhytskyi1, Roza Uteshova2, Meirambek Mukash3 1 Taras Shevchenko National University of Kyiv, Kyiv, Ukraine 2 International Information Technology University, Almaty, Kazakhstan 3 K.Zhubanov Aktobe Regional University, Aktobe, Kazakhstan [email protected], [email protected], [email protected]

Impulsive systems of differential equations serve as mathematical models of objects that, in the course of their evolution, are exposed to the action of short-term forces. A fairly complete theory of such systems is presented in the monograph [1]. Much research has been done on non-fixed impulsive initial value problems. However, in regard to boundary value problems for equations with impulse action, the majority of results concern jumps only at fixed times. This is due to the fact that non-fixed impulses significantly change properties of boundary value problems.

78 Mathematical Analysis, Differential Equation & Applications - MADEA 9 Kyrgyz-Turkish Manas University, Bishkek, Kyrgyz Republic, June 21-25, 2021

In the present paper, we use the averaging method [2,3] to solve the following boundary value problem for a system of differential equations with impulse action at non-ﬁxed moments of time and a small parameter: x˙ = εX(t,x), t 6= ti(x),

∆x|t=ti(x) = εIi(x), (1) F(x(0),x(T/ε)) = 0.

Here ε > 0 is a small parameter, T > 0 is ﬁxed, ti(x) < ti+1(x)(i = 1,2,...) are the moments of impulse action, X, Ii, and F are d-dimensional vector functions. Under the assumption that there exist the limits

T 1 Z 1 X0(x) = lim X(t,x)dt, I0(x) = lim Ii(x), T→∞ T T→∞ T ∑ 0 0

y˙ = ε[X0(y) + I0(y)], F(y(0),y(T/ε)) = 0, (2) or, on the slow time scale τ = εt, dy = X (y) + I (y), F(y(0),y(T)) = 0. dτ 0 0 The main result of our paper is a proof of the following statement: if the averaged boundary value problem (2) has a solution, then, for small values of the parameter ε, the original boundary value problem (1) also has a solution that belongs to a small neighborhood of the solution of the averaged problem.

[1] Samoilenko A. M., Perestyuk M. O. Impulsive Differential equations. Singapore: World Scientiﬁc, 1995. [2] Samoilenko A. M., Petrishin R. I. Averaging method in some boundary value problems, Diff. Equat., 25 (6), 956–964 (1989). [3] Samoilenko A. M. Averaging method in systems with tremors, Math. Phys., 9, 101–117 (1971).

ON STATISTICALLY CONVERGENCE THEOREMS FOR LEBESGUE INTEGRABLEFUNCTIONS Tuncay Tunç, Erdem Alper Mersin University, Mersin, Turkey [email protected], [email protected]

When we have a sequence of integrable functions, “the convergence theorems” depend on this sequence state that the integrability is preserved under the operation of limit. More clearly, let ( fn) be a sequence of integrable functions and the function f be some kind of limit of the sequence. Then we naturally desire integralibility of the function f and the relation Z Z f = lim fn

79 MATHEMATICAL ANALYSIS,DIFFERENTIAL EQUATION &APPLICATIONS - MADEA 9 KYRGYZ-TURKISH MANAS UNIVERSITY,BISHKEK,KYRGYZ REPUBLIC,JUNE 21-25, 2021 holds. We call this type of results as convergence theorems. These theorems are used to show that a given function is integrable or to construct an integrable function. The best known theorem of among the convergence theorems in measure the theory is the Lebesgue dominated convergence theorem. In this work, we prove the certain convergence theorems in σ− ﬁnite measurable spaces, when a given sequence of measurable functions is statistically converges to a function.

