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9 772152 738001 0 3 Applied Mathematics, 2021, 12, 131-239 https://www.scirp.org/journal/am ISSN Online: 2152-7393 ISSN Print: 2152-7385 Table of Contents Volume 12 Number 3 March 2021 Uniqueness of Positive Radial Solutions for a Class of Semipositone Systems on the Exterior of a Ball A. Mohamed, K. A. Abbakar, A. Awad, O. Khalil, B. M. Acyl, A. A. Youssouf, M. Mousa………...............………131 Discrete Model of Plasticity and Failure of Crystalline Materials V. L. Busov…………......................................................................................................…………………………………147 Tuning of Prior Covariance in Generalized Least Squares W. Menke……....................................................................................................................………………………………157 Information Models for Forecasting Nonlinear Economic Dynamics in the Digital Era A. Akaev, V. Sadovnichiy…………........................……………………………………………………………………171 A Stochastic SVIR Model for Measles M. Seydou, O. Moussa Tessa………............................................................……………………………………………209 An Oracle Bone Inscription Detector Based on Multi-Scale Gaussian Kernels G. Y. Liu, S. H. Chen, J. Xiong, Q. J. Jiao……........................................................……………………………………224 Applied Mathematics (AM) Journal Information SUBSCRIPTIONS The Applied Mathematics (Online at Scientific Research Publishing, https://www.scirp.org/) is published monthly by Scientific Research Publishing, Inc., USA. Subscription rates: Print: $89 per copy. To subscribe, please contact Journals Subscriptions Department, E-mail: [email protected] SERVICES Advertisements Advertisement Sales Department, E-mail: [email protected] Reprints (minimum quantity 100 copies) Reprints Co-ordinator, Scientific Research Publishing, Inc., USA. E-mail: [email protected] COPYRIGHT Copyright and reuse rights for the front matter of the journal: Copyright © 2021 by Scientific Research Publishing Inc. 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PRODUCTION INFORMATION For manuscripts that have been accepted for publication, please contact: E-mail: [email protected] Applied Mathematics, 2021, 12, 131-146 https://www.scirp.org/journal/am ISSN Online: 2152-7393 ISSN Print: 2152-7385 Uniqueness of Positive Radial Solutions for a Class of Semipositone Systems on the Exterior of a Ball Alhussein Mohamed1*, Khalid Ahmed Abbakar1,2, Abuzar Awad1, Omer Khalil1,3, Bechir Mahamat Acyl1, Abdoulaye Ali Youssouf1, Mohammed Mousa1 1College of Mathematics and Statistics, Northwest Normal University, Lanzhou, China 2Department of Mathematics and Physics, Faculty of Education, University of Gadarif, Gadarif, Sudan 3Department of Science, College of Education, Sudan University of Science and Technology, Khartoum, Sudan How to cite this paper: Mohamed, A., Abstract Abbakar, K.A., Awad, A., Khalil, O., Acyl, B.M., Youssouf, A.A. and Mousa, M. (2021) In this paper, we study the positive radial solutions for elliptic systems to the Uniqueness of Positive Radial Solutions for −∆u = λ k( x) f( uv,) onΩ , 11 a Class of Semipositone Systems on the −∆v = λ k( x) f( uv,) onΩ , Exterior of a Ball. Applied Mathematics, 12, 22 = = →∞ 131-146. ux( ) vx( ) 0 on x , https://doi.org/10.4236/am.2021.123009 nonlinear BVP: ∂u , where ∆=u div( ∇ u) and +=c10( uu) 0 on x = r , ∂η Received: November 16, 2020 ∂v +=c20( vv) 0, on x = r . Accepted: March 9, 2021 ∂η Published: March 12, 2021 ∆=v div( ∇ v) are the Laplacian of u, λ is a positive parameter, Copyright © 2021 by author(s) and n Ω={x ∈» : N > 2, x > rr00 , > 0} , let i = [1, 2] then Kri :[ 0 ,∞→) ( 0, ∞) Scientific Research Publishing Inc. ∂ This work is licensed under the Creative is a continuous function such that limkr( ) = 0 and is The exter- Commons Attribution International ri→∞ ∂η License (CC BY 4.0). nal natural derivative, and ci :[ 0,∞→) ( 0, ∞) is a continuous function. We http://creativecommons.org/licenses/by/4.0/ Open Access discuss existence and multiplicity results for classes of f with a) fi > 0 , b) fi < 0 , and c) fi = 0 . We base our presence and multiple outcomes via the Sub-solutions method. We also discuss some unique findings. Keywords Elliptic System, Positive Radial Solution, Exterior Domains, Fixed Point Index 1. Introduction In reaction diffusion processes, steady states define the long term dynamics. DOI: 10.4236/am.2021.123009 Mar. 12, 2021 131 Applied Mathematics A. Mohamed et al. Here we consider a steady state reaction diffusion equation on an exterior do- main with a nonlinear boundary condition on the interior boundary. Namely, we study positive radial solutions to: −∆u = λ k( x) f( uv,) onΩ , 11 −∆v = λ k22( x) f( uv,) onΩ , ux( ) = vx( ) =0 on x →∞ , ∂u (1.1) +=c10( uu) 0 on x = r , ∂η ∂v +=c20( vv) 0, on x = r . ∂η where ∆=u div( ∇ u) and ∆=v div( ∇ v) are the Laplacian of u, λ is a positive n parameter, Ω={x ∈» : N > 2, x > rr00 , > 0} , let i = [1, 2] then Kri :[ 0 ,∞→) ( 0, ∞) is a continuous function such that limri→∞ kr( ) = 0 and ∂ is the outward normal derivative, and c :[ 0,∞→) ( 0, ∞) is a is an non ∂η i decreasing (increasing) function. Here the reaction term fRi :[ 0,∞)×[ 0,∞) → 1 is a C function. The case when fi < 0 (see [1] [2], that the study of positive solutions to such problems is considerably more challenging than in the case fi > 0 (positone problems). For a rich history on semipositone problems with Dirichlet boundary conditions on bounded domains, (see [3]-[8], and on do- mains exterior to a ball, see [9] [10] [11]. Such nonlinear boundary conditions occur very naturally in applications see [12] for a detailed description in a model arising in combustion theory. Recently, the existence of a radial positive solution for (1.1) when λ 1 has been established in [13], via the method of subsuper solutions. Here we discuss the uniqueness of this radial solution when some ad- ditional assumptions hold. In [14], the authors study such a uniqueness result for the case of Dirichlet boundary condition on xr= 0 . Our focus in this paper is to consider the uniqueness result for semipositone problem when a class of nonlinear boundary condition is satisfied at xr= 0 . The fact that we have no longer a fixed value of u on xr= 0 results in quite a challenge in extending the results in [15]. Namely, we need to establish a detailed behavior of u at xr= 0 to achieve our goal. Instead of working directly with (1), we note that the change of 2−N r variables rx= and s = transforms (1) into the following boundary r0 value problem: −=u′′ ( t) λ a ( t) f( ut( ), vt( )) t∈[ 0,1] , 11 −=v′′ ( t) λ a22( t) f( ut( ), vt( )) t∈[ 0,1] , N − 2 u′ += cu ( (1)) u( 1) 0, r 1 (1.2) 0 N − 2 v′ += cv ( (1)) v( 1) 0, r 2 0 uv(0) =( 0) = 0. DOI: 10.4236/am.2021.123009 132 Applied Mathematics A. Mohamed et al. −−21( N ) r 2 1 1 a = 0 tNN−−22 k rt k ≤ where ii2 0 . We will only assume i N +µ for (2 − N ) r r 1 and for some µ ∈−(0,N 2) . then ai ∈∞((0,1] ,( 0, )) could be singular at 0.if µ ≥−N 2 , ai will be nonsingular at 0 and it will be an easier case to study. Note that aii= inft∈(0,1] at( ) > 0 and there exists a constant d > 0 such d ( N −−2) µ that ai ≤ α for all t ∈(0,1] where α = . Motivated by the above t N − 2 21 discussion, in this paper, we will study positive solutions in CC(0,1) ∩ [ 0,1] to the following boundary value problems: −=u′′ ( t) λ a( t) f( ut( ), vt( )) t∈[ 0,1] , 11 −=v′′ ( t) λ a22( t) f( ut( ), vt( )) t∈[ 0,1] , u′(1) += cu1 ( ( 1)) u( 1) 0, (1.3) ′ v(1) += cv2 ( ( 1)) v( 1) 0, uv(0) =( 0) = 0. where ci :[ 0,∞) →∞( 0, ) is a continuous function and aCi ∈∞((0,1] ,( 0, )) is such that: (H1) aii= inft∈(0,1] at( ) > 0 ; d (H2) there exists a constant d > 0 such that at( ) = for all t ∈(0,ε ] i tα where a ∈(0,1) and ε ≈ 0 1 (H3) ai is decreasing. We consider various C classes of the reaction term fRi :[ 0,∞)×[ 0,∞) → satisfying the following: fsi ( ) (F1) f < 0 and lim→∞ = 0 ; i = 1, 2 i s s (F2) fi is increasing and limsi→∞ fs( ) = ∞ ; i = 1, 2 (F3) fi is concave on [0,∞) . i = 1, 2 Theorem 1.1. Assume (H1) - (H3) and (F1) - (F3). Then (1.3) has a unique positive solution for all λ sufficiently large. In Section two we will establish important a priori estimates. We will first re- call some important results from [8] where the authors studied the case of Di- richlet boundary condition, or equivalently (1.3) with the boundary condition t = 1 replaced by uv(1) =( 10) = . These results do not depend on the boundary condition at t = 1 and hence it is also true for solutions of (1.3). In view of the readers convenience we include the proofs of these results.