Abstract
Purpose
In this article, the authors discuss the existence and multiplicity of solutions for an anisotropic discrete boundary value problem in T-dimensional Hilbert space. The approach is based on variational methods especially on the three critical points theorem established by B. Ricceri.
Design/methodology/approach
The approach is based on variational methods especially on the three critical points theorem established by B. Ricceri.
Findings
The authors study the existence of results for a discrete problem, with two boundary conditions type. Accurately, the authors have proved the existence of at least three solutions.
Originality/value
An other feature is that problem is with non-local term, which makes some difficulties in the proof of our results.
Keywords
Citation
Ourraoui, A. and Ayoujil, A. (2022), "On a class of non-local discrete boundary value problepm", Arab Journal of Mathematical Sciences, Vol. 28 No. 2, pp. 130-141. https://doi.org/10.1108/AJMS-06-2020-0003
Publisher
:Emerald Publishing Limited
Copyright © 2020, Anass Ourraoui and Abdesslem Ayoujil
License
Published in Arab Journal of Mathematical Sciences. Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) licence. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial and non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this licence may be seen at http://creativecommons.org/licences/by/4.0/legalcode.
1. Introduction
The non-linear difference equations have been of great interest because of their important applications appearing in various fields of research, such as numerical analysis, non-linear differential equations, computer science, mechanical engineering, control systems, artificial or biological neural networks and social sciences, such as economics. To deal with these kind of problems, a various methods such as fixed points theorems, lower and upper solutions, Browder degree, variational approach and critical point theory have been applied by many different authors. For the recent progress in discrete problems, we refer the readers to valuable monograph by Agarwal [1] and the papers [2,3]. Let
In the present paper, we deal with the existence of solutions for the Neumann problem
Equations of this type were suggested by Kirchhoff in 1883. More precisely the following model, which is called Kirchhoff equation, was introduced (see [4])
The study of these problems has received more attention. In [2,5–15], a variety of different methods were applied to obtain the existence results to the discrete boundary value problem of the following type
For example, Jiang and Zhou in [16] employing a three critical point theorem, due to Ricceri, established the existence of at least three solutions for perturbed non-linear difference equations with discrete boundary conditions. Bonanno and Candito [11], employing critical point theorems in the setting of finite dimensional Banach spaces, investigated the multiplicity of solutions for non-linear difference equations involving the p-Laplacian. Cabada et al. in [2], based on three critical points theorems, investigated different sets of assumptions which guarantee the existence and multiplicity of solutions for difference equations involving the discrete p-Laplacian operator. Candito and Giovannelli [12], using variational methods, established the existence of at least three solutions for the problem above. Far from being exhaustive, further details can be found in [13,17–24].
By taking into account the previous papers and inspired by [25], we study problems (1) and (2) and obtain the existence of three weak solutions by employing a kind of Ricceri's theorem [26]. As for the author's best knowledge, the present papers results are not covered in the related literature, and hence, it is original in its own right.
The structure of this paper is outlined as follows. In Section 2, some preliminary results and statement of main results are presented. In Section 3, the proof of the main results is given.
2. Preliminaries
Firstly, we recall some basic properties which will be used in the proof of the precise result.
Through the sequel, we say that the functional
In order to prove our main results, we will use the following Ricceri's theorem.
[26] Let
Finally, setting assume that
Then, for each compact interval
Denoting by
and
Solutions to (1) will be investigated in a space
It can be verified that for all
We list also some inequalities that will be are used later.
([8]) For every
We say that
Define the functionals
and
Let
We make the following assumptions.
Put
and
Now, we provide an example of non-linear term which satisfies
Example:
Set
There exists
For
Similarly,
Since
Therefore,
Besides, for
which means that
Now, we can state the first main result of this article.
Let
Now, suppose that we have:
Solutions to (2) will be investigated in a space
Therefore, the associated norm is defined by
Also, it is useful to introduce other norms on
It can be verified (see [15]) that
We report our second main result.
Let
Example:
Let consider the above example chosen for the function
In addition, for
For function
3. Proof
Proof of Theorem 2.3. It is clear that since the functional
In view of
From the fact that F is bounded on each subset of
Consequently, for
On the other hand, for each
Thus we have
Since
In addition, it is well known that
From
Then the assumptions of Theorem 2.3 are satisfied with
Example
Taking
So
Proof of Theorem 2.4. Let start by defining
Let
Consequently, for
On the other hand, for each
Thus, combining (12) and (13), we get
Since
The following corollary is a direct application of Theorem 2.4.
let
Then for each compact interval
References
1.Agarwal RP. Difference equations and inequalities. New York: Marcel Dekker; 2000.
