EXISTENCE OF SOLUTIONS OF NONLOCAL PERTURBATIONS OF DIRICHLET DISCRETE NONLINEAR PROBLEMS

doi:10.1016/S0252-9602(17)30047-4

Abstract

Abstract:

This paper is devoted to the study of second order nonlinear difference equations. A Nonlocal Perturbation of a Dirichlet Boundary Value Problem is considered. An exhaustive study of the related Green's function to the linear part is done. The exact expression of the function is given, moreover the range of parameter for which it has constant sign is obtained. Using this, some existence results for the nonlinear problem are deduced from monotone iterative techniques, the classical Krasnoselski fixed point theorem or by application of recent fixed point theorems that combine both theories.

Key words: Dirichlet boundary value problem, nonlocal perturbations, Green's function, parameter dependence

Alberto CABADA, Nikolay D. DIMITROV. EXISTENCE OF SOLUTIONS OF NONLOCAL PERTURBATIONS OF DIRICHLET DISCRETE NONLINEAR PROBLEMS[J].Acta mathematica scientia,Series B, 2017, 37(4): 911-926.

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References

[1] Agarwal R P. Difference Equations and Inequalities. Theory, Methods, and Applications. 2nd ed. Monographs and Textbooks in Pure and Applied Mathematics, 228. New York:Marcel Dekker, Inc, 2000
[2] Agarwal R P, Perera K, O'Regan D. Multiple positive solutions of singular discrete p-Laplacian problems via variational methods. Adv Differ Equ, 2005, 2:93-99
[3] Bereanu C, Mawhin J. Existence and multiplicity results for periodic solutions of nonlinear difference equations. J Differ Equ Appl, 2006, 12:677-695
[4] Bereanu C, Mawhin J. Existence and multiplicity results for nonlinear second order difference equations with Dirichlet boundary conditions. Math Bohem, 2006, 131:145-160
[5] Bereanu C, Thompson H B. Periodic solutions of second order nonlinear difference equations with discrete φ-Laplacian. J Math Anal Appl, 2007, 330:1002-1015
[6] Cabada A, Cid J A, Infante G. New criteria for the existence of non-trivial fixed points in cones//Fixed Point Theory and Applications. Springer, 2013
[7] Cabada A, Dimitrov N. Multiplicity results for nonlinear periodic fourth order difference equations with parameter dependence and singularities. J Math Anal Appl, 2010, 371(2):518-533
[8] Cabada A, Iannizzotto A, Tersian S. Multiple solutions for discrete boundary value problems. J Math Anal Appl, 2009, 356:418-428
[9] Cabada A, Otero-Espinar V. Optimal existence results for nth order periodic boundary value difference equations. J Math Anal Appl, 2000, 247:67-86
[10] Cabada A, Otero-Espinar V. Comparison results for n-th order periodic difference equations. Proceedings of the Third World Congress of Nonlinear Analysts, Part 4(Catania, 2000). Nonlinear Anal, 2001, 47:2395-2406
[11] Cabada A, Otero-Espinar V, Pouso R L. Existence and approximation of solutions for discontinuous first order difference problems with nonlinear functional boundary conditions in the presence of lower and upper solutions. Comput Math Appl, 2000, 39:21-33
[12] Cabada A, Souroujon D, Tersian S. Heteroclinic solutions of a second-order difference equation related to the Fisher-Kolmogorov's equation. Appl Math Comput, 2012, 218(18):9442-9450
[13] Candito P, Giovannelli N. Multiple solutions for a discrete boundary value problem involving the pLaplacian. Comput Math Appl, 2008, 56:959-964
[14] Cai X, Guo Z. Existence of solutions of nonlinear fourth order discrete boundary value problem. J Differ Equ Appl, 2006, 12:459-466
[15] Elaydi S N. An Introduction to Difference Equations Undergraduate Texts in Mathematics. New York:Springer-Verlag, 1996
[16] Franco D, O'Regan D, Perán J. Upper and lower solution theory for first and second order difference equations. Dynam Systems Appl, 2004, 13:273-282
[17] Gil' M I. Positive solutions of nonautonomous linear higher order difference equations. Appl Anal, 2005, 84:701-706
[18] Goldberg S. Introduction to Difference Equations. New York:John Wiley & sons, 1960
[19] Graef J R, Kong L, Wang H. A periodic boundary value problem with vanishing Green's function. Appl Math Lett, 2008, 21:176-180
[20] J. R. Graef, L. Kong and H. Wang, Existence, multiplicity, and dependence on a parameter for a periodic boundary value problem. J Differ Equ, 2008, 245:1185-1197
[21] He T, Su Y. On discrete fourth-order boundary value problems with three parameters. J Comput Appl Math, 2010, 233:2506-2520
[22] Krasnoselskii M A. Positive Solutions of Operator Equations. Groningen:Noordhoff, 1964
[23] Kelley W, Peterson A. Difference Equations, An Introduction with Applications. New York:Academic Press, 1991
[24] Lakshmikantham V, Trigiante D. Theory of Difference Equations, Numerical Methods and Applications. Mathematics in Science and Engineering, 181. Boston, MA:Academic Press, Inc, 1988
[25] Liu X, Zhang Y, Shi H. Existence of periodic solutions for a class of nonlinear difference equations. Qual Theory Dyn Syst, 2015, 14(1):51-69
[26] Mawhin J. Bounded solutions of some second order difference equations. Georgian Math J, 2007, 14:315-324
[27] Mendivenci Atici F, Guseinov G Sh. Positive periodic solutions for nonlinear difference equations with periodic coefficients. J Math Anal Appl, 1999, 232:166-182
[28] Reunsumrit J, Sitthiwirattham T. Positive solutions of difference summation boundary value problem for a second-order difference equation. J Comput Anal Appl, 2015, 19(2):319-334
[29] Stehlík P. On variational methods for periodic discrete problems. J Differ Equ Appl, 2008, 14:259-273
[30] Tian Y, Du Z, Ge W. Existence results for discrete Sturm-Liouville problem via variational methods. J Differ Equ Appl, 2007, 13:467-478
[31] Wang Y M. Monotone methods for a boundary value problem of second-order discrete equation. Comput Math Appl, 1998, 36:77-92
[32] Wang H. On the number of positive solutions of nonlinear systems. J Math Anal Appl, 2003, 281:287-306
[33] Zeidler E. Nonlinear Functional Analysis and Its Applications. I. Fixed-Point Theorems. New York:Springer-Verlag, 1986
[34] Zhuang W, Chen Y, Cheng S S. Monotone methods for a discrete boundary problem. Computers & Math Applic, 1996, 32(12):41-49