Show Summary Details
More options …

# International Journal of Nonlinear Sciences and Numerical Simulation

Editor-in-Chief: Birnir, Björn

Editorial Board: Armbruster, Dieter / Chen, Xi / Bessaih, Hakima / Chou, Tom / Grauer, Rainer / Marzocchella, Antonio / Rangarajan, Govindan / Trivisa, Konstantina / Weikard, Rudi

8 Issues per year

IMPACT FACTOR 2017: 1.162

CiteScore 2017: 1.41

SCImago Journal Rank (SJR) 2017: 0.382
Source Normalized Impact per Paper (SNIP) 2017: 0.636

Mathematical Citation Quotient (MCQ) 2017: 0.12

Online
ISSN
2191-0294
See all formats and pricing

Access brought to you by:

provisional account

More options …
Volume 19, Issue 5

# Existence of Solutions to Boundary Value Problems for a Class of Nonlinear Difference Systems

Tao Zhou
/ Xia Liu
• Corresponding author
• Oriental Science and Technology College, Hunan Agricultural University, Changsha 410128, China
• Science College, Hunan Agricultural University, Changsha 410128, Changsha, China
• Email
• Other articles by this author:
/ Haiping Shi
• Modern Business and Management Department, Guangdong Construction Polytechnic, Guangzhou 510440, China
• School of Mathematics and Statistics, Central South University, Changsha 410083, China
• Email
• Other articles by this author:
Published Online: 2018-06-26 | DOI: https://doi.org/10.1515/ijnsns-2017-0138

## Abstract

This paper is devoted to investigate a question of the existence of solutions to boundary value problems for a class of nonlinear difference systems. The proof is based on the notable mountain pass lemma in combination with variational technique. By using the critical point theory, some new existence criteria are obtained.

MSC 2010: 39A10; 47J30; 58E05

## 1 Introduction

In the following and in the sequel, we denote by $\mathbb{N}$, $\mathbb{Z}$ and $\mathbb{R}$ the sets of all natural numbers, integers and real numbers, respectively. For all $a$, $b$ $\in \mathbb{Z}$, we define $\mathbb{Z}\left(a\right)=\left\{a,a+1,\cdots \right\},\text{\hspace{0.17em}}\mathbb{Z}\left(a,b\right)=\left\{a,a+1,\cdots ,b\right\}$ when $a\le b$. Also, the symbol * will denote the transpose of a vector.

Consider the nonlinear second-order difference system $\mathrm{\Delta }\left({p}_{n}\left(\mathrm{\Delta }{x}_{n-1}{\right)}^{\delta }\right)-{q}_{n}{x}_{n}^{\delta }+{f}_{n}\left({x}_{n}\right)=0,\text{\hspace{0.17em}}n\in \mathbb{Z}\left(1,k\right),$(1)

with boundary value conditions $\alpha {x}_{0}-\beta \mathrm{\Delta }{x}_{0}=0=\gamma {x}_{k+1}+\sigma \mathrm{\Delta }{x}_{k},$(2)

where $\mathrm{\Delta }$ is the forward difference operator, $\mathrm{\Delta }{x}_{n}=$ ${x}_{n+1}-{x}_{n}$, $\delta >0$ is the ratio of odd positive integers, ${p}_{n}$ and ${q}_{n}$ are real valued for each $n\in \mathbb{Z}$, $k$ is a given positive integer, $\alpha ,\beta ,\gamma$ and $\sigma$ are constants, $f\in$ $C\left({\mathbb{R}}^{2},$ $\mathbb{R}\right)$. The boundary value problem eq. (1) with eq. (2) contains the following special Dirichlet boundary value conditions, mixed boundary value conditions and Neumann boundary value conditions: $\begin{array}{rl}{x}_{0}& =0,\text{\hspace{0.17em}}{x}_{k+1}=0;\\ {x}_{0}& =0,\text{\hspace{0.17em}}\mathrm{\Delta }{x}_{k}=0;\\ \mathrm{\Delta }{x}_{0}& =0,\text{\hspace{0.17em}}{x}_{k+1}=0;\end{array}$

and $\mathrm{\Delta }{x}_{0}=0,\text{\hspace{0.17em}}\mathrm{\Delta }{x}_{k}=0.$

We may regard eq. (1) as being a discrete analog of the following second-order differential equation ${\left(p\left(t\right)\phi \left({x}^{\prime }\right)\right)}^{\prime }-q\left(t\right){x}^{\prime }\left(t\right)+f\left(t,x\left(t\right)\right)=0,\text{\hspace{0.17em}}t\in \left[1,k\right].$(3)

Equation (3) includes the following equation ${\left(p\left(t\right)\phi \left({x}^{\prime }\right)\right)}^{\prime }+f\left(t,x\left(t\right)\right)=0,\text{\hspace{0.17em}}t\in \mathbb{R},$(4)

which has arose in the study of fluid dynamics, combustion theory, gas diffusion through porous media, thermal self-ignition of a chemically active mixture of gases in a vessel, catalysis theory, chemically reacting systems and adiabatic reactor [1]. Equations similar in structure to eq. (4) arise in the study of periodic solutions and homoclinic orbits [10, 11, 12] of differential equations [21, 27, 28, 31].

Difference equations [1, 2, 3, 4, 5, 8, 9, 13, 14, 15, 16, 17, 18, 22, 23, 24, 25, 26, 29, 30, 32, 33, 34] have attracted the interest of many researchers in the past 20 years since they provide a natural description of several discrete models. Such discrete models are often investigated in various fields of science and technology such as finance insurance, biological populations, disease control, genetic study, physical field and computer application technology. Therefore, it is worthwhile to explore this topic.

When ${f}_{n}\left({x}_{n}\right)=0$, Thandapani and Ravi [26] obtained results on the asymptotic behavior of solutions of $\mathrm{\Delta }\left({p}_{n}\left(\mathrm{\Delta }{x}_{n-1}{\right)}^{\delta }\right)+{q}_{n}{x}_{n}^{\delta }=0$(5)

including sufficient conditions for all solutions to be bounded or unbounded. Some results on the existence and behavior of nonincreasing solutions of eq. (5) are also obtained.

