Periodic and subharmonic solutions for a 2nth-order difference equationcontaining both advance and retardation with ϕ-Laplacian

Abstract

In this paper, by using critical point theory, we obtain some new sufficientconditions on the existence and multiplicity of periodic and subharmonic solutions toa 2n th-order nonlinear difference equation containing both advance andretardation with ϕ-Laplacian. Some previous results have beengeneralized.

1 Introduction

Let N, Z, and R denote the sets of all natural numbers,integers and real numbers, respectively. For $a,b\in Z$, define $Z(a)=\{a,a+1,\dots \}$, $Z(a,b)=\{a,a+1,\dots ,b\}$ when $a\le b$. ${\cdot }^{\mathrm{tr}}$ denotes the transpose of a vector.

Consider the following 2n th-order difference equation containing both advanceand retardation with ϕ-Laplacian of the type:

$${\mathrm{△}}^n\left(r_{k-n}\varphi \left({\mathrm{△}}^n{u}_{k-1}\right)\right)={(-1)}^{n}f(k,u_{k+1},u_k,u_{k-1}),k\in Z,$$

(1.1)

where $n\in Z$, △ is forward difference operator defined by$\mathrm{△}{u}_k=u_{k+1}-u_k$, ${\mathrm{△}}^n{u}_k=\mathrm{△}({\mathrm{△}}^{n-1}{u}_k)$, $\varphi \in C(R,R)$ satisfied $\varphi (0)=0$, $f\in C(Z\times R^3,R)$, $r_k>0$ for each $k\in Z$, $\{r_k\}$ and $\{f(k,v_1,v_2,v_3)\}$ are T-periodic in k and T is agiven positive integer.

In this paper, given positive integer m, we will study the existence ofmT-periodic solutions for (1.1). As usual, such a mT-periodicsolution will be called a subharmonic solution.

We may think of (1.1) as a discrete analog of the following 2n th-orderfunctional differential equation:

$$\frac{d^n}{d{t}^n}[r(t)\varphi \left(\frac{d^{n}u(t)}{d{t}^n}\right)]={(-1)}^{n}f(t,u(t+1),u(t),u(t-1)),t\in R.$$

(1.2)

Equations similar in structure to (1.2) have been studied by many authors. For example,for the case where $\varphi (x)=x$, $n=1$, Smets and Willem [1] have considered solitary waves with prescribed speed on infinite lattices ofparticles with nearest neighbor interaction for the following forward and backwarddifferential difference equation:

$$c^2{u}^{″}(t)=V^{\prime }(u(t+1)-u(t))-V^{\prime }(u(t)-u(t-1)),t\in R.$$

For the case where $\varphi (x)={|x|}^{p-2}x$, $p>1$, $n=1$, Wang [2] has studied the existence of positive solutions of the equation

$${\left({\left|u^{\prime }\right|}^{p-2}{u}^{\prime }\right)}^{\prime }+a(t)f(u)=0,t\in R.$$

For the case where $\varphi (x)={|x|}^{p-2}x$, $p>1$, $n=2$, Agarwal, Lu, and O’Regan [3] have studied the existence of positive solutions of the equation

$${\left({\left|u^{″}\right|}^{p-2}{u}^{″}\right)}^{″}=\lambda q(t)f(u),t\in R.$$

For the case where $\varphi (x)=\frac{x}{\sqrt{1+x^2}}$, $n=1$, Bonheure and Habets [4] have studied classical and non-classical solutions of a prescribed curvatureequation

$$-{\left(\frac{u^{\prime }}{\sqrt{1+u^{\prime 2}}}\right)}^{\prime }=\lambda f(t,u),t\in R.$$

In recent years, many authors have studied the existence of periodic solutions ofdifference equations. To mention a few, see [5–8] for second-order difference equations and [9, 10] for higher-order equations. Since 2003, critical point theory has beenemployed to establish sufficient conditions on the existence of periodic solutions ofdifference equations. By using the critical point theory, Guo and Yu [11–13] and Zhou et al.[14] established sufficient conditions on the existence of periodic solutions ofsecond-order nonlinear difference equations. In 2007, by using the Linking Theorem, Caiand Yu [15] obtained some criteria for the existence of periodic solutions of thefollowing equation:

$${\mathrm{△}}^n\left(r_{k-n}{\mathrm{△}}^n{u}_{k-n}\right)+f(k,u_k)=0,k\in Z,$$

(1.3)

for the case where f grows superlinearly at both 0 and ∞, where$n\in Z(3)$. In 2010, by using the Linking Theorem and the SaddlePoint Theorem, Zhou [16] extended f in (1.3) into sublinear or asymptotically linear andimproved the results of [15] when f is superlinear. In particular, a necessary and sufficientcondition for the existence of the unique periodic solution of (1.3) is also establishedin [16]. In 2013, by using the Linking Theorem, Deng [17] provided some sufficient conditions of the existence and multiplicity ofperiodic solutions and subharmonic solutions of the following equation:

$${\mathrm{△}}^n\left(r_{k-n}{\phi }_p\left({\mathrm{△}}^n{u}_{k-1}\right)\right)={(-1)}^{n}f(k,u_{k+1},u_k,u_{k-1}),k\in Z,$$