ON STANCU GENERALIZATION OF SZASZ-MIRAKYAN-BERNSTEIN OPERATORS Tuncay Tunç, Burcu Fedakar Mersin University, Mersin, Turkey [email protected], [email protected]

In this study, we have introduced a Stancu generalization of the Szasz-Mirakyan-Bernstein Operators [1] ∞ mαn k Ln( f ;x) = ∑ qn,m−1(x) ∑ pmαn,k(x) f , x ∈ [0,1] m=1 k=0 mαn deﬁned on the space of continuous functions deﬁned on a compact interval; where n p (x) = (x)k(1 − x)n−k, 1 ≤ k ≤ n n,k k and (nx)m q (x) = e−nx , m ∈ . n,m m! N0 We have given a general formula for the moments of that operators. We used the Korovkin’s Theorem for uniform approximation under some restrictions. We obtained some results for approximation rates in terms of modulus of contiunity. Finally, we gave some Voronovskaya-type theorems. We have also given some illustrations using Mathematica.

[1] Tunç T.,¸Sim¸sekE. Some approximation properties of Szasz-Mirakyan-Bernstein operators, European Journal of Pure and Applied Mathematics, 7 (4), 419–428 (2014).

SOLUTIONOFSOMESYSTEMSOFLINEARORDINARY DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS Anarkul Urdaletova 1, Syrgak Kydyraliev 2 1 Kyrgyz-Turkish Manas University, Bishkek, Kyrgyz Republic 2 American University of Central Asia, Bishkek, Kyrgyz Republic [email protected], [email protected]

The problem of integrability of ordinary differential equations, an indication of methods for obtaining their exact solution, is one of the urgent problems of the theory of differential equations, which is devoted to a huge number of scientiﬁc works of various researchers. This is because: a) differential equations are widely used for continuous models of dynamical systems in physics, medicine, economics, biology, and other natural sciences, for which the explicit trajectory of the

80 Mathematical Analysis, Differential Equation & Applications - MADEA 9 Kyrgyz-Turkish Manas University, Bishkek, Kyrgyz Republic, June 21-25, 2021 dynamic system’s behavior is very important since the explicit solution formulas contain the maximum information about the behavior of the system; b) an explicit solution of the equation is necessary to confirm the mathematical and physical intuition, to compare the solutions obtained by various approximate methods, and to compare these methods. It is known that on the way of indicating methods for obtaining an explicit form of solving differential equations, an indisputable advantage is given to rather simple algorithms. In this paper, a method is constructed for finding integrable combinations and an explicit form of the solution for one class of systems of first-order linear ordinary differential equations with a variable coefficient at the derivatives. The work also provides examples that were solved by the constructed method. This method is based on the well-known classical methods of integrating ordinary differential equations, such as the Leonard Euler method based on the roots of the charac- teristic equation, and the Jean Leron D’Alembert method of integrable combinations. The method presented in this work is an extension of the method of the authors of this work, from systems of inhomogeneous equations with constant coefficients to a certain class of systems of ordinary differential equations with variable coefficients.

[1] Kydyraliev S. K., Urdaletova A. B. Direct Integration of Systems of Linear Differential and Difference Equa- tions. Filomat, 33 (5), 1453–1461 (2019).

COMMONCOURSE “MATHEMATICS” INTHESCHOOL, ASA SYMBIOSISOFGEOMETRYANDALGEBRA Anarkul Urdaletova 1, Syrgak Kydyraliev 2, Elena Burova 2 1 Kyrgyz-Turkish Manas University, Bishkek,Kyrgyzstan 2 American University of Central Asia, Bishkek,Kyrgyzstan [email protected], [email protected], [email protected]