Cai and Yu [5] in 2006 concerned with the existence of solutions of boundary value problems for nonlinear second-order difference equations of the type $\mathrm{\Delta }\left({p}_{n}\left(\mathrm{\Delta }{x}_{n-1}{\right)}^{\delta }\right)+{q}_{n}{x}_{n}^{\delta }+{f}_{n}\left({x}_{n}\right)=0,\text{\hspace{0.17em}}n\in \mathbb{Z}\left(1,k\right).$

They applied the linking theorem and the mountain pass lemma in the critical point theory and give some new results for the existence of solutions.

Recently, Shi, Liu and Zhang [23] considered the following forward and backward difference equation $\mathrm{\Delta }\left({p}_{n}\left(\mathrm{\Delta }{x}_{n-1}{\right)}^{\delta }\right)+{q}_{n}{x}_{n}^{\delta }+{f}_{n}\left({x}_{n+1},{x}_{n},{x}_{n-1}\right)=0,\text{\hspace{0.17em}}n\in \mathbb{Z},$(6)

and obtained the existence of a nontrivial homoclinic orbit for eq. (6).

Motivated by the recent papers [6, 7, 18], the intention of this paper is to investigate a question of the existence of solutions to boundary value problems for a class of difference systems eq. (1) with eq. (2). The proof is based on the notable mountain pass lemma in combination with variational technique. By using the critical point theory, some new existence criteria are obtained.

Let $F\left(t,z\right)={\int }_{0}^{z}f\left(t,s\right)ds\ge 0,$

and $\begin{array}{rl}{p}_{min}& =min\left\{{p}_{n}:\text{\hspace{0.17em}}n\in \mathbb{Z}\left(1,k+1\right)\right\},\\ {p}_{max}& =max\left\{{p}_{n}:\text{\hspace{0.17em}}n\in \mathbb{Z}\left(1,k+1\right)\right\},\\ {q}_{min}& =min\left\{{q}_{n}:\text{\hspace{0.17em}}=n\in \mathbb{Z}\left(1,k\right)\right\},\\ {q}_{max}& =max\left\{{q}_{n}:\text{\hspace{0.17em}}n\in \mathbb{Z}\left(1,k\right)\right\}.\end{array}$

For basic knowledge of variational methods, the reader is referred to [19, 20].

Let $E$ be a real Banach space, $J\in {C}^{1}\left(E,\mathbb{R}\right)$, i.e. $J$ is a continuously Fréchet-differentiable functional defined on $E$. $J$ is said to satisfy the Palais–Smale condition (P.S. condition for short) if any sequence ${\left\{{x}^{\left(l\right)}\right\}}_{l\in \mathbb{N}}\subset E$ for which ${\left\{J\left({x}^{\left(l\right)}\right)\right\}}_{l\in \mathbb{N}}$ is bounded and ${J}^{\mathrm{\prime }}\left({x}^{\left(l\right)}\right)\to 0\text{\hspace{0.17em}}\left(l\to \mathrm{\infty }\right)$ possesses a convergent subsequence in $E$.

Let ${B}_{\rho }$ denote the open ball in $E$ about 0 of radius $\rho$ and let $\mathrm{\partial }{B}_{\rho }$ denote its boundary.

The rest of this paper is organized as follows. First of all, Section 2 presents variational structure. Next, in Section 3, we shall recall some related fundamental results and present some lemmas. Then, Section 4 is dedicated to the proof of main results. Finally, in Section 5, we shall give two examples to illustrate the applicability of the main results.

## 2 Variational structure

Our main tool is the critical point theory. We shall establish the corresponding variational framework for the boundary value problem (1) with eq. (2). We start by some basic notations for the reader’s convenience.

On one hand, let ${\mathbb{R}}^{k}$ be the real Euclidean space with dimension $k$. ${\mathbb{R}}^{k}$ can be equipped with the inner product $〈x,y〉$ and norm $\parallel x\parallel$ as follows, $〈x,y〉:=\sum _{j=1}^{k}{x}_{j}{y}_{j},\text{\hspace{0.17em}}\mathrm{\forall }x,y\in {\mathbb{R}}^{k},\text{\hspace{0.17em}}\text{\hspace{0.17em}}$(7)

and $\parallel x\parallel :={\left(\sum _{j=1}^{k}{\left|{x}_{j}\right|}^{2}\right)}^{\frac{1}{2}},\text{\hspace{0.17em}}\mathrm{\forall }x\in {\mathbb{R}}^{k}.$(8)

On the other hand, we define the norm $\parallel \cdot {\parallel }_{s}$ on ${\mathbb{R}}^{k}$ as follows: $\parallel x{\parallel }_{s}={\left(\sum _{j=1}^{k}|{x}_{j}{|}^{s}\right)}^{\frac{1}{s}},$

for all $x\in {\mathbb{R}}^{k}$ and $s>1$.

