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Weak homoclinic solutions of anisotropic difference equation with variable exponents
Advances in Difference Equations volume 2012, Article number: 154 (2012)
In this paper, we prove the existence of homoclinic solutions for a family of anisotropic difference equations. The proof of the main result is based on a minimization method and a discrete Hölder type inequality.
MSC:47A75, 35B38, 35P30, 34L05, 34L30.
In this paper, we study the following nonlinear discrete anisotropic problem:
where is the forward difference operator.
The problem (1.1) is a class of partial difference equations which usually describe the evolution of certain phenomena over the course of time. Elementary but relevant examples of partial difference equations are concerned with heat diffusion, heat control, temperature distribution, population growth, cellular neural networks, etc. (see [1–5]). Our interest for problems of the type (1.1) is motivated by major applications of differential and difference operators to various applied fields such as electrorheological (smart) fluids, space technology, robotics, image processing, etc. On the other hand, they are strongly motivated by their applicability to mathematical physics and biology.
The goal of the present paper is to establish the existence of homoclinic solutions for the problem (1.1). In the theory of differential equations, a trajectory , which is asymptotic to a constant as , is called a doubly asymptotic or homoclinic orbit. The notion of a homoclinic orbit was introduced by Poincaré  for continuous Hamiltonian systems. Since we are seeking for solutions u of the problem (1.1) satisfying , then according to the notion of a homoclinic orbit by Poincaré , we are interested in finding homoclinic solutions for the problem (1.1). We remember that boundary value problems involving difference operators with constant exponents have been intensively studied in the last decade (see [7–13] for details). The existence of homoclinic solutions where studied in particular by the authors in . The variable exponent cases were studied by some authors in [14–17] and the references therein. Other very recent applications of variable exponent equations are presented in [18–20]. The study of such kind of problems can be placed at the interface of certain mathematical fields such as nonlinear differential equations and numerical analysis.
The variational approach to the study of homoclinic solutions in the context of variable exponent was firstly done by Mihailescu, Radulescu and Tersian in . They studied the following problem:
where stands for the -Laplace difference operator, that is,
for each .
In this paper, we use the minimization technique to get the existence of homoclinic solutions of (1.1).
The paper is organized as follows. In Section 2, we define the functional spaces and prove some of their useful properties, and finally, in Section 3, we prove the existence of homoclinic solutions of (1.1).
2 Auxiliary results
We will use the following notations from now on:
For the data f and a, we assume the following:
There exists a positive constant such that
for all and , where (a space to be defined later) with .
with , for all .
We now introduce the spaces:
On , we introduce the Luxemburg norm
is a norm on the space .
Remark 2.1 If , then . Indeed, if , then . Let
is necessarily a finite set and for any since .
We also have that , then . As is a finite set, then , which implies that
Proposition 2.1 Under condition (2.5), satisfies:
For , if , we have
and if , we have
For every fixed , is a continuous convex even function in λ, and it increases strictly when .
Proof (a) Let , we have
For , if , we have
We also have .
If , we have
We also have .
For every fixed and , we have
This proves that is convex.
Let such that . We have
Thus, for every fixed , increases strictly when .
For the continuity of , let be a real sequence such that as . We denote . We have
Therefore, we have the continuity of . □
Proposition 2.2 Let , then if and only if .
Proof Let us denote .
Case 1: . Then, there exists a sequence such that .
Therefore, as for all and is continuous with respect to λ, then we get .
Suppose now that . As , then there exists such that . Then
Solving the inequality , we get
then there exists such that , which is a contradiction.
Case 2: . Then, and . Let be a sequence such that .
We suppose that , then
which implies that
which is a contradiction. Thus,
Proposition 2.3 If and , then the following properties hold:
Proof (1) .
Case 1. is proven using Proposition 2.2.
Case 2. . Suppose that , then
Conversely, taking and supposing that , then
which is a contradiction. Therefore, .
Case 3. . Suppose that , then
Conversely, we take and we suppose that . We have
which is a contradiction. Therefore .
Let , then
which is equivalent to
. The proof is similar to that for the Case 2.
Case 1. , then
So as .
Case 2. as , then
So as . □
Proposition 2.4 Let , then .
Proof Case 1. . Then
Therefore, by mimicking the proof of Proposition 2.2, we deduce that .
Case 2. . As in the first case, we get . □
Proposition 2.5 If and , then the following properties hold:
Proof The proof is similar to the proof of Proposition 2.3. □
Theorem 2.1 (Discrete Hölder type inequality)
Let and be such that , then
Proof Let and be such that .
Case 1. or , then the result is true.
Case 2. and . Let us denote and . Then, by Young inequality, we deduce that
Therefore, . □
3 Existence of weak homoclinic solutions
In this section, we investigate the existence of weak homoclinic solutions of (1.1).
Definition 3.1 A weak homoclinic solution of (1.1) is a function such that
for any .
Note that weak solutions are usual solutions of the problem (1.1). It can be seen by taking the test elements with 1 on (k th) place.
The main result of this work is the following.
Theorem 3.1 Assume that (2.1)-(2.5) hold. Then, there exists at least one weak homoclinic solution of (1.1).
The energy functional corresponding to the problem (1.1) is defined by such that
We first present some basic properties of J.
Proposition 3.1 The functional J is well defined on and is of class with the derivative given by
for all .
Proof We denote by
We have by using Young inequality and assumptions (2.1) and (2.3) that
Therefore, J is well defined. Clearly, I, L and Λ are in .
Let us choose . We have
Let us denote .
We get, by using Young inequality,
By the same method, we deduce that
Lemma 3.1 The functional I is weakly lower semi-continuous.
Proof From (2.1), I is convex with respect to the second variable. Thus, by Corollary III.8 in , it is enough to show that I is lower semi-continuous. For this, we fix and . Since I is convex, we deduce that, for any ,
for all with .
Hence, we conclude that I is weakly lower semi-continuous. □
Proposition 3.2 The functional J is bounded from below, coercive and weakly lower semi-continuous.
Proof By Lemma 3.1, J is weakly lower semi-continuous. We will only prove the coerciveness of the energy functional J and its boundedness from below.
To prove the coerciveness of J, we may assume that and we deduce from the above inequality that
As when , then for , there exists such that . For , we have
Thus J is bounded below. □
We can now give the proof of Theorem 3.1.
Proof of Theorem 3.1 By Proposition 3.2, J has a minimizer which is a weak homoclinic solution of (1.1) □
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This work was done within the framework of the visit of the authors at the Abdus Salam International Centre for Theoretical Physics (ICTP), Trieste, Italy. The authors thank the mathematics section of ICTP for their hospitality and for financial support and all facilities.
The authors declare that they have no competing interests.
AG conceived and carried out the study. BK conceived and carried out the study. SO conceived and carried out the study.
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Guiro, A., Kone, B. & Ouaro, S. Weak homoclinic solutions of anisotropic difference equation with variable exponents. Adv Differ Equ 2012, 154 (2012). https://doi.org/10.1186/1687-1847-2012-154
- difference equations
- homoclinic solutions
- discrete Hölder type inequality