(1.4)

where $n\in N$, ${\phi }_p$ is the p-Laplacian operator given by${\phi }_p(u)={|u|}^{p-2}u$ ($1<p<\mathrm{\infty }$) and where f satisfies some growth conditionsnear both 0 and ∞. In 2012, Mawhin [18] considered T-periodic solutions of systems of difference equationsof the form

$$\mathrm{△}\varphi [\mathrm{△}u(k-1)]={▽}_{u}F[k,u(k)]+h(k),k\in Z,$$

(1.5)

under various conditions upon $F:Z\times R^n\to R$ and $h:Z\to R^n$, where $n\in Z$, $\varphi =▽\mathrm{\Phi }$, in which $\mathrm{\Phi }:R^n\to [0,\mathrm{\infty })$ is continuously differentiable and strictly convex,satisfies $\varphi (0)=0$ and is a homeomorphism of $R^n$ onto the ball $B_a\subseteq R^n$ or of $B_a$ onto $R^n$. By using direct variational method, he gave sufficientconditions for the existence of a minimizing sequence for the case of coercivepotential, or some averaged coercivity conditions of the Ahmad-Lazer-Paul type addingthe nonlinearity satisfies some growth conditions, or the convex potential. Using theSaddle Point Theorem, previously obtained results are extended to the case of anaveraged anticoercivity condition in [18]. However, the results on periodic solutions of higher-order nonlineardifference equations involving ϕ-Laplacian are very scarce in theliterature. Furthermore, since (1.1) contains both advance and retardation, there arevery few works dealing with this subject; see [10, 19]. The main purpose of this paper is to give some sufficient conditions for theexistence and multiplicity of periodic and subharmonic solutions of (1.1). Particularly,our results generalize the results in the literature [17, 20]; see Remark 3.4 and Remark 3.5 for details.

2 Preliminaries

Throughout this paper, we assume that,

(F₁) there exists a functional $F(k,v_1,v_2)\in C^1(Z\times R^2,R)$ with $F(k,v_1,v_2)\ge 0$ and satisfies

$$\begin{array}{c}F(k+T,v_1,v_2)=F(k,v_1,v_2),\hfill \\ \frac{\partial F(k-1,v_2,v_3)}{\partial v_2}+\frac{\partial F(k,v_1,v_2)}{\partial v_2}=f(k,v_1,v_2,v_3).\hfill \end{array}$$

In this section, we first establish the variational setting associated with (1.1).

Let S be the set of all two-sided sequences, that is,

$$S=\{u=\{u_k\}|u_k\in R,k\in Z\}.$$

Then S is a vector space with $au+bv=\{a{u}_k+b{v}_k\}$ for $u,v\in S$, $a,b\in R$. For any fixed positive integer m and T,we define the subspace $E_m$ of S as

$$E_m=\{u=\{u_k\}\in S|u_{k+mT}=u_k,k\in Z\}.$$

Obviously, $E_m$ is isomorphic to $R^{mT}$ and hence $E_m$ can be equipped with the inner product$(\cdot ,\cdot )$ and norm $\parallel \cdot \parallel$ as

$$(u,v)=\sum _{j=1}^{mT}{u}_j{v}_j,u,v\in E_m,$$

and

$$\parallel u\parallel ={\left(\sum _{j=1}^{mT}{u}_j^2\right)}^{\frac{1}{2}},u\in E_m.$$

On the other hand, we define the norm ${\parallel \cdot \parallel }_q$ on $E_m$ as follows:

$$\parallel u\parallel ={\left(\sum _{j=1}^{mT}{|u_j|}^q\right)}^{\frac{1}{q}},$$

for all $u\in E_m$ and $q\ge 1$. By Hölder’ inequality and Jensen’inequality, we have

$$\begin{array}{c}{\parallel u\parallel }_2\le {\parallel u\parallel }_q\le {(mT)}^{\frac{2-q}{2q}}{\parallel u\parallel }_2,1\le q<2,\hfill \\ {(mT)}^{\frac{2-q}{2q}}{\parallel u\parallel }_2\le {\parallel u\parallel }_q\le {\parallel u\parallel }_2,2\le q.\hfill \end{array}$$

Let

$$d_{1,q}=\{\begin{array}{ll}1,& 1\le q<2,\\ {(mT)}^{\frac{2-q}{2q}},& 2\le q,\end{array}{d}_{2,q}=\{\begin{array}{ll}{(mT)}^{\frac{2-q}{2q}},& 1\le q<2,\\ 1,& 2\le q.\end{array}$$

Therefore,

$$d_{1,q}{\parallel u\parallel }_2\le {\parallel u\parallel }_q\le d_{2,q}{\parallel u\parallel }_2,u\in E_m.$$

(2.1)

Clearly, $\parallel u\parallel ={\parallel u\parallel }_2$. For all $u\in E_m$, define the functional J on$E_m$ as follows:

$$J(u)=\sum _{k=1}^{mT}{r}_{k-1}\mathrm{\Phi }\left({\mathrm{△}}^n{u}_{k-1}\right)-\sum _{k=1}^{mT}F(k,u_{k+1},u_k),$$

(2.2)

where

$$\mathrm{\Phi }(u)={\int }_0^u\varphi (s)ds$$

is the primitive function of $\varphi (u)$.

