axioms New Trends in Differential and Difference Equations and Applications Edited by Feliz Manuel Minhós and João Fialho Printed Edition of the Special Issue Published in Axioms www.mdpi.com/journal/axioms New Trends in Differential and Difference Equations and Applications New Trends in Differential and Difference Equations and Applications Special Issue Editors Feliz Manuel Minh´os Jo˜ao Fialho MDPI • Basel • Beijing • Wuhan • Barcelona • Belgrade Special Issue Editors Feliz Manuel Minhos´ Joao˜ Fialho University of Evora´ British University Vietnam Portugal Vietnam Editorial Office MDPI St. Alban-Anlage 66 4052 Basel, Switzerland This is a reprint of articles from the Special Issue published online in the open access journal Axioms (ISSN 2075-1680) from 2018 to 2019 (available at: https://www.mdpi.com/journal/axioms/special issues/differential difference equations). For citation purposes, cite each article independently as indicated on the article page online and as indicated below: LastName, A.A.; LastName, B.B.; LastName, C.C. Article Title. Journal Name Year, Article Number, Page Range. ISBN 978-3-03921-538-6 (Pbk) ISBN 978-3-03921-539-3 (PDF) c 2019 by the authors. Articles in this book are Open Access and distributed under the Creative Commons Attribution (CC BY) license, which allows users to download, copy and build upon published articles, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications. The book as a whole is distributed by MDPI under the terms and conditions of the Creative Commons license CC BY-NC-ND. Contents About the Special Issue Editors ..................................... vii Preface to ”New Trends in Differential and Difference Equations and Applications” ................................ ix Feliz Minh´os Heteroclinic Solutions for Classical and Singular φ-Laplacian Non-Autonomous Differential Equations Reprinted from: Axioms 2019, 8, 22, doi:10.3390/axioms8010022 ................... 1 Ravi Agarwal, Snezhana Hristova and Donal O’Regan Lipschitz Stability for Non-Instantaneous Impulsive Caputo Fractional Differential Equations with State Dependent Delays Reprinted from: Axioms 2019, 8, 4, doi:10.3390/axioms8010004 .................... 15 Mina Torabi and Mohammad-Mehdi Hosseini A New Efficient Method for the Numerical Solution of Linear Time-Dependent Partial Differential Equations Reprinted from: Axioms 2018, 7, 70, doi:10.3390/axioms7040070 ................... 32 Vladimir I. Semenov The 3D Navier–Stokes Equations: Invariants, Local and Global Solutions Reprinted from: Axioms 2019, 8, 41, doi:10.3390/axioms8020041 ................... 45 Shin Min Kang, Zain Iqbal, Mustafa Habib and Waqas Nazeer Sumudu Decomposition Method for Solving Fuzzy Integro-Differential Equations Reprinted from: Axioms 2019, 8, 74, doi:10.3390/axioms8020074 ................... 96 Mama Foupouagnigni and Salifou Mboutngam On the Polynomial Solution of Divided-Difference Equations of the Hypergeometric Type on Nonuniform Lattices Reprinted from: Axioms 2019, 8, 47, doi:10.3390/axioms8020047 ...................114 Michael Gil’ Solution Estimates for the Discrete Lyapunov Equation in a Hilbert Space and Applications to Difference Equations Reprinted from: Axioms 2019, 8, 20, doi:10.3390/axioms8010020 ...................125 Jan Andres and Denis Pennequin Note on Limit-Periodic Solutions of the Difference Equation xt+1 − [h(xt)+λ]xt = rt, λ>1 Reprinted from: Axioms 2019, 8, 19, doi:10.3390/axioms8010019 ...................147 Tatyana V. Redkina, Robert G. Zakinyan, Arthur R. Zakinyan, Olesya B. Surneva and Olga S. Yanovskaya Backlund¨ Transformations for Nonlinear Differential Equations and Systems Reprinted from: Axioms 2019, 8, 45, doi:10.3390/axioms8020045 ...................157 Jo˜ao Fialho and Feliz Minh´os First Order Coupled Systems With Functional and Periodic Boundary Conditions: Existence Results and Application to an SIRS Model Reprinted from: Axioms 2019, 8, 23, doi:10.3390/axioms8010023 ...................175 v About the Special Issue Editors Feliz Minh´os, Prof., has a Ph.D. and a Habilitation Degree, both in Mathematics, is the Coordinator of the Research Center on Mathematics and Applications, Director of the Ph.D. Program in Mathematics at the University of Evora,´ Portugal, has published a large number of papers, books, monographs, proceedings, . and is a member of the Editorial Board of several international journals. Prof. Minhos’´ main research interests are in differential equations, boundary value problems with variational and topological methods, fixed point theory, impulsive problems, functional differential equations, nonlocal problems, degree theory, lower and upper solutions, Green’s functions, among others. Joao Fialho, Prof., is a Ph.D. in Mathematics at University of Evora,´ Portugal. He is currently Senior Lecturer and MBA Director at the British University of Vietnam. He has diverse practical working experience and teaching experience, having spent more than 5 years working as a business analyst for private companies in Lisbon, Portugal, and for