Since $\parallel x{\parallel }_{s}$ and $\parallel x{\parallel }_{2}$ are equivalent, there exist constants ${k}_{1},\text{\hspace{0.17em}}{k}_{2}$ such that ${k}_{2}\ge {k}_{1}>0$, and ${k}_{1}\parallel x{\parallel }_{2}\le \parallel x{\parallel }_{s}\le {k}_{2}\parallel x{\parallel }_{2},\text{\hspace{0.17em}}\mathrm{\forall }x\in {\mathbb{R}}^{k}.$(9)

For all $x\in {\mathbb{R}}^{k}$, define the functional $J$ on ${\mathbb{R}}^{k}$ as follows: $\begin{array}{rl}J\left(x\right)& =\frac{1}{\delta +1}\sum _{n=1}^{k+1}{p}_{n}\left(\mathrm{\Delta }{x}_{n-1}{\right)}^{\delta +1}+\frac{1}{\delta +1}\sum _{n=1}^{k}{q}_{n}{x}_{n}^{\delta +1}-\sum _{n=1}^{k}{F}_{n}\left({x}_{n}\right)\\ +& {\left(\frac{\gamma }{\sigma }\right)}^{\delta }\frac{{p}_{k+1}{x}_{k+1}^{\delta +1}}{\delta +1}+{\left(\frac{\alpha }{\beta }\right)}^{\delta }\frac{{p}_{1}{x}_{0}^{\delta +1}}{\delta +1}\end{array}$

where $x=\left\{{x}_{n}{\right\}}_{n=1}^{k}=\left({x}_{1},{x}_{2},\dots ,{x}_{k}{\right)}^{\ast },\text{\hspace{0.17em}}\alpha {x}_{0}-\beta \mathrm{\Delta }{x}_{0}=0=\gamma {x}_{k+1}+\sigma \mathrm{\Delta }{x}_{k}.$

It is easy to see that $J\in {C}^{1}\left({\mathbb{R}}^{k},\mathbb{R}\right)$ and for any $x=\left\{{x}_{n}{\right\}}_{n=1}^{k}=\left({x}_{1},{x}_{2},\dots ,{x}_{k}{\right)}^{\ast }$, by using $\alpha {x}_{0}-\beta \mathrm{\Delta }{x}_{0}=0=\gamma {x}_{k+1}+\sigma \mathrm{\Delta }{x}_{k}$ and the summation by parts $\sum _{n=1}^{k}{y}_{n}\mathrm{\Delta }{x}_{n-1}={y}_{k}{x}_{k}-{y}_{1}{x}_{0}-\sum _{n=1}^{k}\mathrm{\Delta }{y}_{n}{x}_{n},$

we can compute the partial derivative as $\frac{\mathrm{\partial }J}{\mathrm{\partial }{x}_{n}}=-\mathrm{\Delta }\left({p}_{n}\left(\mathrm{\Delta }{x}_{n-1}{\right)}^{\delta }\right)+{q}_{n}{x}_{n}^{\delta }-{f}_{n}\left({x}_{n}\right),\text{\hspace{0.17em}}\mathrm{\forall }n\in \mathbb{Z}\left(1,k\right).$

Thus, $x$ is a critical point of $J$ on ${\mathbb{R}}^{k}$ if and only if $\mathrm{\Delta }\left({p}_{n}\left(\mathrm{\Delta }{x}_{n-1}{\right)}^{\delta }\right)-{q}_{n}{x}_{n}^{\delta }+{f}_{n}\left({x}_{n}\right)=0,\text{\hspace{0.17em}}\mathrm{\forall }n\in \mathbb{Z}\left(1,k\right).$

We reduce the existence of the boundary value problem (1) with eq. (2) to the existence of critical points of $J$ on ${\mathbb{R}}^{k}$. That is, the functional $J$ is just the variational framework of the boundary value problem (1) with eq. (2).

Let $P=\left(\begin{array}{cccccc}1& -1& 0& \cdots & 0& 0\\ -1& 2& -1& \cdots & 0& 0\\ 0& -1& 2& \cdots & 0& 0\\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ 0& 0& 0& \cdots & 2& -1\\ 0& 0& 0& \cdots & -1& 2\end{array}\right)$

be a $k×k$ matrix.

For convenience, we identify $x\in$ ${\mathbb{R}}^{k}$ with $x=$ $\left({x}_{1},$ ${x}_{2},\cdots ,{x}_{k}{\right)}^{\ast }.$

## 3 Main results

In this section, we shall state and prove our main results by using the variational methods.

#### Theorem 3.1:

Suppose that the following assumptions are satisfied:

$\left({B}_{1}\right)$ $\alpha \ge 0,\text{\hspace{0.17em}}\beta >0,\text{\hspace{0.17em}}\gamma \ge 0$ and $\sigma >0$;

$\left(p\right)$ for any $n\in \mathbb{Z}\left(1,k+1\right)$, ${p}_{n}>0$;

$\left({q}_{1}\right)$ for any $n\in \mathbb{Z}\left(1,k\right)$, ${q}_{n}>0$;

$\left({F}_{1}\right)$ there exist constants ${c}_{1}>0,\text{\hspace{0.17em}}{c}_{2}>0$ and $\omega >\delta +1$ such that $F\left(t,y\right)\ge {c}_{1}|y{|}^{\omega }-{c}_{2},\text{\hspace{0.17em}}\mathrm{\forall }\left(t,y\right)\in {\mathbb{R}}^{2}.$

Then, the boundary value problem (1) with eq. (2) possesses at least one solution.