Clearly, $J\in C^1(E_m,R)$ and for any $u={\{u_k\}}_{k\in Z}\in E_m$, by using $u_j=u_{mT+j}$ for $j\in Z(0,mT-1)$, we can compute the partial derivative as

$$\frac{\partial J}{\partial u_k}={(-1)}^n{\mathrm{△}}^n\left(r_{k-n}\varphi \left({\mathrm{△}}^n{u}_{k-1}\right)\right)-f(k,u_{k+1},u_k,u_{k-1}).$$

Thus, u is a critical point of J on $E_m$ if and only if

$${\mathrm{△}}^n\left(r_{k-n}\varphi \left({\mathrm{△}}^n{u}_{k-1}\right)\right)={(-1)}^{n}f(k,u_{k+1},u_k,u_{k-1}),k\in Z(1,mT).$$

Due to the periodicity of $u={\{u_k\}}_{k\in Z}\in E_m$ and $f(k,v_1,v_2,v_3)$ in the first variable k, we reduce the existenceof periodic solutions of (1.1) to the existence of critical points of Jon $E_m$.

Let M be the $mT\times mT$ matrix defined by

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

By matrix theory, we see that the eigenvalues of M are

$${\lambda }_j=2(1-cos\frac{2j}{mT}),j=0,1,2,\dots ,mT-1.$$

Thus, ${\lambda }_0=0$, ${\lambda }_1>0$, ${\lambda }_2>0$, …, ${\lambda }_{mT-1}>0$. Therefore,

$$\begin{array}{c}{\lambda }_{min}=min\{{\lambda }_1,{\lambda }_2,\dots ,{\lambda }_{mT-1}\}=2(1-cos\frac{2}{mT}),\hfill \\ {\lambda }_{max}=max\{{\lambda }_1,{\lambda }_2,\dots ,{\lambda }_{mT-1}\}=\{\begin{array}{ll}4,& \text{when }mT\text{ is even},\\ 2(1+cos\frac{1}{mT}),& \text{when }mT\text{ is odd}.\end{array}\hfill \end{array}$$

For convenience, we identify $u\in E_m$ with $u={(u_1,u_2,\dots ,u_{mT})}^{\mathrm{tr}}$. Let

$${\overline{E}}_m=\{u={(u_1,u_2,\dots ,u_{mT})}^{\mathrm{tr}}\in E_m|{\mathrm{△}}^{n-1}{u}_j=0,j\in Z(1,mT)\}.$$

Then

$${\overline{E}}_m=\{u\in E_m|u=\{a\},a\in R\}.$$

Let ${\tilde{E}}_m$ be the direct orthogonal complement of$E_m$ to ${\overline{E}}_m$, i.e., $E_m={\overline{E}}_m\oplus {\tilde{E}}_m$.

For $u={(u_1,u_2,\dots ,u_{mT})}^{\mathrm{tr}}\in E_m$ and $x={({\mathrm{△}}^{n-1}{u}_1,{\mathrm{△}}^{n-1}{u}_2,\dots ,{\mathrm{△}}^{n-1}{u}_{mT})}^{\mathrm{tr}}$, we have

$${\parallel x\parallel }_2^q={\left[\sum _{k=1}^{mT}{({\mathrm{△}}^{n-2}{u}_{k+1}-{\mathrm{△}}^{n-2}{u}_k)}^2\right]}^{\frac{q}{2}}\le {\left[{\lambda }_{max}\sum _{k=1}^{mT}{\left({\mathrm{△}}^{n-2}{u}_k\right)}^2\right]}^{\frac{q}{2}}\le {\lambda }_{max}^{\frac{(n-1)q}{2}}{\parallel u\parallel }_2^q.$$

(2.3)

For $u={(u_1,u_2,\dots ,u_{mT})}^{\mathrm{tr}}\in {\tilde{E}}_m$ and $x={({\mathrm{△}}^{n-1}{u}_1,{\mathrm{△}}^{n-1}{u}_2,\dots ,{\mathrm{△}}^{n-1}{u}_{mT})}^{\mathrm{tr}}$, we have

$${\parallel x\parallel }_2^q={\left[\sum _{k=1}^{mT}{({\mathrm{△}}^{n-2}{u}_{k+1}-{\mathrm{△}}^{n-2}{u}_k)}^2\right]}^{\frac{q}{2}}\ge {\left[{\lambda }_{min}\sum _{k=1}^{mT}{\left({\mathrm{△}}^{n-2}{u}_k\right)}^2\right]}^{\frac{q}{2}}\ge {\lambda }_{min}^{\frac{(n-1)q}{2}}{\parallel u\parallel }_2^q.$$

(2.4)

Let H be a Hilbert space and $C^1(H,R)$ denote the set of functionals that are Fréchetdifferentiable and their Fréchet derivatives are continuous on H. Let$J\in C^1(H,R)$. A sequence $\{x_j\}\subset H$ is called a Palais-Smale sequence (P. S. sequence forshort) for J if $\{J(x_j)\}$ is bounded and $J^{\prime }(x_j)\to 0$ as $j\to \mathrm{\infty }$. We say J satisfies the Palais-Smale condition(P. S. condition for short) if any P. S. sequence for J possesses a convergentsubsequence.

Let $B_r$ be the open ball in H with radius r andcenter 0, and let $\partial B_r$ denote its boundary. Lemma 2.1 is taken from [21].

Lemma 2.1 (Linking Theorem)

Let H be a real Hilbert space and$H=H_1\oplus H_2$, where$H_1$is a finite-dimensional subspaceof H. Assume that$J\in C^1(H,R)$satisfies the P. S. condition and thefollowing conditions.

(J₁) There exist constants$a>0$and$\rho >0$such that$J{|}_{\partial B_{\rho }\cap H_2}\ge a$;

(J₂) There exist an$e\in \partial B_1\cap H_2$and a constant$R_0>\rho$such that$J{|}_{\partial Q}\le 0$where$Q=({\overline{B}}_{R_0}\cap H_1)\oplus \{re|0<r<R_0\}$.