Mathematics is a science with a rich, centuries-old history of development. It often owes its development to other sciences and practices, which, by their requests, push it towards the development of new mathematical theories. However, new mathematical theories, in turn, also give an impetus for the development of other sciences and expand the horizons of knowledge of the environment around man. On this path of development of sciences, in particular, mathematics, as a symbolic language of other sciences and the phenomenon of human culture, there is a need for reforms in school and higher education. In recent years, Kyrgyzstan has introduced regular, new, state standards of school education, where there is no division of subjects into algebra and geometry. On the basis of these two courses, a common course in Mathematics appears. It presupposes not a mechanical combination of two mathematical courses into one, but the creation of a new course that has absorbed the best features of the original courses. For these purposes, it is necessary to develop new scientiﬁc approaches in teaching, both school mathematics and university mathematics, especially those departments of universities where future mathematics teachers are trained. This paper shows how harmoniously it is possible to solve this problem by using the methods of analytical geometry in algebra and algebraic methods in geometry. Examples of solving problems of school mathematics using the approach adopted by the authors are given. In conclusion, we note that the word symbiosis in the title of the work is used in the sense of "a form of mutually beneﬁcial coexistence."

81 MATHEMATICAL ANALYSIS,DIFFERENTIAL EQUATION &APPLICATIONS - MADEA 9 KYRGYZ-TURKISH MANAS UNIVERSITY,BISHKEK,KYRGYZ REPUBLIC,JUNE 21-25, 2021

ONTHEWIDTHSOFCLASSESOFCOMPLEXVARIABLE ANALYTIC FUNCTIONS Sergii Vakarchuk1, Michailo Vakarchuk2 1Alfred Nobel University, Dnipro, Ukraine 2Oleg Gonchar University, Dnipro, Ukraine [email protected], [email protected]

We represent one of our new results. Let A(G) be the set of analytic functions in G = {z ∈ C : |z| < 1}. By H2,γ (G) we denote the weight space of functions f ∈ A(G) having ﬁnite norm

ZZ 1/2 n 2 o k f k2,γ = γ(z)| f (z)| dxdy . G

Here γ(z) is a weight function which is a constant an each circle |z| = ρ < 1, i.e. it be of the form γ(z) = γ(ρ). We assume that γ is integrable over G and γ is nonzero almost everywhere. B (z) = z j m ∈ A(G) 6= Let a function m ∑ j∈Z+( j>m) β j , Z+, belongs to and let β j 0 for all 1/ j j = m,m + 1,...; lim{|β j| ; j → ∞} = 1 and lim{β j : j → ∞} = ∞. To each function f (z) = c ( f )z j A(G) ∑ j∈Z+( j>m) j of we form the function

j D(Bm, f ;z) = ∑ β jc jz . j∈Z+( j>m) It is the Hadamard composition which is the generalization of the differentiation operation. Let H2,γ (D(Bm)) = { f ∈ A(G) : D(Bm, f ) ∈ H2,γ (G)}. By Φ(t), 0 6 t 6 1, we denote a continuous increasing function such that Φ(0) = 0. Function Φ is the majorant. For f ∈ H2,γ (G) we denote by Ωk( f ,t), k ∈ N and 0 < t < 1, the k-th order generalized modulus of continuity which is formed by the generalized shift operator of the special type. Let H2,γ (D(Bm);Ωk,Φ) = { f ∈ H2,γ (D(Bm)) : Ωk(D(Bm, f ),t) 6 Φ(t) for all 0 < t < 1}. By En( f ), n ∈ N, we denote the best approximation a function f ∈ H2,γ (G) by the space of (n − 1)-th order algebraic polynomials in H2,γ (G). We take En(M) = sup{En( f ) : f ∈ M} for a set M ⊂ H2,γ (G).

Theorem. Let n,k ∈ N, m ∈ Z+ and a majorant Φ satisﬁes the condition

inf{t−kΦ(t) : 0 < t < 1} = limsup{t−kΦ(t) : t → 0+}.

Then for an arbitrary n > m the following equalities hold:

qn(H2,γ (D(Bm);Ωk,Φ),H2,γ (G)) = En(H2,γ (D(Bm);Ωk,Φ)) =

−k −1 −k = n |βn |inf{t Φ(t) : 0 < t < 1}, where qn is any of the next n-widths: Bernstein, Kolmogorov, linear, Gelfand, projection, Fourier.