#### Proof

For any $x=\left({x}_{1},{x}_{2},\cdots ,{x}_{k}{\right)}^{\ast }\in {\mathbb{R}}^{k}$, combining with $\left({F}_{1}\right)$, it is easy to see that $\begin{array}{rl}J\left(x\right)=& \frac{1}{\delta +1}\sum _{n=1}^{k+1}{p}_{n}\left(\mathrm{\Delta }{x}_{n-1}{\right)}^{\delta +1}+\frac{1}{\delta +1}\sum _{n=1}^{k}{q}_{n}{x}_{n}^{\delta +1}-\sum _{n=1}^{k}{F}_{n}\left({x}_{n}\right)\\ & +{\left(\frac{\gamma }{\sigma }\right)}^{\delta }\frac{{p}_{k+1}{x}_{k+1}^{\delta +1}}{\delta +1}+{\left(\frac{\alpha }{\beta }\right)}^{\delta }\frac{{p}_{1}{x}_{0}^{\delta +1}}{\delta +1}\end{array}$ $\le \frac{{p}_{max}{2}^{\delta +1}}{\delta +1}\sum _{n=1}^{k+1}\left({x}_{n}^{\delta +1}+{x}_{n-1}^{\delta +1}\right)+\frac{{q}_{max}{∥x∥}_{\delta +1}^{\delta +1}}{\delta +1}-{c}_{1}\sum _{n=1}^{k}{\left|{x}_{n}\right|}^{\omega }+{c}_{2}k$ $+{\left(\frac{\gamma }{\sigma }\right)}^{\delta }\frac{{p}_{max}{∥x∥}_{\delta +1}^{\delta +1}}{\delta +1}+{\left(\frac{\alpha }{\beta }\right)}^{\delta }\frac{{p}_{max}{∥x∥}_{\delta +1}^{\delta +1}}{\delta +1}$ $\begin{array}{rl}\le & \frac{3{p}_{max}{2}^{\delta +1}}{\delta +1}\parallel x{\parallel }_{\delta +1}^{\delta +1}+\frac{{q}_{max}{∥x∥}_{\delta +1}^{\delta +1}}{\delta +1}+{\left(\frac{\gamma }{\sigma }\right)}^{\delta }\frac{{p}_{max}{∥x∥}_{\delta +1}^{\delta +1}}{\delta +1}\\ & +{\left(\frac{\alpha }{\beta }\right)}^{\delta }\frac{{p}_{max}{∥x∥}_{\delta +1}^{\delta +1}}{\delta +1}-{c}_{1}\sum _{n=1}^{k}{\left|{x}_{n}\right|}^{\omega }+{c}_{2}k\end{array}$ $\begin{array}{rl}\le & \frac{{k}_{2}^{\delta +1}}{\delta +1}\left[3{p}_{max}{2}^{\delta +1}+{q}_{max}+{\left(\frac{\gamma }{\sigma }\right)}^{\delta }{p}_{max}+{\left(\frac{\alpha }{\beta }\right)}^{\delta }{p}_{max}\right]\\ & {∥x∥}^{\delta +1}-{c}_{1}{k}_{1}^{\omega }\parallel x{\parallel }^{\omega }+{c}_{2}k\to -\mathrm{\infty }\end{array}$

as $\parallel x\parallel \to +\mathrm{\infty }$. By the continuity of $J\left(x\right)$, we have from the above inequality that there exist upper bounds of values of functional $J$. Classical calculus shows that $J$ attains its maximal value at some point which is just the critical point of $J$ and the result follows. This completes the proof of Theorem 3.1.   □

#### Remark 3.1:

The results of Theorem 1.1 ensure that the boundary value problem (1) with eq. (2) possesses at least one solution. However, in some cases, we are interested in the existence of nontrivial solutions.

#### Theorem 3.2:

Assume that $\left(p\right)$, $\left({q}_{1}\right)$, $\left({F}_{1}\right)$ and the following assumptions: $\left({B}_{2}\right)$ $\alpha =0,\text{\hspace{0.17em}}\beta >0,\text{\hspace{0.17em}}\gamma =0$ and $\sigma >0$;

$\left({F}_{2}\right)$ for all $\left(t,y\right)\in {\mathbb{R}}^{2}$, $\underset{y\to 0}{lim}\frac{F\left(t,y\right)}{{y}^{\delta +1}}=0.$

Then, the boundary value problem (1) with eq. (2) possesses at least two nontrivial solutions.

#### Lemma 3.1:

(Mountain pass lemma [19, 20]). Let $E$ be a real Banach space and $J\in {C}^{1}\left(E,\mathbb{R}\right)$ satisfy the P.S. condition. If $J\left(0\right)=0$ and

$\left({J}_{1}\right)$ there exist constants $\rho ,\text{\hspace{0.17em}}a>0$ such that

$J{|}_{\mathrm{\partial }{B}_{\rho }}\ge a$, and

$\left({J}_{2}\right)$ there exists $e\in E\setminus {B}_{\rho }$ such that $J\left(e\right)\le 0$.

Then, $J$ possesses a critical value $c\ge a$ given by $c=\underset{g\in \mathrm{Υ}}{inf}\underset{\theta \in \left[0,1\right]}{max}J\left(g\left(\theta \right)\right),$(10)

where $\mathrm{Υ}=\left\{g\in C\left(\left[0,1\right],E\right)|g\left(0\right)=0,\text{\hspace{0.17em}}g\left(1\right)=e\right\}.$(11)

#### Lemma 3.2:

Assume that $\left(p\right)$, $\left({q}_{1}\right)$, $\left({B}_{2}\right)$, $\left({F}_{1}\right)$ and $\left({F}_{2}\right)$ are satisfied. Then $J$ satisfies the P.S. condition.

#### Proof

By $\left({B}_{2}\right)$, we have that $J\left(x\right)=\frac{1}{\delta +1}\sum _{n=1}^{k+1}{p}_{n}\left(\mathrm{\Delta }{x}_{n-1}{\right)}^{\delta +1}+\frac{1}{\delta +1}\sum _{n=1}^{k}{q}_{n}{x}_{n}^{\delta +1}-\sum _{n=1}^{k}{F}_{n}\left({x}_{n}\right).$

It is obvious that $P$ is positive definite. Denote ${\lambda }_{max}=max\left\{{\lambda }_{j}|j=1,2,\cdots ,k\right\},$(12) ${\lambda }_{min}=min\left\{{\lambda }_{j}|j=1,2,\cdots ,k\right\}.$(13)