Then J possesses a critical value$c\ge a$. Moreover, c can be characterized as

$$c=\underset{h\in \mathrm{\Gamma }}{inf}\underset{x\in Q}{sup}J(h(x)),$$

where$\mathrm{\Gamma }=\{h\in C(\overline{Q},H):h{|}_{\partial Q}=\mathrm{id}{|}_{\partial Q}\}$and$\mathrm{id}{|}_{\partial Q}$is the identity operator on ∂Q.

3 Main results

Let

$$\underline{r}=\underset{k\in Z(1,T)}{min}\{r_k\},\overline{r}=\underset{k\in Z(1,T)}{max}\{r_k\}.$$

Here we give some conditions.

(${\mathrm{\Phi }}_1$) There exist constants ${ϵ}_1>0$, $a_1>0$ and $\mu \ge 1$ such that

$$\mathrm{\Phi }(u)\ge a_1{|u|}^{\mu }\text{for }|u|\le {ϵ}_1.$$

(${\mathrm{\Phi }}_2$) There exist constants ${\delta }_1>0$, $b_1>0$, $c_1\ge 0$ and $\nu \ge 1$ such that

$$\mathrm{\Phi }(u)\le b_1{|u|}^{\nu }+c_1\text{for }|u|\ge {\delta }_1.$$

(F₂) There exist constants ${ϵ}_2>0$, $a_2>0$ and $\theta \ge 1$ such that

$$F(k,v_1,v_2)\le a_2{\left(\sqrt{v_1^2+v_2^2}\right)}^{\theta }\text{for }\sqrt{v_1^2+v_2^2}\le {ϵ}_2.$$

(F₃) There exist constants ${\delta }_2>0$, $b_2>0$, $c_2>0$ and $\vartheta \ge 1$ such that

$$F(k,v_1,v_2)\ge b_2{\left(\sqrt{v_1^2+v_2^2}\right)}^{\vartheta }-c_2\text{for }\sqrt{v_1^2+v_2^2}\ge {\delta }_2.$$

(${\text{H}}_{1,s}$) $\mu =\theta =s$ and $\frac{a_1}{a_2}{(\frac{d_{1,s}}{d_{2,s}})}^s\frac{\underline{r}{\lambda }_{min}^{\frac{ns}{2}}}{2^{\frac{s}{2}}}>1$.

(${\text{H}}_{1,p}$) $\nu =\vartheta =p$ and $\frac{b_1}{b_2}{(\frac{d_{2,p}}{d_{1,p}})}^p\frac{\overline{r}{\lambda }_{max}^{\frac{np}{2}}}{2^{\frac{p}{2}}}<1$.

(${\text{H}}_{2,s}$) $\mu <\theta$.

(${\text{H}}_{2,p}$) $\nu <\vartheta$.

Remark 3.1 By (${\mathrm{\Phi }}_2$) it is easy to see that there exists a constant$c_1^{\prime }>0$ such that

$$\mathrm{\Phi }(u)\le b_1{|u|}^{\nu }+c_1^{\prime },u\in R.$$

(3.1)

Remark 3.2 By (F₃) it is easy to see that there exists a constant$c_2^{\prime }>0$ such that

$$F(k,v_1,v_2)\ge b_2{\left(\sqrt{v_1^2+v_2^2}\right)}^{\vartheta }-c_2^{\prime },(k,v_1,v_2)\in Z\times R^2.$$

(3.2)

Remark 3.3 The p-Laplacian operator given by ${\phi }_p(u)={|u|}^{p-2}u$ ($1<p<\mathrm{\infty }$), the curvature-type operator given by${\varphi }_q(u)=\frac{{|u|}^{q-2}u}{\sqrt{1+{|u|}^q}}$ ($2\le q<\mathrm{\infty }$) and the identity operator given by${\varphi }_I(u)=u$ satisfy (${\mathrm{\Phi }}_1$) and (${\mathrm{\Phi }}_2$).

Our main results are as follows.

Theorem 3.1 Assume that (${\mathrm{\Phi }}_1$), (${\mathrm{\Phi }}_2$), (F₁), (F₂), (F₃)are satisfied. If one of the following four cases is satisfied:

• (1) Assume that (${\text{H}}_{2,s}$) and (${\text{H}}_{2,p}$) are satisfied.

• (2) Assume that (${\text{H}}_{1,s}$) and (${\text{H}}_{1,p}$) are satisfied.

• (3) Assume that (${\text{H}}_{1,s}$) and (${\text{H}}_{2,p}$) are satisfied.

• (4) Assume that (${\text{H}}_{2,s}$) and (${\text{H}}_{1,p}$) are satisfied.

Then for any given positive integer m, (1.1) has at leastthree mT-periodic solutions.

Remark 3.4 If $\varphi (u)={|u|}^{p-2}u$ ($1<p<\mathrm{\infty }$), $r_k=1$ and $n=1$, Theorem 3.1 reduces to Theorem 3.1 in [20].

Remark 3.5 If $\varphi (u)={|u|}^{p-2}u$ ($1<p<\mathrm{\infty }$), Theorem 3.1 reduces to Theorem 1.1 in [17].

Corollary 3.1 Assume that (F₁) and the followingconditions are satisfied.