82 Mathematical Analysis, Differential Equation & Applications - MADEA 9 Kyrgyz-Turkish Manas University, Bishkek, Kyrgyz Republic, June 21-25, 2021 ONSIGN-PRESERVING APPROXIMATION OF PERIODIC FUNCTIONSBYTRIGONOMETRICPOLYNOMIALS Viktoriia Voloshyna Taras Shevchenko National University of Kyiv, Ukraine; University of Toulon, La Garde, France [email protected]

2s Let s ∈ N and Ys := {Ys}, where the collections Ys = {yi}i=1 of points yi ∈ R are such that (0) y2s < ··· < y1 < y2s +2π =: y0. By ∆ (Ys) we denote the collection of all such functions f (t) for which the following condition holds.

2s (0) ∆ (Ys) = { f (t) : f (t)∏(t − yi) ≥ 0, t ∈ [y2s,y0].} i=1

r (r− ) (r) W , r ∈ N is the Sobolev space of 2π-periodic functions f ∈ AC 1 (R), such that k f k < g g x +∞, where k k := esssupx∈R| ( )|. Let Tn be the space of trigonometric polynomials of degree ≤ n (of order 2n + 1) and

k k− j k ωk( f ,t) = sup k ∑ (−1) f (· + ih)k, t ≥ 0, h∈[0,t] j=0 j is the modulus of continuity of a function f of order k ∈ N. In case of k = 1, ω1( f ,t) =: ω( f ,t). The approximation of f ∈ ∆(0)(Y) by an algebraic polynomial was described in [1, p. 336-341]. However, in the trigonometrical case same methods cannot be used. We have proved that for any natural r,k and n ≥ N(Y,k) = const, and any function f ∈ W (r) T (0) T (0) ∆ (Ys), there exists a polynomial Rn ∈ Tn ∆ (Ys) such that c(r,k,s) || f − R || ≤ ω ( f (r),1/n), f ∈ W (r). n nr k [1] Gilewicz J., Shevchuk I. A. Comonotone approximation, Fundam. Prikl. Mat., 2 (2), 319–363 (1996).

83 MATHEMATICAL ANALYSIS,DIFFERENTIAL EQUATION &APPLICATIONS - MADEA 9 KYRGYZ-TURKISH MANAS UNIVERSITY,BISHKEK,KYRGYZ REPUBLIC,JUNE 21-25, 2021

LIST OF PARTICIPANTS

• Abdiev A. (Kyrgyz Republic) ...... 26

• Abdullayev F. (Kyrgyz Republic, Turkey) ...... 12, 13, 14

• Abylaeva E. (Kyrgyz Republic) ...... 15, 58

• Afanas’eva E. (Ukraine) ...... 16, 42

• Akmatbekova A. (Kyrgyz Republic)...... 17

• Amanalieva M. (Kyrgyz Republic) ...... 18

• Amin R. (Bangladesh) ...... 18, 29, 46

• Akta¸sB. (Turkey) ...... 35

• Aktay M. (Turkey) ...... 17

• Alaoui H. (Morocco) ...... 67

• Alper E. (Turkey) ...... 79

• Altınta¸s I.˙ (Kyrgyz Republic, Turkey) ...... 20, 20

• Parlatıcı H. (Turkey) ...... 20

• Asanov A. (Kyrgyz Republic) ...... 19

• Baigesekov A. (Kyrgyz Republic) ...... 44

• Belyaev A. (Ukraine) ...... 21

• Benaissa L. (Algeria )...... 25

• Bihun D. (Ukraine) ...... 24

• Bilalov B. (Azerbaijan) ...... 22

• Bilet V. (Ukraine) ...... 23

• Blatt H.-P. (Germany)...... 24

• Boichuk O. (Ukraine) ...... 24

• Burova E. (Kyrgyz Republic) ...... 81

• Celik E. (Turkey) ...... 54

• Chaichenko S. (Ukraine) ...... 13

• Chaouchi C. (Algeria ) ...... 25

• Chekeev A. (Kyrgyz Republic) ...... 26, 26, 27

• Chowdhury L. (Bangladesh) ...... 29

• Chuiko S. (Ukraine) ...... 29, 30, 31, 32

• Dai F. (Canada) ...... 33

84 Mathematical Analysis, Differential Equation & Applications - MADEA 9 Kyrgyz-Turkish Manas University, Bishkek, Kyrgyz Republic, June 21-25, 2021