the last 7 years he has been working in higher education, in several different countries, as Head of the Mathematics Department at College of the Bahamas, Nassau, Bahamas and as Assistant Professor of Mathematics at the American University of the Middle East, Kuwait, before joining the British University of Vietnam. Dr. Fialho has a vast research record of international publications, a book, and several presentations in international conferences. His current research interests are in differential equations, boundary value problems, mathematical modelling, and big data analytics in sports and industry. vii Preface to ”New Trends in Differential and Difference Equations and Applications” Differential and difference equations, their methods, their techniques, and their huge variety of applications have attracted interest in different fields of science in the last few years. Not only their solvability and the study of qualitative properties have been the aim of many research papers, but, also, their role in different types of boundary value problems have allowed the study of many real-world phenomena. This Special Issue provides examples of some new methods and techniques on research topics, such as sufficient conditions to obtain heteroclinic solutions for phi-Laplacian equations, invariants, local and global solutions, stability theory (asymptotic, exponential, Lipsichtz, etc.), numerical methods for partial differential equations, fuzzy integro-differential equations, divided-difference equations, limit-periodic solutions for difference equations, Backlund¨ transformations for nonlinear equations and systems, and coupled systems with functional boundary conditions that include the periodic case. These topics, which encompass several areas of mathematical research, give the reader a comprehensive and quick overview of the trends and recent research results, which may be useful in their research or in future research topics. Feliz Manuel Minh´os, Jo˜ao Fialho Special Issue Editors ix axioms Article Heteroclinic Solutions for Classical and Singular φ-Laplacian Non-Autonomous Differential Equations Feliz Minhós Departamento de Matemática, Escola de Ciências e Tecnologia, Centro de Investigação em Matemática e Aplicações (CIMA), Instituto de Investigação e Formação Avançada, Universidade de Évora. Rua Romão Ramalho, 59, 7000-671 Évora, Portugal; [email protected] Received: 28 December 2018; Accepted: 11 February 2019; Published: 15 February 2019 Abstract: In this paper, we consider the second order discontinuous differential equation in the − + real line, (a (t, u) φ (u )) = f (t, u, u ) , a.e.t ∈ R, u(−∞)=ν , u(+∞)=ν , with φ an increasing homeomorphism such that φ(0)=0 and φ(R)=R, a ∈ C(R2, R) with a(t, x) > 0 for (t, x) ∈ R2, − + − + f : R3 → R a L1-Carathéodory function and ν , ν ∈ R such that ν < ν . The existence and localization of heteroclinic connections is obtained assuming a Nagumo-type condition on the real line and without asymptotic conditions on the nonlinearities φ and f . To the best of our knowledge, this result is even new when φ(y)=y, that is for equation(a (t, u(t)) u (t)) = f (t, u(t), u (t)) , a.e.t ∈ R. Moreover, these results can be applied to classical and singular φ-Laplacian equations and to the mean curvature operator. Keywords: φ-Laplacian operator; mean curvature operator; heteroclinic solutions; problems in the real line; lower and upper solutions; Nagumo condition on the real line; fixed point theory 2010 Mathematics Subject Classification: 34C37; 34B40; 34B15; 47H10 1. Introduction In this paper, we study the second order non-autonomous half-linear equation on the whole real line, a (t, u) φ u = f t, u, u , a.e.t ∈ R, (1) with φ an increasing homeomorphism, φ(0)=0 and φ(R)=R, a ∈ C(R2, R) such that a(t, x) > 0 for (t, x) ∈ R2, and f : R3 → R a L1-Carathéodory function, together with the asymptotic conditions: − + u(−∞)=ν , u(+∞)=ν , (2) + − − + with ν , ν ∈ R such that ν < ν . Moreover, an application to singular φ-Laplacian equations will be shown. This problem (1) and (2) was studied in [1,2] . This last paper contained several results and criteria. For example, Theorem 2.1 in [2] guarantees the existence of heteroclinic solutions under, in short, the following main assumptions: • φ grows at most linearly at infinity; − + • f (t, ν ,0) ≤ 0 ≤ f (t, ν ,0) for a.e.t ∈ R; Axioms 2019, 8, 22; doi:10.3390/axioms80100221 www.mdpi.com/journal/axioms Axioms 2019, 8,22 + + • there exist constants L, H > 0, a continuous function θ : R → R and a function λ ∈ Lp([−L, L]), with 1 ≤ p ≤ ∞, such that: − + | f (t, x, y)| ≤ λ(t) θ (a(t, x) |y|) , for a.e. |t|≤L, every x ∈ ν , ν , +∞ − 1 s1 q |y| > H, ds =+∞; θ(s) • > η ∈ 1(R) Λ ∈ 1 ([ +∞)) [ ] for every C 0, there exist functions C L , C Lloc 0, , null in 0, L and positive [ +∞) ( ) ∈ 1(R) in L, , and NC t L such that: f (t, x, y) ≤−Λ (t)φ (|y|) , C (− ) ≥ Λ ( )φ (| |) ≥ ∈ ν− ν+ f t, x, y C t y , for a.e. t L, every x , , |y|≤N (t), C | ( )| ≤ η ( ) ∈ ν− ν+ | |≤ ( ) ∈ R f t, x, y C t if x , , y NC t , for a.e.t .