A Variational Approach to Perturbed Discrete Anisotropic Equations

Salari, Amjad; Caristi, Giuseppe; Barilla, David; Puglisi, Alfio

doi:https://doi.org/10.1155/2016/5676138

Abstract and Applied Analysis

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Research Article | Open Access

Volume 2016 | Article ID 5676138 | https://doi.org/10.1155/2016/5676138

A Variational Approach to Perturbed Discrete Anisotropic Equations

Amjad Salari,¹Giuseppe Caristi,²David Barilla,²and Alfio Puglisi²

Academic Editor: Patricia J. Y. Wong

Received02 Jul 2016

Accepted10 Oct 2016

Published20 Nov 2016

Abstract

We continue the study of discrete anisotropic equations and we will provide new multiplicity results of the solutions for a discrete anisotropic equation. We investigate the existence of infinitely many solutions for a perturbed discrete anisotropic boundary value problem. The approach is based on variational methods and critical point theory.

1. Introduction

The aim of this paper is to investigate the existence of infinitely many solutions for the following perturbed discrete anisotropic problem:where is a fixed positive integer number, is the discrete interval , , , and are some fixed functions, and are two continuous functions. Let

Many problems in applied mathematics lead to the study of discrete boundary value problems and difference equations. Indeed, common among many fields of research, such as computer science, mechanical engineering, control systems, artificial or biological neural networks, and economics, is the fact that the mathematical modelling of fundamental questions is usually tended towards considering discrete boundary value problems and nonlinear difference equations. Regarding these issues, a thoroughgoing overview has been given in, as an example, the monograph [1] and the reference therein. On the other hand, in recent years some researchers have studied the existence and multiplicity of solutions for equations involving the discrete -Laplacian operator by using various fixed point theorems, lower and upper solutions method, critical point theory and variational methods, Morse theory, and the mountain-pass theorem. For background and recent results, we refer the reader to [2–20] and the references therein. For example, Atici and Guseinov in [3] investigated the existence of positive periodic solutions for nonlinear difference equations with periodic coefficients by employing a fixed point theorem in cone. Atici and Cabada in [2], by using the upper and lower solution method, obtained the existence and uniqueness results for a discrete boundary value problem. Henderson and Thompson in [12] gave conditions on the nonlinear term involving pairs of discrete lower and discrete upper solutions which led to the existence of at least three solutions of discrete two-point boundary value problems, and in a special case of the nonlinear term they gave growth conditions on the function and applied their general result to show the existence of three positive solutions. Chu and Jiang in [9] based on fixed point theorem in a cone, due to Krasnoselskii, characterized the eigenvalues and showed the existence of positive solutions to a discrete boundary value problem. Jiang and Zhou in [13] employing a three-critical point theorem, due to Ricceri, established the existence of at least three solutions for perturbed nonlinear difference equations with discrete boundary conditions. Wang and Guan in [20], by using the Five Functionals Fixed Point Theorem, obtained the existence criteria for three positive solutions of a -Laplacian difference equation. Bonanno and Candito [5], employing critical point theorems in the setting of finite dimensional Banach spaces, investigated the multiplicity of solutions for nonlinear difference equations involving the -Laplacian. Cabada et al. in [6], 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 -Laplacian operator. In [4] Bian et al. by using critical point theory studied a class of discrete -Laplacian periodic boundary value problems; some results were obtained for the existence of two positive solutions, three solutions, and multiple pairs of solutions of the problem when the parameter lies in some suitable infinite or finite intervals. In [15], by using variational methods and critical point theory, the existence of infinitely many solutions for perturbed nonlinear difference equations with discrete Dirichlet boundary conditions was discussed.

There seems to be increasing interest in the existence of solutions to discrete anisotropic equations, because of their applications in many fields such as models in physics [21–24], biology [25, 26], and image processing (see, for example, Weickert’s monograph [27]). We also mention Fragalà et al. [28] and El Hamidi and Vétois [29] as essential references in treating the nonlinear anisotropic problems. Besides, Mihăilescu et al. (see [30, 31]) were the first authors who studied anisotropic elliptic problems with variable exponents. On the other hand, numerous researches have been undertaken on the existence of solutions for discrete anisotropic boundary value problems (BVPs) in recent years. As to the background and latest results, the readers can refer to [32–38] and the references therein. For example, Mihăilescu et al. in [36], by using critical point theory, obtained the existence of a continuous spectrum for a family of discrete boundary value problems. Galewski and Wieteska in [34] investigated the existence of solutions of the system of anisotropic discrete boundary value problems using critical point theory, while in [33], using variational methods, they derived the intervals of the numerical parameter for which the problem has at least 1, exactly 1, or at least 2 positive solutions. They also derived some useful discrete inequalities. Molica Bisci and Repovš in [37] considered advantage of a recent critical point theorem to establish the existence of infinitely many solutions for anisotropic difference equation: where is a positive parameter, is a continuous function for every (with ), and is the forward difference operator, assuming that the map satisfies as well as . Stegliński in [38], based on critical point theory, obtained the existence of infinitely many solutions for the parametric version of the problem , in the case where .