• Deger˘ U. (Turkey) ...... 38

• Djurciˇ c´ D. (Serbia) ...... 50

• Dmytryshyn R. (Ukraine) ...... 34

• Dobrodzii T. (Ukraine)...... 48

• Donchev T. (Bulgaria) ...... 34

• Dovgoshey O. (Ukraine) ...... 23

• Dudkin M. (Ukraine) ...... 36

• Durmaz O. (Turkey) ...... 35

• Dyuzhenkova O. (Ukraine) ...... 36

• Dziuba M. (Ukraine) ...... 29

• Dzyubenko G. (Ukraine) ...... 37

• Ellik I.(Turkey)...... 38˙

• Elmalı C. (Turkey) ...... 53

• Esengul kyzy P. (Kyrgyz Republic)...... 59

• Fedakar B. (Turkey) ...... 80

• Fedunyk-Yaremchuk O. (Ukraine) ...... 38

• Feruk V. (Ukraine) ...... 24

• Fonseka N. (USA)...... 39

• Gaiko V. (Belarus)...... 40

• Georgieva A. (Bulgaria)...... 41

• Goel S. (India) ...... 49

• Gokta¸sS. (Turkey) ...... 41

• Golberg A. (Israel) ...... 42

• Gün C. (Turkey) ...... 43, 61

• Gündogan˘ H. (Turkey) ...... 35

• Hajj L. (UAE) ...... 43

• Hembars’ka S. (Ukraine) ...... 38

• Hossain S. (Bangladesh) ...... 18

• Hrabova U. (Ukraine) ...... 72

• Hristova S. (Bulgaria)...... 44

• Imankulova˙ A.˙ (Kyrgyz Republic) ...... 20

85 MATHEMATICAL ANALYSIS,DIFFERENTIAL EQUATION &APPLICATIONS - MADEA 9 KYRGYZ-TURKISH MANAS UNIVERSITY,BISHKEK,KYRGYZ REPUBLIC,JUNE 21-25, 2021

• Iskandarov S. (Kyrgyz Republic) ...... 44, 45

• Islam R. (Bangladesh) ...... 46

• Kadıoglu E. (Turkey) ...... 46

• Kakharman N. (Kazakhstan)...... 60

• Kalinichenko Ya. (Ukraine) ...... 30

• Kal’chuk I. (Ukraine) ...... 49

• Kamali M. (Kyrgyz Republic) ...... 46

• Karasheva T. (Kyrgyz Republic) ...... 47

• Karpenko O. (China) ...... 48

• Kasymova T. (Kyrgyz Republic) ...... 27, 28

• Kathuria L. (India) ...... 49

• Khalilov A. (Kyrgyz Republic) ...... 45

• Khanna N. (India) ...... 49

• Kharkevych Yu. (Ukraine) ...... 49

• Kocinacˇ L. (Serbia) ...... 50

• Kopotun K. (Canada) ...... 50

• Kopuzlu A. (Turkey)...... 69

• Kozachenko Yu. (Ukraine) ...... 51

• Kuduk G. (Poland) ...... 52

• Kumam P. (Thailand) ...... 65

• Kumar D. (India) ...... 53

• Kunduracı T. (Turkey) ...... 53

• Kuzmina V. (Ukraine) ...... 31

• Kydyraliev S. (Kyrgyz Republic) ...... 80, 81

• Lanbaran N. (Turkey) ...... 54

• Lazu A. (Romania) ...... 34

• Mahmudov N. (T. R. Northen Cyprus) ...... 54

• Matanova K. (Kyrgyz Republic) ...... 19

• Mogylova V. (Ukraine) ...... 48

• Motorna O. (Ukraine)...... 55

• Muhametjanova G. (Kyrgyz Republic)...... 17

86 Mathematical Analysis, Differential Equation & Applications - MADEA 9 Kyrgyz-Turkish Manas University, Bishkek, Kyrgyz Republic, June 21-25, 2021