Let ${\left\{{x}^{\left(l\right)}\right\}}_{l\in \mathbb{N}}\subset {\mathbb{R}}^{k}$ be such that ${\left\{J\left({x}^{\left(l\right)}\right)\right\}}_{l\in \mathbb{N}}$ is bounded and ${J}^{\prime }\left({x}^{\left(l\right)}\right)\to 0$ as $l\to \mathrm{\infty }.$ Then, there exists a positive constant $A$ such that $\left|J\left({x}^{\left(l\right)}\right)\right|\le A$. It comes from $\left({F}_{2}\right)$, eq. (12) that $\begin{array}{rl}-A\le J\left({x}^{\left(l\right)}\right)\phantom{\rule{negativethinmathspace}{0ex}}=& \frac{1}{\delta +1}\sum _{n=1}^{k+1}{p}_{n}{\left(\mathrm{\Delta }{x}_{n-1}^{\left(l\right)}\right)}^{\delta +1}+\phantom{\rule{negativethinmathspace}{0ex}}\frac{1}{\delta +1}\sum _{n=1}^{k}{q}_{n}{\left({x}_{n}^{\left(l\right)}\right)}^{\delta +1}\\ & -\sum _{n=1}^{k}{F}_{n}\left({x}_{n}^{\left(l\right)}\right)\end{array}$ $\begin{array}{rl}\le & \frac{{p}_{max}{2}^{\delta +1}}{\delta +1}\sum _{n=1}^{k+1}\left[{\left({x}_{n}^{\left(l\right)}\right)}^{\delta +1}+{\left({x}_{n-1}^{\left(l\right)}\right)}^{\delta +1}\right]+\frac{{q}_{max}}{\delta +1}{∥{x}^{\left(l\right)}∥}_{\delta +1}^{\delta +1}\\ & -{c}_{1}\sum _{n=1}^{k}{\left|{x}_{n}^{\left(l\right)}\right|}^{\omega }+{c}_{2}k\end{array}$ $\le \phantom{\rule{negativethinmathspace}{0ex}}\frac{3{p}_{max}{2}^{\delta +1}{k}_{2}^{\delta +1}}{\delta +1}{∥{x}^{\left(l\right)}∥}^{\delta +1}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}+\phantom{\rule{negativethinmathspace}{0ex}}\frac{{q}_{max}{k}_{2}^{\delta +1}}{\delta +1}{∥{x}^{\left(l\right)}∥}^{\delta +1}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}-{c}_{1}{k}_{1}^{\omega }{∥{x}^{\left(l\right)}∥}^{\omega }\phantom{\rule{negativethinmathspace}{0ex}}+{c}_{2}k.$

That is, ${c}_{1}{k}_{1}^{\omega }{∥{x}^{\left(l\right)}∥}^{\omega }-\frac{\left(3{p}_{max}{2}^{\delta +1}+{q}_{max}\right){k}_{2}^{\delta +1}}{\delta +1}{∥{x}^{\left(l\right)}∥}^{\delta +1}\le A+{c}_{2}k.$

Since $\omega >\delta +1$, there exists a positive constant $B$ such that $∥{x}^{\left(l\right)}∥\le B,\text{\hspace{0.17em}}\mathrm{\forall }n\in \mathbb{N}.$

Therefore, ${\left\{{x}^{\left(l\right)}\right\}}_{n\in \mathbb{N}}$ is bounded on ${\mathbb{R}}^{k}$. As a consequence, ${\left\{{x}^{\left(l\right)}\right\}}_{n\in \mathbb{N}}$ possesses a convergent subsequence in ${\mathbb{R}}^{k}$. Therefore, the P.S. condition is satisfied.   □

#### Proof of Theorem 3.2

It follows from $\left({F}_{2}\right)$ that for any $\epsilon =\frac{{p}_{min}{\lambda }_{min}^{\frac{\delta +1}{2}}+{q}_{min}}{2\left(\delta +1\right)}{k}_{1}^{\delta +1}$, there exists $\delta >0$, such that $\left|F\left(t,y\right)\right|\le \frac{{p}_{min}{\lambda }_{min}^{\frac{\delta +1}{2}}+{q}_{min}}{2\left(\delta +1\right)}{k}_{1}^{\delta +1}{y}^{\delta +1},$

for $|y|\le \rho$.

For any $x=\left({x}_{1},{x}_{2},\cdots ,{x}_{k}{\right)}^{\ast }\in {\mathbb{R}}^{k}$ and $\parallel x\parallel \le \rho$, we have $|{x}_{n}|\le \rho ,\text{\hspace{0.17em}}n\in \mathbb{Z}\left(1,k\right)$. Then, $\begin{array}{rl}J\left(x\right)& =\frac{1}{\delta +1}\sum _{n=1}^{k+1}{p}_{n}\left(\mathrm{\Delta }{x}_{n-1}{\right)}^{\delta +1}+\frac{1}{\delta +1}\sum _{n=1}^{k}{q}_{n}{x}_{n}^{\delta +1}-\sum _{n=1}^{k}{F}_{n}\left({x}_{n}\right)\\ & \ge \frac{{p}_{min}}{\delta +1}\sum _{n=1}^{k+1}\left(\mathrm{\Delta }{x}_{n-1}{\right)}^{\delta +1}+\frac{{q}_{min}}{\delta +1}\parallel x{\parallel }_{\delta +1}^{\delta +1}-\frac{{p}_{min}{\lambda }_{min}^{\frac{\delta +1}{2}}+{q}_{min}}{2\left(\delta +1\right)}\\ & \sum _{n=1}^{k}{\left|{x}_{n}\right|}^{\delta +1}\\ \ge & \frac{{p}_{min}{k}_{1}^{\delta +1}}{\delta +1}{\left({x}^{\ast }Px\right)}^{\frac{\delta +1}{2}}+\frac{{q}_{min}{k}_{1}^{\delta +1}}{\delta +1}\parallel x{\parallel }^{\delta +1}-\frac{{p}_{min}{\lambda }_{min}^{\frac{\delta +1}{2}}+{q}_{min}}{2\left(\delta +1\right)}\\ & {k}_{1}^{\delta +1}\parallel x{\parallel }^{\delta +1}\\ & \ge \frac{{p}_{min}{k}_{1}^{\delta +1}{\lambda }_{min}^{\frac{\delta +1}{2}}}{\delta +1}\parallel x{\parallel }^{\delta +1}+\frac{{q}_{min}{k}_{1}^{\delta +1}}{\delta +1}\parallel x{\parallel }^{\delta +1}\\ & -\frac{{p}_{min}{\lambda }_{min}^{\frac{\delta +1}{2}}+{q}_{min}}{2\left(\delta +1\right)}{k}_{1}^{\delta +1}\parallel x{\parallel }^{\delta +1}\\ & =\frac{{p}_{min}{\lambda }_{min}^{\frac{\delta +1}{2}}+{q}_{min}}{2\left(\delta +1\right)}{k}_{1}^{\delta +1}\parallel x{\parallel }^{\delta +1}.\end{array}$