2.Cabada A, Iannizzotto A, Tersian S. Multiple solutions for discrete boundary value problems. J. Math. Anal. Appl. 2009; 356: 418-28.
3.Mihăilescu M, Rădulescu V, Tersian. Eigenvalue problems for anisotropic discrete boundary value problems. J. Difference Equ. Appl. 2009; 15: 557-67.
4.Kirchhoff G. Vorlesungen uber mathematische physik: mechanik. Leipzig: Teubner. 1883.
5.Agarwal RP, Perera K, O'Regan D. Multiple positive solutions of singular and nonsingular discrete problems via variational methods. Nonlinear Anal. 2004; 58: 69-73.
6.Avci M, Pankov A. Nontrivial solutions of discrete nonlinear equations with variable exponent. J. Math. Anal. Appl. 2015; 431: 22-33.
7.Avci M. Existence results for anisotropic discrete boundary value problems. EJDE. 2016; (148): 1-11.
8.Ayoujil A. On class of nonhomogenous discrete dirichlet problems, Acta Univ. Apulensis Math. Inform. 2014; 39: 1-15.
9.Ayoujil A. On class of discrete boundary value problem via variational methods. Afrika Mat. 2014: 1349-57.
10.Bisci GM, Repovs DD. Existence of solutions for p−Laplacian discrete equations. Appl. Math. Comput. 2014; 242: 454-61.
11.Bonanno G, Candito P. Nonlinear difference equations investigated via critical point methods. Nonlinear Anal. 2009; 70: 3180-86.
12.Candito P, Giovannelli N. Multiple solutions for a discrete boundary value problem involving the p−Laplacian. Comput. Math. Appl. 2008; 56: 959-64.
13.Kone B, Ouaro S. Weak solutions for anisotropic discrete boundary value problems, J. Difference Equ. Appl. 2010; 18(February): 1-11.
14.Ourraoui A. Existence of solution to a semilinear discrete problem involving p− Laplacian. Acta Univ. Apulensis Math. Inform. 2015; (43): 45-51.
15.Yucedag Z. Existence of solution for anisotrpic discrete boundary value problems of Kirchhoff type. Int. J. Differ. Equ. Appl. 2014; 13(1): 1-15.
16.Jiang L, Zhou Z. Three solutions to Dirichlet boundary value prolems for p-Laplacian difference equations. Adv. Difference Equ. 2008. Article ID 345916: 10. doi: 10.1155/2008/345916.
17.Anderson DR, Rachunkova I, Tisdell CC. Solvability of discrete Neumann boundary value problems. J. Math. Anal. Appl. 2007; 331: 736-41.
18.Bereanu C, Jebelean P, Serban C. Periodic and Neumann problems for discrete p(.)− laplacian. J. Math. Anal. Appl. 2013; 399(1): 75-87.
19.Bian LH, Sun HR, Zhang QG. Solutions for discrete p−Laplacian periodic boundary value problems via critical point theory. J. Difference Equ. Appl. 2012; 18(3): 345-55.
20.Cai X, Yu J. Existence theorems of periodic solutions for second-order nonlinear difference equations. Adv. Difference Equ. 2008. Article ID 247071.
21.Galewski M, Głkçb S. On the discrete boundary value problem for anisotropic equation. J. Math. Anal. Appl. 2012; 386: 956-65.
22.Galewski M, Wieteska R. Existence and multiplicity of positive solutions for discrete anisotropic equations. Turk. J. Math. 2014; 38(2): 297-310. doi: 10.3906/mat-1303-6.
23.Ružička M. Flow of shear dependent electrorheological fluids. CR Math. Acad. Sci. Paris. 1999; 329: 393-98.
24.Yang Y, Zhang J. Existence of solution for some discrete value problems with a parameter. Appl. Math. Comput. 2009; 211: 293-302.
25.Ji C. Remarks on the existence of three solutions for the p(x)−Laplacian equations, Nonlinear Analysis. 2011; 74: 2908-15.
26.Ricceri B. A further three critical points theorem. Nonlinear Anal. 2009; 71(9): 4151-157.
Acknowledgements
The authors would like to thank the anonymous referee for the valuable comments and constructive suggestions.