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# Open Mathematics

### formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

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Volume 15, Issue 1

# Triple solutions for a Dirichlet boundary value problem involving a perturbed discrete p(k)-Laplacian operator

• Corresponding author
• Department of Basic Sciences, Sari Agricultural Sciences and Natural Resources University, Sari, Iran
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• Other articles by this author:
/ Johnny Henderson
Published Online: 2017-08-19 | DOI: https://doi.org/10.1515/math-2017-0090

## Abstract

Triple solutions are obtained for a discrete problem involving a nonlinearly perturbed one-dimensional p(k)-Laplacian operator and satisfying Dirichlet boundary conditions. The methods for existence rely on a Ricceri-local minimum theorem for differentiable functionals. Several examples are included to illustrate the main results.

MSC 2010: 39A10; 39A12; 39A70; 34B15; 65Q10

## 1 Introduction

There is an increasing interest in the existence of solutions to boundary value problems for finite difference equations with the p(x)–Laplacian operator in the last decades. These kinds of problems like (1) play a fundamental role in different fields of research, because of their applications in many fields, they can model various phenomena arising from the study of elastic mechanics [33], electrorheological fluids [15] and image restoration [14] and other fields such as biological neural networks, cybernetics, ecology, control systems, economics, computer science, physics, finance, artificial and many others.

Important tools in the study of nonlinear difference equations are fixed point methods in cone; see [2, 19], and upper and lower solution techniques; see, for instance, [24, 25] and references therein. It is well known that critical point theory is an important tool to deal with the problems for differential equations. For background and recent results for nonlinear discrete boundary value problems, we refer the reader to [57, 10, 12, 13, 16, 17, 20, 27] and the seminal papers [3, 4] and references therein.

The aim of this paper is to establish the existence of three solutions for the following discrete boundary-value problem $−Δ(w(k−1)ϕp(k−1)(Δu(k−1))+q(k)ϕp(k)(u(k))=λf(k,u(k))+μg(k,u(k)),u(0)=u(T+1)=0,$(1) for every k ∈ [1, T], where T ≥ 2 is a fixed positive integer, [1, T] is the discrete interval {1, …, T}, f, g : [1, T] × ℝ → ℝ are two continuous functions in the second variable, λ > 0 and μ ≥ 0 are two parameters, w : [0, T] → [1, ∞) is a fix function such that $w−:=mink∈[0,T]w(k)andw+:=maxk∈[0,T]w(k),$ and Δ u(k) = u(k + 1) − u(k) is the forward difference operator, and ϕp(⋅)(s) = |s|p(⋅)−2s is as one-dimensional discrete p(⋅)-Laplacian operator such that $−Δ(w(k−1)ϕp(k−1)(Δu(k−1))=w(k−1)|Δu(k−1)|p(k−1)−2Δu(k−1)−w(k)|Δu(k)|p(k)−2Δu(k),$ the function p : [0, T + 1] → [2, ∞) is bounded, we denote for short $p+:=maxk∈[0,T+1]p(k)andp−:=mink∈[0,T+1]p(k),$ and the function q : [0, T + 1] →[1, ∞) is bounded such that $q+:=maxk∈[0,T+1]q(k).$

We note that problem (1) is the discrete variant of a type of the variable exponent anisotropic problem $−∑i=1N∂∂xi(wi(x)|∂u∂xi|pi(x)−2∂u∂xi)+q(x)|u|pi(x)−2u=λf(k,u)+μg(k,u),x∈Ω,u=0,x∈∂Ω,$ where Ω ⊂ ℝN, N ≥ 3, is a bounded domain with smooth boundary, f, gC(Ω × ℝ, ℝ) are two given functions that satisfy certain properties, and pi(x), wi(x) ≥ 1 and q(x) ≥ 1 are continuous functions on Ω, with 2 ≤ pi(x), for each x ∈ Ω and every i ∈ {1, 2, …, N}, λ > 0 and μ are real numbers.

In [31] the authors studied (1) depending on a parameter with λ = −1 and μ = 0 by the mountain pass method. This method has been applied for (1) with q(k) = 0 and μ = 0 in [18]. In 2003, by using the upper and lower solution method, Atici and Cabada [1] considered Eq. (1) with w(k) = 1, p(k) = 2, and λ = −1 subject to the boundary value conditions u(0) = u(T), Δ u(0) = Δ u(T). In [11] the authors studied Eq. (1) with p(x) = 2 and w(k) = 1 and μ = 0 and Neumann boundary condition Δ u(0) = Δ u(T) = 0. In [23, 26], the authors considered Eq. (1) in the special case, w(k) = 1 and p(x) = p. In [21], the authors applying variational methods, studied the existence of solutions of the following Kirchhoff-type discrete boundary value problems $−M(∥u∥p)Δ(ϕp(Δu(k−1))+q(k)ϕp(u(k))=f(k,u(k)),k∈[1,T]Δu(0)=Δu(T+1)=0,$ where M : [0, ∞) → ℝ is a continuous function.

In this paper, based on a local minimum theorem (Theorem 2.5) due to Ricceri [30], we ensure an exact interval of the parameter λ, in which the problem (1) admits at least three solutions.

As an example, here, we point out the following special case of our main results.

#### Theorem 1.1

Assume T ≥ 2 is a fixed integer number and there exists a positive constant d < 1 such that,

(A0)$\begin{array}{}{\int }_{0}^{d}\phantom{\rule{thinmathspace}{0ex}}f\left(t\right)dt>{d}^{2}\left(2{T}^{2}+6T+4\right).\end{array}$

(A1)$\begin{array}{}max\phantom{\rule{thinmathspace}{0ex}}\left\{lim\underset{|t|\to +\mathrm{\infty }}{sup}\frac{{\int }_{0}^{t}f\left(\xi \right)d\xi }{{t}^{2}},\phantom{\rule{thinmathspace}{0ex}}lim\underset{|t|\to 0}{sup}\frac{{\int }_{0}^{t}f\left(\xi \right)d\xi }{{t}^{2}}\right\}<1.\end{array}$

Then, there exists r > 0 with the following property: for every continuous function g : [1, T] × ℝ → ℝ, there exists δ > 0 such that, for each μ ∈ [0, δ], the problem $−Δ2(u(k−1))+u(k)=14T(T+1)f(u(k))+μg(k,u(k)),k∈[1,T],u(0)=u(T+1)=0,$(2) has at least three solutions whose norms are less than r.