Take $a\triangleq \frac{{p}_{min}{\lambda }_{min}^{\frac{\delta +1}{2}}+{q}_{min}}{2\left(\delta +1\right)}{k}_{1}^{\delta +1}{\rho }^{\delta +1}>0.$ Therefore, $J\left(x\right)\ge a>0,\text{\hspace{0.17em}}\mathrm{\forall }x\in \mathrm{\partial }{B}_{\rho }.$ At the same time, we have also proved that there exist constants $a>0$ and $\rho >0$ such that $J{|}_{\mathrm{\partial }{B}_{\rho }}\ge a$. That is to say, $J$ satisfies the condition $\left({J}_{1}\right)$ of the mountain pass lemma.

The rest of the proof is similar to that of [16, Theorem 1.2], but for the sake of completeness, we give the details.

For our setting, clearly $J\left(0\right)=0$. In order to exploit the mountain pass lemma in critical point theory, we need to verify other conditions of the mountain pass lemma. By Lemma 3.2, $J$ satisfies the P.S. condition. So it suffices to verify the condition $\left({J}_{2}\right)$.

From the proof of the P.S. condition, we know $J\left(x\right)\le \frac{\left(3{p}_{max}{2}^{\delta +1}+{q}_{max}\right){k}_{2}^{\delta +1}}{\delta +1}\parallel x{\parallel }^{\delta +1}-{c}_{1}{k}_{1}^{\omega }\parallel x{\parallel }^{\omega }+{c}_{2}k.$ Since $\omega >\delta +1$, we can choose $\stackrel{ˉ}{x}$ large enough to ensure that $J\left(\stackrel{ˉ}{x}\right)<0$. By the mountain pass lemma, $J$ possesses a critical value $c\ge a>0$, where $c=\underset{g\in \mathrm{Υ}}{inf}\underset{s\in \left[0,1\right]}{sup}J\left(g\left(\theta \right)\right)$ and $\mathrm{Υ}=\left\{g\in C\left(\left[0,1\right],{\mathbb{R}}^{k}\right)\mid g\left(0\right)=0,\text{\hspace{0.17em}}g\left(1\right)=\stackrel{ˉ}{x}\right\}.$

Let $\stackrel{˜}{x}\in {\mathbb{R}}^{k}$ be a critical point associated to the critical value $c$ of $J$, i.e., $J\left(\stackrel{˜}{x}\right)=c$. Similar to the proof of the P.S. condition, we know that there exists $\stackrel{ˆ}{x}\in {\mathbb{R}}^{k}$ such that $J\left(\stackrel{ˆ}{x}\right)={c}_{max}=\underset{\theta \in \left[0,1\right]}{max}J\left(g\left(\theta \right)\right).$

Clearly, $\stackrel{ˆ}{x}\ne 0$. If $\stackrel{˜}{x}\ne \stackrel{ˆ}{x}$, then the conclusion of Theorem 1.2 holds. Otherwise, $\stackrel{˜}{x}=\stackrel{ˆ}{x}$. Then, $c=J\left(\stackrel{˜}{x}\right)={c}_{max}=\underset{\theta \in \left[0,1\right]}{max}J\left(g\left(\theta \right)\right)$. That is, $\underset{x\in {\mathbb{R}}^{k}}{sup}J\left(x\right)=\underset{g\in \mathrm{Υ}}{inf}\underset{\theta \in \left[0,1\right]}{sup}J\left(g\left(\theta \right)\right).$ Therefore, ${c}_{max}=\underset{\theta \in \left[0,1\right]}{max}J\left(g\left(\theta \right)\right),\text{\hspace{0.17em}}\mathrm{\forall }g\in \mathrm{Υ}.$

By the continuity of $J\left(g\left(\theta \right)\right)$ with respect to $\theta$, $J\left(0\right)=0$ and $J\left(\stackrel{ˉ}{X}\right)<0$ imply that there exists ${\theta }_{0}\in \left(0,1\right)$ such that $J\left(g\left({\theta }_{0}\right)\right)={c}_{max}.$ Choose ${g}_{1},\text{\hspace{0.17em}}{g}_{2}\in \mathrm{Υ}$ such that $\left\{{g}_{1}\left(\theta \right)\mid \theta \in \left(0,1\right)\right\}\cap \left\{{g}_{1}\left(\theta \right)\mid \theta \in \left(0,1\right)\right\}$ is empty, then there exists ${\theta }_{1},\text{\hspace{0.17em}}{\theta }_{2}\in \left(0,1\right)$ such that $J\left({g}_{1}\left({\theta }_{1}\right)\right)=J\left({g}_{2}\left({\theta }_{2}\right)\right)={c}_{max}.$ Thus, we get two different critical points of $J$ on ${\mathbb{R}}^{k}$ denoted by ${x}^{1}={g}_{1}\left({\theta }_{1}\right),\text{\hspace{0.17em}}{x}^{2}={g}_{2}\left({\theta }_{2}\right).$ The above argument implies that the boundary value problem (1) with eq. (2) possesses at least two nontrivial solutions.   □

#### Theorem 3.3:

Assume that $\left(p\right)$, $\left({q}_{1}\right)$, $\left({B}_{2}\right)$ and the following assumptions are satisfied:

$\left({q}_{2}\right)$ for any $n\in \mathbb{Z}\left(1,k\right)$, ${q}_{n}\le 0$;

$\left({F}_{3}\right)$ $yf\left(t,y\right)>0,$ for $y\ne 0,\text{\hspace{0.17em}}\mathrm{\forall }t\in \mathbb{R}$.