• Mukash M. (Kazakhstan) ...... 78

• Myslo Ju. (Ukraine) ...... 56

• Nesmelova O. (Ukraine) ...... 32

• Ogul˘ B. (Turkey) ...... 57

• Omuraliev A. (Kyrgyz Republic) ...... 58, 59

• Omurov T. (Kyrgyz Republic) ...... 69

• Otelbaev M. (Kazakhstan)...... 60

• Özdemir M. (Turkey)...... 17, 65

• Özkartepe P. (Turkey)...... 61

• Pahirya M. (Ukraine) ...... 56

• Parlatıcı H. (Turkey) ...... 20

• Parﬁnovych N. (Ukraine) ...... 62

• Petrova I. (Ukraine) ...... 62

• Petrova T. (Ukraine) ...... 62

• Pokutnyi O. (Ukraine) ...... 24

• Pozharska K. (Ukraine) ...... 63

• Qasim M. (India) ...... 64, 64

• Rehman H. (Thailand) ...... 65

• Romanyuk A. (Ukraine) ...... 65

• Rovenska O. (Ukraine) ...... 66

• Sadek L. (Morocco) ...... 67

• Sadigova S. (Azerbaijan) ...... 68

• Salimov A. (Azerbaijan) ...... 68

• Sarı H. (Turkey)...... 69

• Sarkelova Zh. (Kyrgyz Republic) ...... 69

• Savas E. (Turkey) ...... 70

• Savchuk M. (Ukraine) ...... 71

• Savchuk V. (Ukraine) ...... 14, 51, 66, 71

• Serdyuk A. (Ukraine)...... 72, 73, 74, 75

• Shaha S. (Bangladesh) ...... 46

• Sharma N. (India) ...... 76

87 MATHEMATICAL ANALYSIS,DIFFERENTIAL EQUATION &APPLICATIONS - MADEA 9 KYRGYZ-TURKISH MANAS UNIVERSITY,BISHKEK,KYRGYZ REPUBLIC,JUNE 21-25, 2021

• Shchehlov M. (Ukraine) ...... 76

• Shevchuk I. (Ukraine) ...... 55

• Shidlich A. (Ukraine) ...... 13, 73

• Shivaji R. (USA) ...... 39

• Shvai O. (Ukraine) ...... 77

• ¸Sim¸sekD. (Turkey)...... 57

• Sokolenko I. (Ukraine)...... 74

• Son R. (USA) ...... 39

• Spasova M. (Bulgaria) ...... 41

• Spetzer K. (USA) ...... 39

• Stanzhytskyi A. (Ukraine) ...... 78

• Stanzhytskyi O. (Ukraine) ...... 78

• Stepaniuk T. (Ukraine, Germany) ...... 75

• Sukochev F. (Australia)

• Sultanova T. (Azerbaijan) ...... 68

• Symotyuk M. (Ukraine) ...... 52

• Tulenov K. (Kazakhstan)

• Tunç T. (Turkey) ...... 79, 80

• UgurT.(Turkey)...... 53˘

• Urdaletova A. (Kyrgyz Republic) ...... 80, 81

• Uteshova R. (Kazakhstan) ...... 78

• Vakarchuk M. (Ukraine) ...... 82

• Vakarchuk S. (Ukraine) ...... 82

• Voloshyna V. (Ukraine, France) ...... 37, 83

• Yanchenko S. (Ukraine) ...... 65

• YarA.(Turkey)...... 41

• Yılmaz E. (Turkey) ...... 41

• Zanin D. (Australia)

• Zhusupbekova E. (Kyrgyz Republic) ...... 28