Nonlinear difference equations investigated via critical point methods

Abstract

The multiplicity of solutions for nonlinear difference equations involving the $p$-Laplacian is investigated. The approach is based on critical point theorems in the setting of finite dimensional Banach spaces.

Introduction

It is well known that in different fields of research, such as computer science, mechanical engineering, control systems, artificial or biological neural networks, economics and many others, the mathematical modelling of important questions leads naturally to the consideration of nonlinear difference equations. For this reason, in recent years, many authors have widely developed various methods and techniques, such as fixed points theorems or upper and lower solutions methods, to study discrete problems. On these subjects, a complete overview is given in, for instance, the monographs [1], [16] and the references given therein.

Here, we are interested in investigating nonlinear discrete boundary value problems by using variational methods. This approach has been recently adopted in, for instance, [2], [3], [5], [9], [12], [14], [17], [20]. It is worth noting that, in this type of variational framework, the functional associated with a given discrete problem is defined on a Banach space, which has finite dimension. On the basis of ideas, we present some novel critical point results (Theorem 2.1, Theorem 2.2, Theorem 2.3) for real functionals defined on a finite dimensional Banach space, in Section 2. However, we stress that Theorem 2.1, Theorem 2.2 also work in the framework of an infinite dimensional Banach space (Remark 2.1), whereas Theorem 2.3 only holds in the finite dimensional setting.

The main aim of this paper is to establish, on the basis of the results obtained in Section 2, multiple solutions for the following problem:

$$\left(P_{\lambda }^f\right)\{\begin{array}{c}-\Delta \left({\varphi }_p\left(\Delta u(k-1)\right)\right)=\lambda f(k,u\left(k\right))\text{,}k\in [1,T]\text{,}\hfill \\ u\left(0\right)=u(T+1)=0\text{,}\hfill \end{array}$$

where $T$ is a fixed positive integer, $[1,T]$ is the discrete interval $\{1,\dots ,T\}$, $\lambda$ is a positive real parameter, $\Delta u\left(k\right)≔u(k+1)-u\left(k\right)$ is the forward difference operator, ${\varphi }_p\left(s\right)={\left|s\right|}^{p-2}s$, $1<p<+\infty$ and $f:[1,T]\times \mathbb{R}\to \mathbb{R}$ is a continuous function.

To be precise, for a suitable behavior of $f$ in the set $[1,T]\times [0,c_2]$, where $c_2$ is a positive real constant, and without requiring asymptotic conditions on $f$, we establish the existence of at least three positive solutions to problem $\left(P_{\lambda }^f\right)$ for each $\lambda$ belonging to a well-defined interval of parameters (Theorem 3.1). Moreover, the existence of at least four solutions, two positive and two negative, for $\lambda$ large enough, is achieved on requiring a suitable growth for $f$ at zero and at infinity (Corollary 3.1). Further, the existence of at least three nontrivial solutions is ensured, for $\lambda$ small enough, when $f$ has a suitable behavior in $[1,T]\times \mathbb{R}$ with a growth at infinity which is opposite that of the previous corollary (Theorem 3.2). Finally, some examples of applications of previous results are given.

We also remark that the problem $\left(P_{\lambda }^f\right)$ has been previously studied in, for instance, [2], [9], [10], [15] and, for $p=2$, in [5], [13]. It is very easy to verify that the results obtained in these papers and ours are mutually independent.