Then, the boundary value problem (1) with eq. (2) has no nontrivial solution.

#### Proof

For the sake of contradiction, suppose that the boundary value problem (1) with eq. (2) has a nontrivial solution. Then, $J$ has a nonzero critical point ${x}^{\star }$. On one hand, since $\frac{\mathrm{\partial }J}{\mathrm{\partial }{x}_{n}^{\star }}=-\mathrm{\Delta }\left[{p}_{n}{\left(\mathrm{\Delta }{x}_{n-1}^{\star }\right)}^{\delta }\right]+{q}_{n}{\left({x}_{n}^{\star }\right)}^{\delta }-{f}_{n}\left({x}_{n}^{\star }\right),$

we have $\sum _{n=1}^{k}{f}_{n}\left({x}_{n}^{\star }\right){x}_{n}^{\star }=-\sum _{n=1}^{k}\mathrm{\Delta }\left[{p}_{n}{\left(\mathrm{\Delta }{x}_{n-1}^{\star }\right)}^{\delta }\right]{x}_{n}^{\star }+\sum _{n=1}^{k}{q}_{n}{\left({x}_{n}^{\star }\right)}^{\delta +1}$ $=-\sum _{n=1}^{k}{p}_{n}{\left(\mathrm{\Delta }{x}_{n-1}^{\star }\right)}^{\delta +1}+\sum _{n=1}^{k}{q}_{n}{\left({x}_{n}^{\star }\right)}^{\delta +1}\le 0.$(14)

On the other hand, it follows from $\left({F}_{3}\right)$ that $\sum _{n=1}^{k}{f}_{n}\left({x}_{n}^{\star }\right){x}_{n}^{\star }>0.$(15)

This contradicts eq. (14) and hence Theorem 3.3 is proved.   □

## 4 Examples

As an application of Theorems 3.2 and 3.3, we give two examples to illustrate our main results.

#### Example 4.1:

For $n\in \mathbb{Z}\left(1,k\right)$, consider the boundary value problem: $\mathrm{\Delta }\left(\left(\mathrm{\Delta }{x}_{n-1}{\right)}^{\delta }\right)-{n}^{2}{x}_{n}^{\delta }+\left(\alpha +1\right){x}_{n}^{\alpha }=0$(16)

with boundary value conditions (2) and $\alpha \ge 0,\text{\hspace{0.17em}}\beta >0,\text{\hspace{0.17em}}\gamma \ge 0$, $\sigma >0$, where $\delta >0$ is the ratio of odd positive integers, $\alpha >\delta +1$. We have ${p}_{n}\equiv 1,\text{\hspace{0.17em}}{q}_{n}={n}^{2}$

and ${F}_{n}\left({x}_{n}\right)={x}_{n}^{\alpha +1}.$

It is easy to verify that all the conditions of Theorem 3.2 are satisfied and then the boundary value problem (16) with eq. (2) possesses at least two nontrivial solutions.

#### Example 4.2:

For $n\in \mathbb{Z}\left(1,k\right)$, consider the boundary value problem: $\mathrm{\Delta }\left({e}^{n}\left(\mathrm{\Delta }{x}_{n-1}{\right)}^{\delta }\right)+{n}^{6}{x}_{n}^{\delta }+8{x}_{n}^{7}=0$(17)

with boundary value conditions (2) and $\alpha =0,\text{\hspace{0.17em}}\beta >0,\text{\hspace{0.17em}}\gamma =0$, $\sigma >0$, where $\delta >0$ is the ratio of odd positive integers.

We have ${p}_{n}={e}^{n},\text{\hspace{0.17em}}{q}_{n}=-{n}^{6}$

and ${f}_{n}\left({x}_{n}\right)=8{x}_{n}^{7}.$

It is easy to verify that all the conditions of Theorem 3.3 are satisfied and then the boundary value problem (17) with eq. (2) has no nontrivial solution.

## 5 Conclusions

Difference equations occur widely in numerous settings and forms both in mathematics itself and in its applications. The boundary value problem discussed in this paper has important analog in the continuous case of the second-order differential equation. Such problem is of special significance for the study of a result that describes dynamically changing phenomena, evolution and variation. The problem discussed in this paper can be extended to boundary value problem for 2$n$-order ${\varphi }_{c}$-Laplacian [30] difference equations.

## Acknowledgements

The author Haiping Shi wishes to thank Professor Xianhua Tang for his invitation.

## References

• [1]

R. P. Agarwal, D. O’Regan, P. J. Y. Wong, Positive solutions of differential, difference and integral equations, Kluwer Academic, Dordrecht, 1999. Google Scholar

• [2]

C. D. Ahlbrandt, A. C. Peterson. Discrete hamiltonian systems: difference equations, continued fraction and Riccati equations, Kluwer Academic Publishers, Dordrecht, 1996. Google Scholar

• [3]

Z. AlSharawi, J. M. Cushing, S. Elaydi, Theory and applications of difference equations and discrete dynamical systems, Springer, New York, 2014. Google Scholar

• [4]

G. M. Bisci, D. Repovš, Existence of solutions for p-Laplacian discrete equations, Appl. Math. Comput. 242 (2014), 454–461. Google Scholar

• [5]

X. C. Cai, J. S. Yu, Existence theorems for second-order discrete boundary value problems, J. Math. Anal. Appl. 320 (2) (2006), 649–661.

• [6]

P. Chen, X. F. He, X. H. Tang, Infinitely many solutions for a class of fractional Hamiltonian systems via critical point theory, Math. Methods Appl. Sci. 39 (5) (2016), 1005–1019.

• [7]

P. Chen, X. H. Tang, Existence of solutions for a class of second-order p-Laplacian systems with impulsive effects, Appl. Math. 59 (5) (2014), 543–570.