(${\mathrm{\Phi }}_1^{\prime }$) There exists constant$\mu \ge 1$such that${lim}_{|u|\to 0}\frac{\mathrm{\Phi }(u)}{{|u|}^{\mu }}=d>0$.

(${\mathrm{\Phi }}_2^{\prime }$) There exist constants${\delta }_1>0$and$\nu \ge 1$such that

$$0<\varphi (u)u\le \nu \mathrm{\Phi }(u),|u|\ge {\delta }_1.$$

(${\mathrm{F}}_2^{\prime }$) There exists constant$\theta \ge \mu$such that

$$\underset{(v_1,v_2)\to (0,0)}{lim}\frac{F(k,v_1,v_2)}{{(v_1^2+v_2^2)}^{\frac{\theta }{2}}}=0,(k,v_1,v_2)\in Z\times R^2.$$

(${\mathrm{F}}_3^{\prime }$) There exist constants${\delta }_2>0$and$\vartheta >\nu$such that

$$0<\vartheta F(k,v_1,v_2)\le \frac{\partial F(k,v_1,v_2)}{\partial v_1}{v}_1+\frac{\partial F(k,v_1,v_2)}{\partial v_2}{v}_2,\sqrt{v_1^2+v_2^2}\ge {\delta }_2.$$

Then for any given positive integer m, (1.1) has at leastthree mT-periodic solutions.

4 Proof of the main results

Lemma 4.1 Assume that (${\mathrm{\Phi }}_2$), (F₁), (F₃), and(${\text{H}}_{2,p}$) are satisfied. Then thefunctional J is bounded from above in$E_m$.

Proof By (2.1), (2.3), (3.1), and (3.2), for any $u\in E_m$, we have

$$\begin{array}{rl}J(u)& =\sum _{k=1}^{mT}{r}_{k-1}\mathrm{\Phi }\left({\mathrm{△}}^n{u}_{k-1}\right)-\sum _{k=1}^{mT}F(k,u_{k+1},u_k)\\ \le \sum _{k=1}^{mT}{r}_{k-1}{b}_1{\left|{\mathrm{△}}^n{u}_{k-1}\right|}^{\nu }+mT{c}_1^{\prime }-\sum _{k=1}^{mT}{b}_2{\left(\sqrt{u_{k+1}^2+u_k^2}\right)}^{\vartheta }+mT{c}_2^{\prime }\\ \le b_1\overline{r}{d}_{2,\nu }^{\nu }{\left(\sum _{k=1}^{mT}{\left|{\mathrm{△}}^n{u}_{k-1}\right|}^2\right)}^{\frac{\nu }{2}}-b_2{\left[{\left(\sum _{k=1}^{mT}{\left(\sqrt{u_{k+1}^2+u_k^2}\right)}^{\vartheta }\right)}^{\frac{1}{\vartheta }}\right]}^{\vartheta }+mT(c_1^{\prime }+c_2^{\prime })\\ \le b_1\overline{r}{d}_{2,\nu }^{\nu }{\left(x^{\mathrm{tr}}Mx\right)}^{\frac{\nu }{2}}-b_2{d}_{1,\vartheta }^{\vartheta }{\left(2{\parallel u\parallel }^2\right)}^{\frac{\vartheta }{2}}+mT(c_1^{\prime }+c_2^{\prime })\\ \le b_1\overline{r}{d}_{2,\nu }^{\nu }{\lambda }_{max}^{\frac{\nu }{2}}{\parallel x\parallel }^{\nu }-2^{\frac{\vartheta }{2}}{b}_2{d}_{1,\vartheta }^{\vartheta }{\parallel u\parallel }^{\vartheta }+mT(c_1^{\prime }+c_2^{\prime })\\ \le b_1\overline{r}{d}_{2,\nu }^{\nu }{\lambda }_{max}^{\frac{n\nu }{2}}{\parallel u\parallel }^{\nu }-2^{\frac{\vartheta }{2}}{b}_2{d}_{1,\vartheta }^{\vartheta }{\parallel u\parallel }^{\vartheta }+mT(c_1^{\prime }+c_2^{\prime })\\ \le \underset{\parallel u\parallel \le {\rho }_0}{max}\{b_1\overline{r}{d}_{2,\nu }^{\nu }{\lambda }_{max}^{\frac{n\nu }{2}}{\parallel u\parallel }^{\nu }-2^{\frac{\vartheta }{2}}{b}_2{d}_{1,\vartheta }^{\vartheta }{\parallel u\parallel }^{\vartheta }\}+mT(c_1^{\prime }+c_2^{\prime }),\end{array}$$

(4.1)

where $x={({\mathrm{△}}^{n-1}{u}_1,{\mathrm{△}}^{n-1}{u}_2,\dots ,{\mathrm{△}}^{n-1}{u}_{mT})}^{\mathrm{tr}}$ and ${\rho }_0={(\frac{b_1\overline{r}{d}_{2,\nu }^{\nu }{\lambda }_{max}^{\frac{n\nu }{2}}}{2^{\frac{\vartheta }{2}}{b}_2{d}_{1,\vartheta }^{\vartheta }})}^{\frac{1}{\vartheta -\nu }}$.

The proof of Lemma 4.1 is complete. □

Remark 4.1 The case $mT=1$ is trivial. For the case $mT=2$, M has a different form, namely,

$$M=\left(\begin{array}{cc}2& -2\\ -2& 2\end{array}\right).$$

However, in this special case, the argument need not be changed and we omit it.