The rest of this paper is arranged as follows. In Section 2, we recall some basic definitions and the main tool (Theorem 2.5.), and in Section 3, we provide our main results that contain several theorems, and finally, we illustrate the results by giving examples.

## 2 Preliminaries

Let us introduce some notations that will be used later. Let T ≥ 2 be a fixed positive integer, [1, T] is the discrete interval {1, …, T}. In order to give the variational formulation of the problem (1), we introduce, T-dimensional Banach space $W:={u:[0,T+1]→R:u(0)=u(T+1)=0},$ equipped with the norm $∥u∥:=∑k=1T+1w(k−1)|Δu(k−1)|p−+q(k)|u(k)|p−1/p−.$

If $∥u∥+=∑k=1T+1w(k−1)|Δu(k−1)|p++q(k)|u(k)|p+1/p+,$ then by Weighted Hölder’s inequality, one can conclude that $L∥u∥+≤∥u∥≤2p+−p−p+p−L∥u∥+,$ where, $L={(T+1)max{w+,q+}}p+−p−p+p−.$

In the space W we can also consider the Luxemburg norm [8], $∥u∥p(⋅)=inf{μ>0:∑k=1T+1w(k−1)|Δu(k−1)μ|p(k−1)+q(k)|u(k)μ|p(k)≤1}.$

Since W has finite dimension, the two last norms are equivalent. Therefore there exist constants L1 > 0 and L2 > 1 such that $L1∥u∥p(⋅)≤∥u∥≤L2∥u∥p(⋅).$(3)

Now, let φ : W → ℝ be given by the formula $φ(u):=∑k=1T+1[w(k−1)|Δu(k−1)|p(k−1)+q(k)|u(k)|p(k)].$(4)

It is easy to check that for any uW the following properties hold: $∥u∥p(⋅)<1⇒∥u∥p(⋅)p+≤φ(u)≤∥u∥p(⋅)p−,$ $∥u∥p(⋅)>1⇒∥u∥p(⋅)p−≤φ(u)≤∥u∥p(⋅)p+.$(5)

Let, $\begin{array}{}F\left(k,\phantom{\rule{thinmathspace}{0ex}}t\right):={\int }_{0}^{t}f\left(k,\xi \right)d\xi ,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}G\left(k,\phantom{\rule{thinmathspace}{0ex}}t\right):={\int }_{0}^{t}g\left(k,\xi \right)d\xi ,\end{array}$ for every k ∈ [1, T] and t ∈ ℝ.

In the sequel, we will use the following inequality

#### Lemma 2.1

Let uW, Then $∥u∥∞≤c1∥u∥,$(6) where $∥u∥∞=maxk∈[1,T]|u(k)|,c1=(2T+2)p−−1p−.$

#### Proof

Let u be in W. Then by the discrete Hölder inequality, for every k ∈ [1, T], we get $|u(k)|≤∑k=1T+1|u(k)|+∑k=1T+1|Δu(k−1)|≤(T+1)p−−1p−∑k=1T+1|u(k)|p−1p−+(T+1)p−−1p−∑k=1T+1|Δu(k−1)|p−1p−≤(T+1)p−−1p−∑k=1T+1q(k)|u(k)|p−1p−+∑k=1T+1w(k−1)|Δu(k−1)|p−1p−≤2p−−1p−(T+1)p−−1p−∑k=1T+1w(k−1)|Δu(k−1)|p−+q(k)|u(k)|p−1p−=(2T+2)p−−1p−∥u∥,$ for all k ∈ [1, T]. □

#### Lemma 2.2

1. There exists a positive constant C1 such that for all uW withu∥ > 1, $∑k=1T+1[w(k−1)p(k−1)|Δu(k−1)|p(k−1)+q(k)p(k)|u(k)|p(k)]≥∥u∥p−p+−C1.$

2. for all uW withu∥ < 1, $∑k=1T+1[w(k−1)p(k−1)|Δu(k−1)|p(k−1)+q(k)p(k)|u(k)|p(k)]≥2p−−p+p−p+Lp+∥u∥p+.$

#### Proof

Let uW be fixed. By a similar argument as in [28], we set $A<:={k∈[0,T+1],|Δu(k)|<1},B<:={k∈[0,T+1],|u(k)|<1},A>:={k∈[0,T+1],|Δu(k)|>1},B>:={k∈[0,T+1],|u(k)|>1},A=:={k∈[0,T+1],|Δu(k)|=1},B=:={k∈[0,T+1],|u(k)|=1}.$

We define for each k ∈ [0, T + 1], $αk:=p+, if k∈A<,p−, if k∈A>,1, if k∈A=,andβk:=p+, if k∈B<,p−, if k∈B>,1, if k∈B=.$

Thus, if ∥u∥ > 1, we have $∑k=1T+1[w(k−1)p(k−1)|Δu(k−1)|p(k−1)+q(k)p(k)|u(k)|p(k)]≥1p+∑k=1T+1[w(k−1)|Δu(k−1)|p(k−1)+q(k)|u(k)|p(k)]=1p+[(∑k∈A<+∑k∈A>+∑k∈A=)w(k−1)|Δu(k−1)|p(k−1)+(∑k∈B<+∑k∈B>+∑k∈B=)q(k)|u(k)|p(k)]≥1p+[∑k=1T+1w(k−1)|Δu(k−1)|αk−1+∑k=1T+1q(k)|u(k)|βk]≥1p+[∑k=1T+1w(k−1)|Δu(k−1)|p−−∑k∈A<(w(k−1)(|Δu(k−1)|p−−|Δu(k−1)|p+))+∑k=1T+1q(k)|u(k)|p−−∑k∈B<(q(k)[|u(k)|p−−|u(k)|p+])]≥1p+[∑k=1T+1w(k−1)|Δu(k−1)|p−−w+∑k∈A<(|Δu(k−1)|p−−|Δu(k−1)|p+)+∑k=1T+1q(k)|u(k)|p−−q+∑k∈B<([|u(k)|p−−|u(k)|p+])]≥1p+[∑k=1T+1w(k−1)|Δu(k−1)|p−−(T+1)w++∑k=1T+1q(k)|u(k)|p−−q+(T+1)]=∥u∥p−p+−(T+1)(w++q+)p+.$

If ∥u∥ < 1, then A> = A= = B> = B= = ∅. It follows that |Δ u(k − 1 |, |u(k)| < 1 for each k ∈ [1, T + 1]. Hence, we have $∑k=1T+1[w(k−1)p(k−1)|Δu(k−1)|p(k−1)+q(k)p(k)|u(k)|p(k)]≥1p+∑k=1T+1[w(k−1)|Δu(k−1)|p++q(k)|u(k)|p+]=1p+∥u∥+p+≥2p−−p+p−p+Lp+∥u∥p+.$ □

To study the problem (1), we consider the functional Iλ,μ : W → ℝ defined by $Iλ,μ(u)=∑k=1T+1[w(k−1)p(k−1)|Δu(k−1)|p(k−1)+q(k)p(k)|u(k)|p(k)]−λ∑k=1TF(k,u)−μ∑k=1TG(k,u).$

An easy computation ensures that Iλ,μ is of class C1 on W with $Iλ,μ′(u)(v)=∑k=1T+1[w(k−1)ϕp(k−1)(Δu(k−1))Δv(k−1)+q(k)|u(k)|p(k)−2u(k)v(k)]−∑k=1T[λf(k,u(k))+μg(k,u(k)]v(k),$ for all u, vW.

#### Lemma 2.3

The critical points of Iλ,μ are exactly the solutions of the problem (1).

#### Proof

Let u be a critical point of Iλ,μ in W. Then u(0) = u(T + 1) = 0 and for all vW, $\begin{array}{}{I}_{\lambda ,\mu }^{\prime }\end{array}$(u)(v) = 0. Thus, for every vW, and taking v(0) = v(T + 1) = 0 and summation by parts into account, one has $0=Iλ,μ′(u¯)(v)=∑k=1T+1w(k−1)ϕp(k−1)(Δu¯(k−1))Δv(k−1)+∑k=1T[q(k)|u¯(k)|p(k)−2u¯(k)v(k)−λf(k,u¯(k))v(k)−μg(k,u¯(k))v(k)]=w(T+1)ϕp(k−1)(Δu¯(T+1))v(T+1)−w(0)ϕp(k−1)(Δu¯(0))v(0)−∑k=1T+1Δ(w(k−1)ϕp(k−1)(Δu¯(k−1)))v(k)+∑k=1T[q(k)|u¯(k)|p(k)−2u¯(k)v(k)−λf(k,u¯(k))v(k)−μg(k,u¯(k))v(k)]=−∑k=1T[Δ(w(k−1)ϕp(k−1)(Δu¯(k−1)))−q(k)|u¯(k)|p(k)−2u¯(k)+λf(k,u¯(k))+μg(k,u¯(k))]v(k).$

Since vW is arbitrary, one has $−Δ(w(k−1)ϕp(k−1)(Δu¯(k−1))+q(k)ϕp(k)(u¯(k))=λf(k,u¯(k))+μg(k,u¯(k),$ for every k ∈ [1, T]. Therefore, u is a solution of (1). So by bearing in mind that u is arbitrary, we conclude that every critical point of the functional Iλ,μ in W is exactly a solution of the problem (1).

Also, if uW is a solution of problem (1), one can show that u is a critical point of Iλ,μ. Indeed, by multiplying the difference equation in problem (1) by v(k) as an arbitrary element of W and summing and using the fact that $∑k=1T+1w(k−1)ϕp(k−1)(Δu¯(k−1))Δv(k−1)=−∑k=1TΔ(w(k−1)ϕp(k−1)(Δu¯(k−1)))v(k),$ we have $\begin{array}{}{I}_{\lambda ,\mu }^{\prime }\end{array}$ (u)(v) = 0, hence u is a critical point for Iλ,μ. Thus the vice versa holds and the proof is completed. □

Our main tool is a local minimum theorem (Theorem 2.5) due to Ricceri (see [30, Theorem 2]), which is recalled below. We refer to the papers [22, 30, 32] in which Theorem 2.5 has been successfully employed for the existence of at least three solutions for two-point boundary value problems. First, we give the following definition.

#### Definition 2.4

If X is a real Banach space, we denote by 𝓦X the class of all functionals Φ : X → ℝ possessing the following property: if {un} is a sequence in X converging weakly to uX and lim infn→∞ Φ(un) ≤ Φ(u), then {un} has a subsequence converging strongly to u. For instance, if X is uniformly convex and g : [0, +∞[→ ℝ is a continuous, strictly increasing function, then, by a classical result, the functional ug(∥u∥) belongs to the class 𝓦X.

Our main tool reads as follows:

#### Theorem 2.5

([30, Theorem 2]). Let X be a separable and reflexive real Banach space and Φ : X → ℝ be a coercive, sequentially weakly lower semicontinuous C1 functional, belonging to 𝓦X, bounded on each bounded subset of X and whose derivative admits a continuous inverse on X*; J : X → ℝ a C1 functional with compact derivative. Assume that Φ has a strict local minimum x0 with Φ(x0) = J(x0) = 0. Finally, setting $α=max{0,lim sup∥x∥→+∞J(x)Φ(x),lim sup∥x∥→0J(x)Φ(x)},β=supx∈Φ−1(]0+∞[)J(x)Φ(x),$ assume that α < β. Then, for each compact interval [a, b]⊂] $\begin{array}{}\frac{1}{\beta },\frac{1}{\alpha }\end{array}$[(with the conventions $\begin{array}{}\frac{1}{0}=+\mathrm{\infty },\frac{1}{+\mathrm{\infty }}\end{array}$ = 0 ) there exists r > 0 with the following property: for every λ ∈ [a, b] and every C1 functional Ψ : X → ℝ with compact derivative, there exists δ > 0 such that, for each μ ∈ [0, δ], the equation Φ′(x) = λ J′(x) + μ Ψ′(x) has at least three solutions whose norms are less than r.

Also we state the following consequence of the Strong comparison principle [3, Lemma 2.3] (see also ([5, Theorem 2.2]) which we will use in the sequel in order to obtain positive solutions to the problem (1), i.e. u(k) > 0 for each k ∈ [1, T].

#### Lemma 2.6

If uW and $−Δ(w(k−1)ϕp(k−1)(Δu(k−1)))+q(k)ϕp(k)(u(k))≥0,k∈[1,T],u(0)≥0,u(T+1)≥0,$(7) then either u is positive or u ≡ 0.

#### Proof

First, we can prove that u ≥ 0. Assume that u(k)≱ 0 for any k ∈ [0, T + 1]. Then, there would exist k0 ∈ [0, T + 1] such that mink∈ [1, T] u(k) = u(k0) < 0 and Δ u(k0−1) ≤ 0. Hence from (7), one has $0>w(k0−1)ϕp(k0−1)(Δu(k0−1))+q(k0)|u(k0)|p(k0)u(k0)≥w(k0)ϕp(k0)(Δu(k0)).$

Thus u(k0+1) < u(k0). This is a contradiction. Therefore one can conclude u(k) ≥ 0 for any k ∈ [0, T + 1]. Finally, arguing as in the proof of [3, Lemma 2.2], we observe that if u(k) = 0 satisfying (7), then by positivity of w(k) and w(k − 1) and non-negativity of u(k + 1), u(k − 1), one has $0≤−w(k)|u(k+1)|p(k)−2u(k+1)−w(k−1)|u(k−1)|p(k−1)−2u(k−1)≤0,$ from which it follows that, u(k + 1) = u(k − 1) = 0. Thus either u is positive or u ≡ 0. Hence, the result follows. □

By a similar argument used in [5, Theorem 2.3], we obtain the next result which guarantees the same conclusion of the preceding Strong maximum principle, independently of the sign of the operator.

#### Lemma 2.7

Fix uW such that, if u(k) ≤ 0, it follows that $−Δ(w(k−1)ϕp(k−1)(Δu(k−1)))+q(k)ϕp(k)(u(k))=0.$ Then either u is positive or u ≡ 0.

## 3 Main results

#### Theorem 3.1

Assume that there exist constants M > 0, m > 0, d ∈ (0, 1), and a, b > 0 and s > p+ such that

(B0) $\begin{array}{}{p}^{+}Tmax\left\{\overline{b}{c}_{1}^{{p}^{-}},{L}^{{p}^{+}}{2}^{\frac{{p}^{+}-{p}^{-}}{{p}^{-}}}\overline{a}{c}_{1}^{{p}^{+}}\right\}<\frac{{p}^{-}\sum _{k=1}^{T}F\left(k,d\right)}{d{\phantom{\rule{thinmathspace}{0ex}}}^{{p}^{-}}\left(w\left(0\right)+w\left(T\right)+\sum _{k=1}^{T}q\left(k\right)\right)}.\end{array}$

(B1) F(k, t) < a(|t|s + |t|p+); for every (k, |t|) ∈ [1, T] × [0, m].

(B2) F(k, t) < b(1 + |t|p); for every (k, |t|) ∈ [1, T] × [M, ∞).

Then, for each compact interval [a, b] ⊂ Λ :=]λ1, λ2[, where $λ1=dp−(w(0)+w(T)+∑k=1Tq(k))p−∑k=T1F(k,d),λ2=1p+Tmax{b¯c1p−,Lp+2p+−p−p−a¯c1p+},$ there exists r > 0 with the following property: for every λ ∈ [a, b] and for every continuous function g : [1, T] × ℝ → ℝ, there exists δ > 0 such that, for each μ ∈ [0, δ], the problem (1) has at least three solutions whose norms are less than r.

#### Proof

Our aim it to apply Theorem 2.5 to our problem. To this end, we observe that due to (B0) the interval ]λ1, λ2[ is non-empty, so fix λ in [a, b] ⊂]λ1, λ2[, take X = W, and put Φ, Ψ as follows: $Φ(u):=∑k=1T+1[w(k−1)p(k−1)|Δu(k−1)|p(k−1)+q(k)p(k)|u(k)|p(k)],$(8) $J(u):=∑k=1TF(k,u(k)),andΨ(u):=∑k=1TG(k,u(k)),$(9) for every uW. So Iλ,μ = Φ − λ Jμ Ψ. An easy computation ensures that Φ, J and Ψ turn out to be of class C1 on W with $Φ′(u)(v)=∑k=1T+1[w(k−1)ϕp(k−1)(Δu(k−1))Δv(k−1)+q(k)|u(k)|p(k)−2u(k)v(k)]=−∑k=1T[Δ(w(k−1)ϕp(k−1)(Δu(k−1))v(k)−q(k)|u(k)|p(k)−2u(k)v(k)],$ and $J′(u)(v)=∑k=1Tf(k,u(k))v(k),Ψ′(u)(v)=∑k=1Tg(k,u(k))v(k),$ for all u, vW. By Lemma 2.3, the solutions of the equation $\begin{array}{}{I}_{\overline{\lambda },\overline{\mu }}^{\prime }={\mathrm{\Phi }}^{\prime }-\overline{\lambda }{J}^{\prime }-\overline{\mu }{\mathrm{\Psi }}^{\prime }=0\end{array}$ are exactly the solutions for problem (1). Hence, to prove our result, it is enough to apply Theorem 2.5. We know Φ is a nonnegative continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on X*, and Ψ is a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact. Also the functional Φ is coercive. Indeed, by (3) for ∥u∥ > L2, we have $\begin{array}{}\parallel u{\parallel }_{p\left(\cdot \right)}>\frac{\parallel u\parallel }{{L}_{2}}>1.\end{array}$ Also by (5), $\begin{array}{}\phi \left(u\right)>\parallel u{\parallel }_{p\left(\cdot \right)}^{{p}^{-}}>\frac{\parallel u{\parallel }^{{p}^{-}}}{{L}_{2}^{{p}^{-}}}.\end{array}$ Therefore by (4), one has $Φ(u)>φ(u)p+>∥u∥p(⋅)p−p+>∥u∥p−p+L2p−→+∞ as ∥u∥→+∞.$

In addition, Φ has a strict local minimum 0 with Φ(0) = J(0) = 0.

We show that for λ ∈ [a, b] ⊂ Λ :=]λ1, λ2[ to be fixed, $\begin{array}{}\alpha <\frac{1}{\overline{\lambda }},\phantom{\rule{thinmathspace}{0ex}}\beta >\frac{1}{\overline{\lambda }}.\end{array}$ Let d ∈ (0, 1) be fixed and put v(k) = d for every k ∈ [1, T] and v(0) = v(T + 1) = 0. Clearly vW and $0<Φ(v¯)<φ(v¯)p−=1p−w(0)dp(0)+w(T)dp(T)+∑k=1Tdp(k)q(k)(10) and $J(v¯)=∑k=1TF(k,v¯(k))=∑k=1TF(k,d).$

Therefore, $β=supu∈Φ−1(]0,∞[)J(u)Φ(u)>J(v¯)Φ(v¯)>p−∑k=T1F(k,d)dp−(w(0)+w(T)+∑k=1Tq(k))=1λ1>1λ¯.$(11)

On the other hand, from the conditions (B1) and (B2), and bearing (6) in mind, we get $∑k=1TF(k,u(k))

Hence, by Lemma 2.2(1) $lim sup∥u∥→∞J(u)Φ(u)(12) and by Lemma 2.2(2) $lim sup∥u∥→0J(u)Φ(u)(13)

By (12) and (13), we get, $α=max{0,lim sup∥u∥→∞J(u)Φ(u),lim sup∥u∥→0J(u)Φ(u)}(14)

By (11) and (14), we deduce that α < β. All the assumptions of Theorem 2.5 are satisfied, so by applying that theorem, the conclusion follows. Hence, the proof is complete. □

Now, we present an example to illustrate the results of Theorem 3.1.

#### Example 3.2

Choose m = 0.198, M = 250, $\begin{array}{}d=\mathrm{ln}\left(1+\sqrt{2}\right)\in \left(0,1\right),\phantom{\rule{thinmathspace}{0ex}}s=4.5,\phantom{\rule{thinmathspace}{0ex}}p\left(k\right)=\frac{2k}{11}+2,\end{array}$ a = e−19, b = e−11, T = 10, q(k) = 1, $\begin{array}{}w\left(k\right)={e}^{\frac{k\left(10-k\right)}{{k}^{2}+1}},\end{array}$ hence w(0) = 1, w(10) = 1, w+ = e4.5, $\begin{array}{}L=\sqrt[4]{11{e}^{45}},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\sum _{k=1}^{T}q\left(k\right)=10,\phantom{\rule{thinmathspace}{0ex}}{p}^{+}=4,\phantom{\rule{thinmathspace}{0ex}}{p}^{-}=2,\phantom{\rule{thinmathspace}{0ex}}{c}_{1}=\left(2T+2{\right)}^{\frac{{p}^{-}-1}{{p}^{-}}}=\sqrt{22}.\end{array}$ Let $f(k,t)=2cosh⁡tsinh3⁡te−1sinh2⁡t,f(k,0)=0,∀t≠0,k∈[1,10],F(k,t)=e−1sinh2⁡t,$ for any k ∈ [1, 10], where t+ = max{0, t}. Therefore, $p+Tmax{b¯c1p−,Lp+2p+−p−p−a¯c1p+}=40×223e−14.5≃0.2148117565,p−∑k=T1F(k,d)dp−(w(0)+w(T)+∑k=1Tq(k))=53e(ln⁡(1+2)2≃0.7892856465,$ it follows that (B0) holds. Also by the graph of the function $h(x):=e−1sinh(x)2−|x|4.5+x4e19,$ on [−0.198, 0.198], where it is under x axis (see Figure 1), one can conclude (B1) holds. It is clear that (B2) holds.

Figure 1

The graph of the function h

We see that all assumptions of Theorem 3.1 are satisfied, hence Theorem 3.1 implies that for each compact interval $[a,b]⊂3e(ln⁡(1+2))25,e14.540×223,$ there exists r > 0 with the following property: for every λ ∈ [a, b] and every positive continuous function g : [1, 10] × ℝ → ℝ, there exists δ > 0 such that, for each μ ∈ [0, δ], the problem $−Δ(ek(10−k)k2+1ϕp(k−1)(Δu(k−1))+ϕp(k)(u(k))=λf(k,u+(k))+μg(k,u(k)),u(0)=u(11)=0,$ k ∈ [1, 10], has at least three solutions whose norms are less than r, hence by Lemma 2.6 whose signs are positive.

The next result reads as follows.

#### Theorem 3.3

Assume that there exists a constant d > 1 such that,

(C0) $\begin{array}{}{p}^{+}T{c}_{1}^{{p}^{+}}{2}^{\frac{{p}^{+}-{p}^{-}}{{p}^{-}}}{L}^{{p}^{+}}<\frac{{p}^{-}\sum _{k=1}^{T}F\left(k,d\right)}{d{\phantom{\rule{thinmathspace}{0ex}}}^{{p}^{+}}\left(w\left(0\right)+w\left(T\right)+\sum _{k=1}^{T}q\left(k\right)\right)}.\end{array}$

(C1) $\begin{array}{}max\left\{\underset{|t|\to +\mathrm{\infty }}{lim sup}\frac{F\left(k,t\right)}{|t{|}^{{p}^{-}}},\phantom{\rule{thinmathspace}{0ex}}lim\underset{|t|\to 0}{sup}\frac{F\left(k,t\right)}{|t{|}^{{p}^{+}}}\right\}<1\mathit{;}\end{array}$ for every k ∈ [1, T].

Then, for each compact interval [a, b] ⊂ Λ := ]λ1, λ2[, where $λ1=dp+(w(0)+w(T)+∑k=1Tq(k))p−∑k=1TF(k,d),λ2=1p+Tc1p+2p+−p−p−Lp+,$ there exists r > 0 with the following property: for every λ ∈ [a, b] and every continuous function g : [1, T] × ℝ → ℝ, there exists δ > 0 such that, for each μ ∈ [0, δ], the problem (1) has at least three solutions whose norms are less than r.

#### Proof

Our aim it to apply Theorem 2.5 to our problem. To this end, we observe that due to (C0) the interval] λ1, λ2 [is non-empty, so fix λ in]λ1, λ2 [, take X = W, and put Φ, Ψ and J, as given in (8) and (9). So $\begin{array}{}{I}_{\overline{\lambda },\overline{\mu }}=\mathrm{\Phi }-\overline{\lambda }J-\overline{\mu }\mathrm{\Psi }.\end{array}$ By Lemma 2.3, the solutions of the equation $\begin{array}{}{I}_{\overline{\lambda },\overline{\mu }}^{\prime }={\mathrm{\Phi }}^{\prime }-\overline{\lambda }{J}^{\prime }-\overline{\mu }{\mathrm{\Psi }}^{\prime }=0\end{array}$ are exactly the solutions for problem (1). We know that Φ, Ψ and J satisfy the regularity assumptions of Theorem 2.5. Hence, to prove our result, it is enough to apply Theorem 2.5.

We show that for $\begin{array}{}\overline{\lambda }\in \left[a,b\right]\subset \mathrm{\Lambda }:=\right]{\lambda }_{1},{\lambda }_{2}\left[\text{to be fixed},\alpha <\frac{1}{\overline{\lambda }},\beta >\frac{1}{\overline{\lambda }}.\end{array}$ Let d > 1 be fixed and put v(k) = d for every k ∈ [1, T] and v(0) = v(T + 1) = 0. Clearly vW and $0<Φ(v¯)<φ(v¯)p−<1p−(w(0)dp(0)+w(T)dp(T)+∑k=1Tdp(k)q(k))

Therefore $β=supu∈Φ−1(]0,∞[)J(u)Φ(u)>J(v¯)Φ(v¯)>p−∑k=1TF(k,d)dp+(w(0)+w(T)+∑k=1Tq(k))=1λ1>1λ¯.$(15)

In view of (C1), there exist 0 < r1 < 1 < r2 such that $F(k,t)<|t|p+, for any (k,t)∈[1,T]×[−r1,r1],F(k,t)<|t|p−<|t|p+, for any (k,t)∈[1,T]×R∖([−r2,r2]).$

Since F is continuous, then F(k, t) is bounded for any (k, t) ∈ [1, T] × ([r1, r2] ∪ [−r2, −r1]), so we can choose C2 > 0 and s > p+ such that $F(k,t)<|t|p++C2|t|s, for all (k,t)∈[1,T]×R,$ and bearing (6) in mind, we get and by Lemma 2.2(2) $lim sup||u||→0J(u)Φ(u)(16)

Again since F is continuous, then F(k, t) is bounded for any (k, t) ∈ [1, T] × [−r2, r2], so we can choose C3 > 0 such that $F(k,t)<|t|p−+C3, for all (k,t)∈[1,T]×R,$ and bearing (6) in mind, we get $J(u)=∑k=1TF(k,u(k)) hence, by Lemma 2.2(1) $lim sup||u||→∞J(u)Φ(u)(17)

By (16) and (17), we get $α=max{0,lim sup||u||→∞J(u)Φ(u),lim sup0||u||→0J(u)Φ(u)}(18)

By (15) and (18), we deduce that α < β. All the assumptions of Theorem 2.5 are satisfied, so by applying that theorem, the conclusion follows. Hence, the proof is complete. □

Now, we present an example to illustrate the results of Theorem 3.3.

#### Example 3.4

Let T = 10, and for every k ∈ [1, 10 ], put $f(k,t)=0,ift=0,t3[sin⁡(ln⁡|t|)+cos⁡(ln⁡|t|)],if|t|∈(0,1],t2+(2+e6π)t+e6π,ift∈[−e6π,−1],−(t)2+(2+e6π)t−e6π,ift∈[1,e6π],t[sin⁡(ln⁡|t|)+cos⁡(ln⁡|t|)],if|t|∈[e6π,∞),$ as an odd continuous function on ℝ. Let $\begin{array}{}d={e}^{\frac{\pi }{4}},\phantom{\rule{thinmathspace}{0ex}}p\left(k\right)=\frac{-2}{11}k+4,q\left(k\right)=1,w\left(k\right)=e\frac{k\left(10-k\right)}{{k}^{2}+5k+6}\end{array}$ for k = 1, 2, 3, … 10. Hence p = 2, p+ = 4, $\begin{array}{}\sum _{k=1}^{T}q\left(k\right)=10,{q}^{+}=1,{w}^{+}={e}^{0.8},L=\sqrt[4]{11{\mathrm{e}}^{0.08}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}and\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{c}_{1}=\left(2T+2{\right)}^{\frac{{p}^{-}-1}{{p}^{-}}}=\sqrt{22}.\end{array}$ Simple calculations show that $F(k,t)=t417[3cos⁡(ln⁡|t|+5sin⁡(ln⁡|t|)],if|t|∈[0,1]−t33+(2+e6π)t22−e6πt−23+12e6π+317,if|t|∈[1,e6π],t25[3sin⁡(ln⁡|t|)+cos⁡(ln⁡|t|)]+16e18π−15e12π+12e6π−23+317,if|t|[e6π,∞).lim sup|t|→+∞F(k,t)|t|2=105<1,lim sup|t|→0F(k,t)|t|4=3417<1.$

By using software Maple, one can conclude that $p−∑k=1TF(k,d)dp+(w(0)+w(T)+∑k=1Tq(k))=2012eπ{−e3π43+eπ2+e132π2−e25π4−2551+e6π2}≈7873816.449>p+Tc1p+2p+−p−p−Lp+=40⋅223e0.8≃947902.4.$

Hence, by using Theorem 3.3, for each compact interval [a, b] ∈ ] 0.00000012, 0.000001 [there exists r > 0 with the following property: for every λ ∈ [a, b] and every continuous function g : [1, 10] × ℝ → ℝ, there exists δ > 0 such that, for each μ ∈ [0, δ], the problem $−Δ(ek2+5k+6k(10−k)ϕp(k−1)(Δu(k−1))+ϕp(k)(u(k))=λf(k,u(k))+μg(k,u(k)),u(0)=u(11),$ for every k ∈ [1, 10], has at least three solutions whose norms are less than r.

#### Remark 3.5

By Lemma 2.6, the ensured solutions in the conclusions of Theorems 3.1 and 3.3 are either zero or positive. Now, let f(k, 0)+g(k, 0) = 0 for all k ∈ [0, T], by putting $f∗(k,t)+g∗(k,t)=f(k,t)+g(k,t),t>0,0,t≤0,$ and due to Lemma 2.7, the ensured solutions are positive (see [5, Remark 2.1]).

Finally we present the problem (1), in which the function f(k, u) has separable variables and λ = 1.

#### Theorem 3.6

Let f0 : [1, T] → ℝ be a non-negative, non-zero and essentially bounded function such that $∑k=1Tf0(k)=1p+2p+−p−p−Lp+c1p+,$ and f1 : ℝ → ℝ be a non-negative and continuous function and $\begin{array}{}{F}_{1}\left(\xi \right)={\int }_{0}^{\xi }{f}_{1}\left(x\right)dx\phantom{\rule{thinmathspace}{0ex}}for\phantom{\rule{thinmathspace}{0ex}}every\phantom{\rule{thinmathspace}{0ex}}\xi \in \mathbb{R}.\end{array}$ Assume that there exists a positive constant d < 1 such that,

(DO) $F1(d)dp−>p+2p+−p−p−Lp+c1p+p−w(0)+w(T)+∑k=1Tq(k)$

(D1) $max{lim sup|t|→+∞F1(t)|t|p−,lim sup|t|→0F1(t)|t|p+}<1.$

Then, there exists r > 0 with the following property: for every continuous function g : [1, T] × ℝ → ℝ, there exists δ > 0 such that, for each μ ∈ [0, δ], the problem $−Δ(w(k−1)ϕp(Δu(k−1)))+q(k)ϕp(u(k))=f0(k)f1(u(k))+μg(k,u(k)),u(0)=u(T+1)=0,$(19) for every k ∈ [1, T], has at least three solutions whose norms are less than r.

#### Proof

We can choose λ = 1 ∈ [a, b] such that [a, b] ⊂ ($\begin{array}{}\frac{1}{\beta },\frac{1}{\alpha }\end{array}$). Take X = W, and put Φ, Ψ and J as given in (8) and (9), so I1,μ = Φ − JμΨ and the solutions of the equation $\begin{array}{}{I}_{1,\overline{\mu }}^{\prime }\end{array}$ = Φ′ − J′ − μΨ′ = 0 are exactly the solutions for problem (19). Hence, to prove our result, it is enough to apply Theorem 2.5.

We show that, α < 1, β > 1. Put v(k) = d ∈ (0, 1) for every k ∈ [1, T] and v(0) = v(T + 1) = 0. In view of (D0) and (10), $β>J(v¯)Φ(v¯)>p−F1(d)∑k=1Tf0(k)dp−(w(0)+w(T)+∑k=1Tq(k))>p+2p+−p−p−Lp+c1p+∑k=1Tf0(k)=1.$(20)

Applying similar argument as in the proof of Theorem 3.3, and taking into account (D1) and Lemma 2.2 $lim sup∥u∥→0J(u)Φ(u)≤p+2p+−p−p−Lp+c1p+∑k=1Tf0(k),lim sup∥u∥→∞J(u)Φ(u)≤p+c1p−∑k=1Tf0(k),$ therefore, considering c1 > $\begin{array}{}\sqrt{6}\end{array}$ and L ≥ 1, we have $α=max{0,lim sup∥u∥→∞J(u)Φ(u),lim sup∥u∥→0J(u)Φ(u)}≤p+2p+−p−p−Lp+c1p++∑k=1Tf0(k)=1.$(21)

By (20) and (21), we deduce that α < β. All the assumptions of Theorem 2.5 are satisfied, so by applying that theorem, the conclusion follows. Hence, the proof is complete. □

#### Remark 3.7

Theorem 1.1 follows from Theorem 3.6 taking into account that p(k) = 2, f1(t) = f(t) and w(k − 1) = 1 for every k ∈ [1, T + 1], q(k) = 1, $\begin{array}{}{f}_{0}\left(k\right)=\frac{1}{4T\left(T+1\right)}\end{array}$ for every k ∈ [1, T].

Finally, we present an example of Theorem 1.1.

#### Example 3.8

Let T = 4 be a fixed positive integer Consider the problem (2) where $f(t)=0,ift=0,t[sin⁡(2ln⁡|t|)+cos⁡(2ln⁡|t|)],if0 as an odd continuous function on ℝ. Simple calculations show that $F(t)=t22[(sin⁡(2ln⁡|t|)],ift where $K=2π1+(2π)2−e−4π2,L=K−2πe2π1+(2π)2+12.$

Thus $lim sup|t|→0,+∞F(k,t)|t|2=12<1.$

Hence, (A1) holds. Let $\begin{array}{}d={e}^{-\frac{1}{2}}\in \left({e}^{-2\pi },1\right),\end{array}$ so one has, $F(d)=e2π−12(2π)1+(2π)2+e−12+K≃50.75439760>d2(2T2+6T+4)≃36.39183958.$

Thus, (A2) holds. Therefore, by using Theorem 1.1, there exists r > 0 such that for every continuous function g : [1, T] × ℝ → ℝ, there exists δ > 0 such that, for each μ ∈ [0, δ], the problem (2) has at least three solutions whose norms are less than r.

## Acknowledgement

The authors express their gratitude to Professor Biagio Ricceri for his helpful suggestions. The authors are thankful to the anonymous referee for helpful suggestions and comments.

## References

• [1]

Atici F.M., Cabada A., Existence and uniqueness results for discrete second-order periodic boundary value problems, Comput. Math. Appl. 2003, 45, 1417–1427.

• [2]

Avery R., Henderson J., Existence of three positive pseudo-symmetric solutions for a one dimensional discrete p-Laplacian, J. Difference Equ. Appl. 2004, 10, 529–539.

• [3]

Agarwal R.P., Perera K., O’Regan D., Multiple positive solutions of singular discrete p-Laplacian problems via variational methods, Adv. Difference Equ. 2005, 2, 93–99. Google Scholar

• [4]

Agarwal R.P., Perera K., O’Regan D., Multiple positive solutions of singular and nonsingular discrete problems via variational methods, Nonlinear Anal. 2004, 58, 69–73.

• [5]

Bonanno G., Candito P., Infinitely many solutions for a class of discrete non-linear boundary value problems, Appl. Anal. 2009, 884, 605–616.

• [6]

Bonanno G., Candito P., Nonlinear difference equations investigated via critical points methods, Nonlinear Anal. 2009, 70, 3180– 3186.

• [7]

Bonanno G., Jebelean P., Şerban C., Three solutions for discrete anisotropic periodic and neumann problems, Dynamic Sys. Appl. 2013, 22, 183–196. Google Scholar

• [8]

Bereanu C., Jebelean P., Şerban C., Periodic and Neumann problems for discrete p(⋅)-Laplacian, J. Math. Anal. Appl. 2013, 399, 75–87.

• [9]

Bonanno G., Pizzimenti P.F., Neumann boundary value problems with not coercive potential, Mediterr. J. Math. 2012, 9, 601–609.

• [10]

Candito P., D’Aguì G., Three solutions to a perturbed nonlinear discrete Dirichlet problem, J. Math. Anal. Appl. 2011, 375, 594– 601.

• [11]

Candito P., D’Aguì G., Constant-sign solutions for a nonlinear neumann problem involving the discrete p-Laplacian, Opuscula Math. 2014, 344, 683–690. Google Scholar

• [12]

Candito P., D’Aguì G., Three Solutions for a Discrete Nonlinear Neumann Problem Involving the p-Laplacian, Adv. Differ. Equ. 2010, 2010, 1–11.

• [13]

Candito P., Giovannelli N., Multiple solutions for a discrete boundary value problem, Comput. Math. Appl. 2008, 56, 959–964.

• [14]

Chen Y., Levine S., Rao M., Variable exponent, linear growth functionals in image processing, SIAM J. Appl. Math. 2006, 664, 1383–1406. Google Scholar

• [15]

Diening L., Theoretical and numerical results for electrorheological fluids, Ph.D. thesis, University of Frieburg, Germany (2002). Google Scholar

• [16]

Elaydi S., An introduction to difference equations, 3rd ed., Springer Publishing, New York, 1996. Google Scholar

• [17]

Faraci F., Iannizzotto A., Multiplicity theorems for discrete boundary problems, Aequationes Math. 2007, 74, 111–118.

• [18]

Galewski M., Wieteska R., Existence and multiplicity of positive solutions for discrete anisotropic equations, Turkish J. Math. 2014, 38, 297–310.

• [19]

He Z., On the existence of positive solutions of p-Laplacian difference equations, J. Comput. Appl. Math. 2003, 161, 193–201.

• [20]

Heidarkhani S., Afrouzi A., Caristi G., Hendersonc J., Moradi S., A variational approach to difference equations, J. Difference Equ. Appl., 2016, 22, 1761–1776.

• [21]

Heidarkhani S., Afrouzi A., Hendersonc J., Moradi S., Caristi G., Variational approaches to p-Laplacian discrete problems of Kirchhoff-type, J. Difference Equ. Appl., (in press) .

• [22]

Heidarkhani S., Caristi G., Salari A., Perturbed Kirchhoff-type p-Laplacian discrete problems, Collect. Math., (in press) .

• [23]

Heidarkhani S., Khaleghi Moghadam M., Existence of Three solutions for Perturbed nonlinear difference equations, Opuscula Math. 2014, 344, 747–761. Google Scholar

• [24]

Henderson J., Thompson H.B., Existence of multiple solutions for second order discrete boundary value problems, Comput. Math. Appl. 2002, 43, 1239–1248.

• [25]

Jiang D., Chu J., O’Regan D., Agarwal R.P., Positive solutions for continuous and discrete boundary value problems to the onedimensional p-Laplacian, Math. Inequal. Appl. 2004, 7, 523–534. Google Scholar

• [26]

Khaleghi Moghadam M., Heidarkhani S., Henderson J., Infinitely many solutions for Perturbed difference equations, J. Difference Equ. Appl. 2014, 207, 1055–1068.

• [27]

Khaleghi Moghadam M., Avci M., Existence Results to a nonlinear p(k)-Laplacian difference equation, J. Difference Equ. Appl. (impress),

• [28]

Mihăilescu M., Rădulescu V., Tersian S., Eigenvalue problems for anisotropic discrete boundary value problems, J. Difference Equ. Appl. 2009, 15, 557–567.

• [29]

Ricceri B., A general variational principle and some of its applications, J. Comput. Appl. Math. 2000, 113, 401–410.

• [30]

Ricceri B., A further three critical points theorem, Nonlinear Anal. 2009, 71, 4151–4157.

• [31]

Smejda J., Wieteska R., On the depending on parameters for second order discrete boundary value problems with the p(k)-Laplacian, Opuscula Math. 2014, 344, 851–870. Google Scholar

• [32]

Yanga L., Chena H., Yanga X., The multiplicity of solutions for fourth-order equations generated from a boundary condition, Appl. Math. Lett. 2011, 24, 1599–1603.

• [33]

Zhikov V., Averaging of functionals in the calculus of variations and elasticity, Math. USSR Izv. 1987, 29, 33–66.

Accepted: 2017-05-29

Published Online: 2017-08-19

Citation Information: Open Mathematics, Volume 15, Issue 1, Pages 1075–1089, ISSN (Online) 2391-5455,

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