• [8]

P. Chen, Z. M. Wang, Infinitely many homoclinic solutions for a class of nonlinear difference equations, Electron. J. Qual. Theory Differ. Equ. 2012 (47) (2012), 1–18. Google Scholar

• [9]

X. Q. Deng, Nonexistence and existence results for a class of fourth-order difference mixed boundary value problems, J. Appl. Math. Comput. 45 (1) (2014), 1–14.

• [10]

C. J. Guo, R. P. Agarwal, C. J. Wang, D. O’Regan, The existence of homoclinic orbits for a class of first order superquadratic Hamiltonian systems, Mem. Differential Equations Math. Phys. 61 (2014), 83–102. Google Scholar

• [11]

C. J. Guo, D. O’Regan, Y. T. Xu, R. P. Agarwal. Existence of periodic solutions for a class of second-order superquadratic delay differential equations. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 21 (5) (2014) 405–419. Google Scholar

• [12]

C. J. Guo, D. O’Regan, Y. T. Xu, R. P. Agarwal, Existence of homoclinic orbits of a class of second-order differential difference equations, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms 20 (2013), 675–690. Google Scholar

• [13]

M. Jia, Standing waves for the discrete nonlinear Schrödinger equations, Electron. J. Differ. Equ. 2016 (183) (2016), 1–9. Google Scholar

• [14]

J. H. Leng, Existence of periodic solutions for a higher order nonlinear difference equation, Electron. J. Differ. Equ. 2016 (134) (2016), 1–10. Google Scholar

• [15]

J. H. Leng, Periodic and subharmonic solutions for 2nth-order ϕc-Laplacian difference equations containing both advance and retardation, Indag. Math. (N.S.) 27 (4) (2016), 902–913.

• [16]

X. Liu, Y. B. Zhang, H. P. Shi, Nonexistence and existence results for a class of fourth-order difference Neumann boundary value problems, Indag. Math. (N.S.) 26 (1) (2015), 293–305.

• [17]

X. Liu, Y. B. Zhang, H. P. Shi, Existence of periodic solutions for a class of nonlinear difference equations, Qual. Theory Dyn. Syst. 14 (1) (2015), 51–69.

• [18]

X. Liu, Y. B. Zhang, H. P. Shi, Nonexistence and existence results for a class of fourth-order difference Dirichlet boundary value problems, Math. Methods Appl. Sci. 38 (4) (2015), 691–700.

• [19]

J. Mawhin, M. Willem, Critical point theory and hamiltonian systems, Springer, New York, 1989. Google Scholar

• [20]

P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, Amer. Math. Soc., Providence, RI, New York, 1986. Google Scholar

• [21]

S. Salahshour, A. Ahmadian, M. Senu, D. Baleanu, P. Agarwal, On analytical solutions of the fractional differential equation with uncertainty: application to the Basset problem, Entropy 17 (2) (2015), 885–902.

• [22]

H. P. Shi, X. Liu, Y. B. Zhang, Periodic solutions for a class of nonlinear difference equations, Hokkaido Math. J. 45 (1) (2016), 109–126.

• [23]

H. P. Shi, X. Liu, Y. B. Zhang, Homoclinic orbits for a class of nonlinear difference equations, Azerb. J. Math. 6 (1) (2016), 2218–6816. Google Scholar

• [24]

H. P. Shi, Y. B. Zhang, Standing wave solutions for the discrete nonlinear Schrödinger equations with indefinite sign subquadratic potentials, Appl. Math. Lett. 58 (2016), 95–102.

• [25]

A. N. Sharkovsky, Y. L. Maistrenko, E. Y. Romanenko. Difference equations and their applications, Kluwer Academic Publishers, Dordrecht, 1993. Google Scholar

• [26]

T. Thandapani, K. Ravi, Bounded and monotone properties of solutions of second-order quasilinear forced difference equations, Comput. Math. Appl. 38 (2) (1999), 113–121.

• [27]

X. M. Zhang, P. Agarwal, Z. H. Liu, H. Peng, The general solution for impulsive differential equations with Riemann-Liouville fractional-order q ∈ (1,2), Open Math. 13 (1) (2015), 908–923.

• [28]

X. M. Zhang, P. Agarwal, Z. H. Liu, H. Peng, F. You, Y. J. Zhu, Existence and uniqueness of solutions for stochastic differential equations of fractional-order q > 1 with finite delays, Adv. Difference Equ. 2017 (123) (2017), 1–18.

• [29]

Z. Zhou, D. F. Ma, Multiplicity results of breathers for the discrete nonlinear Schrödinger equations with unbounded potentials, Sci. China Math. 58 (4) (2015), 781–790.

• [30]

Z. Zhou, M. T. Su, Boundary value problems for 2n-order ϕc-Laplacian difference equations containing both advance and retardation, Appl. Math. Lett. 41 (2015), 7–11.

• [31]

H. Zhou, L. Yang, P. Agarwal, Solvability for fractional p-Laplacian differential equations with multipoint boundary conditions at resonance on infinite interval, J. Appl. Math. Comput. 53 (1–2) (2017), 51–76.

• [32]

Z. Zhou, J. S. Yu, Homoclinic solutions in periodic nonlinear difference equations with superlinear nonlinearity, Acta Math. Sin. (Engl. Ser.) 29 (9) (2013), 1809–1822.

• [33]

Z. Zhou, J. S. Yu, On the existence of homoclinic solutions of a class of discrete nonlinear periodic systems, J. Differ. Equ. 249 (5) (2010), 1199–1212.

• [34]

Z. Zhou, J. S. Yu, Y. M. Chen, Homoclinic solutions in periodic difference equations with saturable nonlinearity, Sci. China Math. 54 (1) (2011), 83–93.

Accepted: 2018-05-20

Published Online: 2018-06-26

Published in Print: 2018-07-26

This project is supported by the National Natural Science Foundation of China (No. 11501194). This work was carried out while visiting Central South University.

Citation Information: International Journal of Nonlinear Sciences and Numerical Simulation, Volume 19, Issue 5, Pages 531–537, ISSN (Online) 2191-0294, ISSN (Print) 1565-1339,

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston.