Weak homoclinic solutions of anisotropic difference equation with variable exponents
© Guiro et al.; licensee Springer 2012
Received: 22 May 2012
Accepted: 9 August 2012
Published: 5 September 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.
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.
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
is a norm on the space .
is necessarily a finite set and for any since .
- (b)For , if , we have
For every fixed , is a continuous convex even function in λ, and it increases strictly when .
- (b)For , if , we have
We also have .
- (c)For every fixed and , we have
This proves that is convex.
Thus, for every fixed , increases strictly when .
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 .
then there exists such that , which is a contradiction.
Case 2: . Then, and . Let be a sequence such that .
Proof (1) .
Case 1. is proven using Proposition 2.2.
which is a contradiction. Therefore, .
. The proof is similar to that for the Case 2.
So as .
So as . □
Proposition 2.4 Let , then .
Therefore, by mimicking the proof of Proposition 2.2, we deduce that .
Case 2. . As in the first case, we get . □
Proof The proof is similar to the proof of Proposition 2.3. □
Theorem 2.1 (Discrete Hölder type inequality)
Proof Let and be such that .
Case 1. or , then the result is true.
Therefore, . □
3 Existence of weak homoclinic solutions
In this section, we investigate the existence of weak homoclinic solutions of (1.1).
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).
We first present some basic properties of J.
for all .
Therefore, J is well defined. Clearly, I, L and Λ are in .
Let us denote .
Lemma 3.1 The functional I is weakly lower semi-continuous.
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.
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) □
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.
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