Motivated by the above works, in the present paper, by employing a smooth version of [39, Theorem 2.1], which is more precise version of Ricceri’s Variational Principle [40, Theorem 2.5] under some hypotheses on the behavior of the nonlinear terms at infinity, we prove the existence of definite intervals about and in which the problem admits a sequence of solutions which is unbounded in the space which will be introduced later (Theorem 6). Furthermore, some consequences of Theorem 6 are listed. A partial case of main result is formulated as Theorem 5. Replacing the conditions at infinity on the nonlinear terms, by a similar one at zero, we obtain a sequence of pairwise distinct solutions strongly converging at zero; see Theorem 14. Three examples of applications are pointed out (see Examples 8, 13, and 16).

2. Preliminaries

Our main tool to ensure the existence of infinitely many solutions for the problem is a smooth version of Theorem 2.1 of [39] which is a more precise version of Ricceri’s Variational Principle [40] that we now recall here.

Theorem 1. Let be a reflexive real Banach space and let be two Gâteaux differentiable functionals such that is sequentially weakly lower semicontinuous, strongly continuous, and coercive and is sequentially weakly upper semicontinuous. For every , let one put Then, one has the following: (a)For every and every , the restriction of the functional to admits a global minimum, which is a critical point (local minimum) of in .(b)If , then, for each , the following alternative holds: either() possesses a global minimum or()there is a sequence of critical points (local minima) of such that (c)If , then, for each , the following alternative holds: either()there is a global minimum of which is a local minimum of or()there is a sequence of pairwise distinct critical points (local minima) of which weakly converges to a global minimum of .

We refer the reader to the paper [41–47] in which Theorem 1 was successfully employed to ensure the existence of infinitely many solutions for boundary value problems.

Here and in the sequel we take the -dimensional Banach space endowed with the norm

Remark 2. We consider that whenever is a finite dimensional Banach space in Theorem 1, in order to show the regularity of the derivative of and , it is merely enough to indicate that and are two continuous functionals on .

Lemma 3 (see [48, Section ]). For every with we have For every with we have For every and for any we have If there exists positive constant such that for every For every and for any we have For every and for any such that we have

Put

Remark 4. We recall that a map is continuous if it is continuous as a map of the topological space into the topological space . In this paper, the topology on will be the discrete topology.

A special case of our main result is the following theorem.

Theorem 5. Let be a continuous function and put for all . Assume that Then, for every continuous function whose , for every , is a nonnegative function satisfying the conditionand for every , where , the problem has an unbounded sequence of solutions.

3. Main Results

We present our main result as follows.

Theorem 6. Assume that
.
Then, for each , where for every continuous function whose for every , is a nonnegative function satisfying the conditionand for every , where the problem has an unbounded sequence of solutions.

Proof. Fix and let be a function satisfying the condition (18). Since , one has . Fix and put and . If , clearly, , , and . If , since , we obtain , and so , namely, . Hence, since , one has . Now, set for all . Take and define on two functionals and as follows: Since is a finite dimensional Banach space, is a Gâteaux differentiable functional and sequentially weakly upper semicontinuous whose Gâteaux derivative at the point is the functional , given byfor every , and is a compact operator. Moreover, is a Gâteaux differentiable functional of which Gâteaux derivative at the point is the functional , given by for every . Furthermore, is sequentially weakly lower semicontinuous (see [5, Remarks 2.2 and 2.3]). Put . We observe that the solutions of the problem are exactly the solutions of the equation . So, our end is to apply Theorem 1 to and . Now, we wish to prove that , where is defined in Theorem 1. Let be a real sequence such that for all and as and Put for all . Let be such that for all . We claim thatIndeed, if and , one has Then, for every . Consequently, since , we deduce by easy induction that for every and this gives (24). Hence, taking into account the fact that , for every large enough, one has Moreover, it follows from Assumption thatwhich concludes thatThen, in view of (18) and (30), we have which followsTherefore,Sincetaking (18) into account, one hasMoreover, since is nonnegative, we haveTherefore, from (35) and (36) and from Assumption and (33), one hasFor fixed , inequality (33) assures that condition (b) of Theorem 1 can be used and either has a global minimum or there exists a sequence of solutions of the problem such that .
The other step is to verify that the functional has no global minimum. Since we can consider a real sequence with for all and a positive constant such that as andfor each large enough. Thus, we consider a sequence in defined by settingThus On the other hand, since is nonnegative, we observeSo, from (39), (42), and (43), we conclude that for every large enough. Hence, the functional is unbounded from below, and it follows that has no global minimum. Therefore, Theorem 1 assures that there is a sequence of critical points of such that , which from Lemma 3 follows that . Hence, we have the conclusion.

Remark 7. Under the conditions Theorem 6 assures that for every and for each the problem admits infinitely many solutions. Moreover, if , the result holds for every and .

Now, we give an application of Theorem 6 as follows.

Example 8. Let , let for all , let for all , and let be a sequence defined by and let be a sequence such that and for all . Consider the problemwhere for all with where denotes the characteristic function of the interval . According to the above data we have , , , , and for all and it is easy to verify that and for all . Then, one has , and it is easy to see thatHence, using Theorem 6, since problem (47) for every and has an unbounded sequence of solutions in the space

Remark 9. Assumption in Theorem 6 could be replaced by the following more general condition:
there exist two sequences and with for every and for all and such that Indeed, clearly, by choosing for all , from we obtain . Moreover, if we assume instead of and choose for all , applying the same argument in the proof of Theorem 6, we obtain where is the same as (40) but is replaced by . We have the same conclusion as in Theorem 6 with the interval ] replaced by the interval

Here, we point out a simple consequence of Theorem 6.

Corollary 10. Assume that
;
.
Then, for every arbitrary function whose for every is a nonnegative function satisfying condition (18) and for every , where the problem has an unbounded sequence of solutions.

Remark 11. Theorem 5 is an immediate consequence of Corollary 10.

We here give the following consequence of the main result.

Corollary 12. Let and let for all . Assume that
;
.
Then, for every function , denoting for all for , satisfying for each for every arbitrary function whose for every , is a nonnegative function satisfying condition (18) and for every , wherethe problem has an unbounded sequence of solutions.

Proof. Set for all . Assumption along with the condition ensures that Moreover, Assumption , together with the condition implies that Hence, the conclusion follows from Theorem 6.

Here we give the following example to illustrate Corollary 12.

Example 13. Let , for all , and for all . Let be defined by respectively, where are two nonnegative continuous functions. Consider the problemA direct calculation shows , , and Put Then So, Moreover, Hence, all assumptions of Corollary 12 with are satisfied. So, for every , problem (66) has an unbounded sequence of solutions in the space

Arguing as in the proof of Theorem 6 but using conclusion (c) of Theorem 1 instead of (b), one establishes the following result.

Theorem 14. Assume that
.
Then, for each , wherefor every arbitrary function whose for every is a nonnegative function satisfying the conditionand for every where the problem has a sequence of pairwise distinct solutions which strongly converges to in .

Proof. Fix and let be the function satisfying condition (74). Since , one has . Fix and set and . If , clearly, , , and If , since , one has and so namely, . Hence, recalling that , one has .
Now, put for all and . Since taking (74) into account, one hasMoreover, since is nonnegative, from Assumption , we haveTherefore, from (79) and (80), we obtainWe take , , , and as in the proof of Theorem 6. We prove that . For this, let be a sequence of positive numbers such that as and Put for all . Let us show that the functional has not a local minimum at zero. For this, let be a sequence of positive numbers and such that as andfor each large enough. Let be a sequence in defined by (40). So, owing to (42), (43), and (83), we obtain for every large enough. Since , that means that is not a local minimum of the functional . Hence, part (c) of Theorem 1 ensures that there exists a sequence in of critical points of such that as , and the proof is complete.

Remark 15. Applying Theorem 14, results similar to Remark 9 and Corollaries 10 and 12 can be obtained.

We end this paper by giving the following example as an application of Theorem 14.

Example 16. Let , for all , and for all and let be the function defined by Let for all . Direct calculations show , , , , for all , and Thus Hence, using Theorem 14, since the problem for every has a sequence of pairwise distinct solutions which strongly converges to in the space

Remark 17. If and are nonnegative then the nontrivial solutions of the problem are positive. Indeed, let be a nontrivial solution of the problem ; assume that the discrete interval is nonempty. Put for . Clearly, and one has Since and are nonnegative, by choosing , one hasthat is,which contradicts with the fact that is a nontrivial solution. Hence, is positive.

Competing Interests

The authors declare that there are no competing interests regarding the publication of this paper.

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