Lemma 4.2 Assume that (${\mathrm{\Phi }}_2$), (F₁), (F₃), and(${\text{H}}_{2,p}$) are satisfied. Then thefunctional J satisfies the P. S. conditionin$E_m$.

Proof Let $\{u^{(j)}\}$ be a P. S. sequence, then there exists a positive constant$M_1$ such that

$$-M_1\le J\left(u^{(j)}\right),j\in N.$$

By (4.1), it is easy to see that

$$-M_1\le J\left(u^{(j)}\right)\le b_1\overline{r}{d}_{2,\nu }^{\nu }{\lambda }_{max}^{\frac{n\nu }{2}}{\parallel u^{(j)}\parallel }^{\nu }-2^{\frac{\vartheta }{2}}{b}_2{d}_{1,\vartheta }^{\vartheta }{\parallel u^{(j)}\parallel }^{\vartheta }+mT(c_1^{\prime }+c_2^{\prime }),j\in N.$$

Therefore,

$$2^{\frac{\vartheta }{2}}{b}_2{d}_{1,\vartheta }^{\vartheta }{\parallel u^{(j)}\parallel }^{\vartheta }-b_1\overline{r}{d}_{2,\nu }^{\nu }{\lambda }_{max}^{\frac{n\nu }{2}}{\parallel u^{(j)}\parallel }^{\nu }\le M_1+mT(c_1^{\prime }+c_2^{\prime }),j\in N.$$

Since $\vartheta >\nu$, it is not difficult to see that $\{u^{(j)}\}$ is a bounded sequence in $E_m$. As a consequence, $\{u^{(j)}\}$ possesses a convergence subsequence in$E_m$. Thus the P. S. condition is verified. □

Lemma 4.3 Assume that (${\mathrm{\Phi }}_2$), (F₁), (F₃), and(${\text{H}}_{1,p}$) are satisfied. Then thefunctional J is bounded from above in$E_m$.

Proof Similar to the proof of Lemma 4.1, we have

$$J(u)\le b_1\overline{r}{d}_{2,p}^p{\lambda }_{max}^{\frac{np}{2}}{\parallel u\parallel }^p-2^{\frac{p}{2}}{b}_2{d}_{1,p}^p{\parallel u\parallel }^p+mT(c_1^{\prime }+c_2^{\prime }),$$

(4.2)

where $x={({\mathrm{△}}^{n-1}{u}_1,{\mathrm{△}}^{n-1}{u}_2,\dots ,{\mathrm{△}}^{n-1}{u}_{mT})}^{\mathrm{tr}}$. Since $\frac{b_1}{b_2}{(\frac{d_{2,p}}{d_{1,p}})}^p\frac{\overline{r}{\lambda }_{max}^{\frac{np}{2}}}{2^{\frac{p}{2}}}<1$, we have

$$J(u)\le mT(c_1^{\prime }+c_2^{\prime }).$$

The proof of Lemma 4.3 is complete. □

Lemma 4.4 Assume that (${\mathrm{\Phi }}_2$), (F₁), (F₃), and(${\text{H}}_{1,p}$) are satisfied. Then thefunctional J satisfies the P. S. conditionin$E_m$.

Proof Let $\{u^{(j)}\}$ be a P. S. sequence, then there exists a positive constant$M_2$ such that

$$-M_2\le J\left(u^{(j)}\right),j\in N.$$

By (4.2), it is easy to see that

$$-M_2\le J\left(u^{(j)}\right)\le b_1\overline{r}{d}_{2,p}^p{\lambda }_{max}^{\frac{np}{2}}{\parallel u^{(j)}\parallel }^p-2^{\frac{p}{2}}{b}_2{d}_{1,p}^p{\parallel u^{(j)}\parallel }^p+mT(c_1^{\prime }+c_2^{\prime }),j\in N.$$

Therefore,

$$2^{\frac{p}{2}}{b}_2{d}_{1,p}^p{\parallel u^{(j)}\parallel }^p-b_1\overline{r}{d}_{2,p}^p{\lambda }_{max}^{\frac{np}{2}}{\parallel u^{(j)}\parallel }^p\le M_2+mT(c_1^{\prime }+c_2^{\prime }),j\in N.$$

Since $\frac{b_1}{b_2}{(\frac{d_{2,p}}{d_{1,p}})}^p\frac{\overline{r}{\lambda }_{max}^{\frac{np}{2}}}{2^{\frac{p}{2}}}<1$, we know that $\{u^{(j)}\}$ is a bounded sequence in $E_m$. As a consequence, $\{u^{(j)}\}$ possesses a convergence subsequence in$E_m$. Thus the P. S. condition is verified. □

Proof of Theorem 3.1 Assumptions (F₁) and (F₂) imply that$F(k,0)=0$ and $f(k,0)=0$ for $k\in Z$. Adding $\varphi (0)=0$, then $u=0$ is a trivial mT-periodic solution of (1.1).

By Lemma 4.1 or Lemma 4.3, J is bounded from above on $E_m$. We define ${\alpha }_0={sup}_{u\in E_m}J(u)$. Equation (4.1) implies ${lim}_{\parallel u\parallel \to \mathrm{\infty }}J(u)=-\mathrm{\infty }$. This means that −J is coercive. By thecontinuity of J, there exists $\overline{u}\in E_m$ such that $J(\overline{u})={\alpha }_0$. Clearly, $\overline{u}$ is a critical point of J.

Case 1. Assume that (${\text{H}}_{2,s}$) and (${\text{H}}_{2,p}$) are satisfied. We claim that ${\alpha }_0>0$.

Let

$$\rho =min\{{ϵ}_1{\lambda }_{max}^{-\frac{n}{2}},{ϵ}_2,\frac{1}{2}{\left(\frac{a_1\underline{r}{d}_{1,\mu }^{\mu }{\lambda }_{min}^{\frac{n\mu }{2}}}{2^{\frac{\theta }{2}}{a}_2{d}_{2,\theta }^{\theta }}\right)}^{\frac{1}{\theta -\mu }}\}.$$

By (${\mathrm{\Phi }}_1$), (F₂), and (${\text{H}}_{2,s}$), for any $u\in {\tilde{E}}_m$, $\parallel u\parallel \le \rho$, we have

$$\begin{array}{rl}J(u)& =\sum _{k=1}^{mT}{r}_{k-1}\mathrm{\Phi }\left({\mathrm{△}}^n{u}_{k-1}\right)-\sum _{k=1}^{mT}F(k,u_{k+1},u_k)\\ \ge \sum _{k=1}^{mT}{r}_{k-1}{a}_1{\left|{\mathrm{△}}^n{u}_{k-1}\right|}^{\mu }-\sum _{k=1}^{mT}{a}_2{\left(\sqrt{u_{k+1}^2+u_k^2}\right)}^{\theta }\\ \ge a_1\underline{r}{d}_{1,\mu }^{\mu }{\left(\sum _{k=1}^{mT}{\left|{\mathrm{△}}^n{u}_{k-1}\right|}^2\right)}^{\frac{\mu }{2}}-a_2{\left[{\left(\sum _{k=1}^{mT}{\left(\sqrt{u_{k+1}^2+u_k^2}\right)}^{\theta }\right)}^{\frac{1}{\theta }}\right]}^{\theta }\\ \ge a_1\underline{r}{d}_{1,\mu }^{\mu }{\left(x^{\mathrm{tr}}Mx\right)}^{\frac{\mu }{2}}-a_2{d}_{2,\theta }^{\theta }{\left(2{\parallel u\parallel }^2\right)}^{\frac{\theta }{2}}\ge a_1\underline{r}{d}_{1,\mu }^{\mu }{\lambda }_{min}^{\frac{\mu }{2}}{\parallel x\parallel }^{\mu }-2^{\frac{\theta }{2}}{a}_2{d}_{2,\theta }^{\theta }{\parallel u\parallel }^{\theta }\\ \ge a_1\underline{r}{d}_{1,\mu }^{\mu }{\lambda }_{min}^{\frac{n\mu }{2}}{\parallel u\parallel }^{\mu }-2^{\frac{\theta }{2}}{a}_2{d}_{2,\theta }^{\theta }{\parallel u\parallel }^{\theta },\end{array}$$

(4.3)

where $x={({\mathrm{△}}^{n-1}{u}_1,{\mathrm{△}}^{n-1}{u}_2,\dots ,{\mathrm{△}}^{n-1}{u}_{mT})}^{\mathrm{tr}}$.

Take $\sigma =a_1\underline{r}{d}_{1,\mu }^{\mu }{\lambda }_{min}^{\frac{n\mu }{2}}{\rho }^{\mu }-2^{\frac{\theta }{2}}{a}_2{d}_{2,\theta }^{\theta }{\rho }^{\theta }$. Then $\sigma \ge \frac{1}{2}{a}_1\underline{r}{d}_{1,\mu }^{\mu }{\lambda }_{min}^{\frac{n\mu }{2}}{\rho }^{\mu }>0$ and

$$J(u)\ge \sigma ,u\in {\tilde{E}}_m\cap \partial B_{\rho }.$$

(4.4)

Therefore, ${\alpha }_0={sup}_{u\in E_m}J(u)\ge \sigma >0$. From (4.4), we have also proved that J satisfiesthe condition (J₁) of the Linking Theorem.

For all $u\in {\overline{E}}_m$, we have

$$J(u)=-\sum _{k=1}^{mT}F(k,u_{k+1},u_k)\le 0.$$

Thus, the critical point $\overline{u}$ of J corresponding to the critical value${\alpha }_0$ is a nontrivial mT-periodic solution of (1.1). Inthe following, we will verify the condition (J₂).

Take $e\in \partial B_1\cap {\tilde{E}}_m$, for any $z\in {\overline{E}}_m$ and $r>0$, let $u=re+z$. Then

$$\begin{array}{rl}J(u)& =\sum _{k=1}^{mT}{r}_{k-1}\mathrm{\Phi }\left({\mathrm{△}}^n{u}_{k-1}\right)-\sum _{k=1}^{mT}F(k,u_{k+1},u_k)\\ =\sum _{k=1}^{mT}{r}_{k-1}\mathrm{\Phi }\left(r{\mathrm{△}}^n{e}_{k-1}\right)-\sum _{k=1}^{mT}F(k,u_{k+1},u_k)\\ \le \sum _{k=1}^{mT}{r}_{k-1}{b}_1{\left|r{\mathrm{△}}^n{e}_{k-1}\right|}^{\nu }+mT{c}_1^{\prime }-\sum _{k=1}^{mT}{b}_2{\left(\sqrt{u_{k+1}^2+u_k^2}\right)}^{\vartheta }+mT{c}_2^{\prime }\\ \le b_1\overline{r}{r}^{\nu }{d}_{2,\nu }^{\nu }{\left(\sum _{k=1}^{mT}{\left|{\mathrm{△}}^n{e}_{k-1}\right|}^2\right)}^{\frac{\nu }{2}}-b_2{\left[{\left(\sum _{k=1}^{mT}{\left(\sqrt{u_{k+1}^2+u_k^2}\right)}^{\vartheta }\right)}^{\frac{1}{\vartheta }}\right]}^{\vartheta }+mT(c_1^{\prime }+c_2^{\prime })\\ \le b_1\overline{r}{r}^{\nu }{d}_{2,\nu }^{\nu }{\left(y^{\mathrm{tr}}My\right)}^{\frac{\nu }{2}}-b_2{d}_{1,\vartheta }^{\vartheta }{\left(2{\parallel u\parallel }^2\right)}^{\frac{\vartheta }{2}}+mT(c_1^{\prime }+c_2^{\prime })\\ \le b_1\overline{r}{r}^{\nu }{d}_{2,\nu }^{\nu }{\lambda }_{max}^{\frac{\nu }{2}}{\parallel y\parallel }^{\nu }-2^{\frac{\vartheta }{2}}{b}_2{d}_{1,\vartheta }^{\vartheta }(r^{\vartheta }+{\parallel z\parallel }^{\vartheta })+mT(c_1^{\prime }+c_2^{\prime })\\ \le b_1\overline{r}{d}_{2,\nu }^{\nu }{\lambda }_{max}^{\frac{n\nu }{2}}{r}^{\nu }-2^{\frac{\vartheta }{2}}{b}_2{d}_{1,\vartheta }^{\vartheta }{r}^{\vartheta }-2^{\frac{\vartheta }{2}}{b}_2{d}_{1,\vartheta }^{\vartheta }{\parallel z\parallel }^{\vartheta }+mT(c_1^{\prime }+c_2^{\prime }),\end{array}$$

(4.5)

where $y={({\mathrm{△}}^{n-1}{e}_1,{\mathrm{△}}^{n-1}{e}_2,\dots ,{\mathrm{△}}^{n-1}{e}_{mT})}^{\mathrm{tr}}$.

Let $g_1(t)=b_1\overline{r}{d}_{2,\nu }^{\nu }{\lambda }_{max}^{\frac{n\nu }{2}}{t}^{\nu }-2^{\frac{\vartheta }{2}}{b}_2{d}_{1,\vartheta }^{\vartheta }{t}^{\vartheta }$, $g_2(t)=-2^{\frac{\vartheta }{2}}{b}_2{d}_{1,\vartheta }^{\vartheta }{t}^{\vartheta }+mT(c_1^{\prime }+c_2^{\prime })$. We have ${lim}_{t\to +\mathrm{\infty }}{g}_1(t)=-\mathrm{\infty }$ and ${lim}_{t\to +\mathrm{\infty }}{g}_2(t)=-\mathrm{\infty }$, and $g_1(t)$, $g_2(t)$ are bounded from above, and $J(z)\le 0$ for $z\in {\overline{E}}_m$. Thus there exists a constant $R_0>\rho$ such that $J{|}_{\partial Q}\le 0$ where $Q=({\overline{B}}_{R_0}\cap {\tilde{E}}_m)\oplus \{re|0<r<R_0\}$.

Case 2. Assume that (${\text{H}}_{1,s}$) and (${\text{H}}_{1,p}$) are satisfied. We claim that ${\alpha }_0>0$.

Let $\rho =min\{{ϵ}_1{\lambda }_{max}^{-\frac{n}{2}},{ϵ}_2\}$. By (${\text{H}}_{1,s}$), (${\mathrm{\Phi }}_1$), and (F₂), for any $u\in {\tilde{E}}_m$, $\parallel u\parallel \le \rho$, we have

$$\begin{array}{rl}J(u)& =\sum _{k=1}^{mT}{r}_{k-1}\mathrm{\Phi }\left({\mathrm{△}}^n{u}_{k-1}\right)-\sum _{k=1}^{mT}F(k,u_{k+1},u_k)\\ \ge \sum _{k=1}^{mT}{r}_{k-1}{a}_1{\left|{\mathrm{△}}^n{u}_{k-1}\right|}^s-\sum _{k=1}^{mT}{a}_2{\left(\sqrt{u_{k+1}^2+u_k^2}\right)}^s\\ \ge a_1\underline{r}{d}_{1,s}^s{\left(\sum _{k=1}^{mT}{\left|{\mathrm{△}}^n{u}_{k-1}\right|}^2\right)}^{\frac{s}{2}}-a_2{\left[{\left(\sum _{k=1}^{mT}{\left(\sqrt{u_{k+1}^2+u_k^2}\right)}^s\right)}^{\frac{1}{s}}\right]}^s\\ \ge a_1\underline{r}{d}_{1,s}^s{\left(x^{\mathrm{tr}}Mx\right)}^{\frac{s}{2}}-a_2{d}_{2,s}^s{\left(2{\parallel u\parallel }^2\right)}^{\frac{s}{2}}\ge a_1\underline{r}{d}_{1,s}^s{\lambda }_{min}^{\frac{s}{2}}{\parallel x\parallel }^s-2^{\frac{s}{2}}{a}_2{d}_{2,s}^s{\parallel u\parallel }^s\\ \ge a_1\underline{r}{d}_{1,s}^s{\lambda }_{min}^{\frac{ns}{2}}{\parallel u\parallel }^s-2^{\frac{s}{2}}{a}_2{d}_{2,s}^s{\parallel u\parallel }^s,\end{array}$$

(4.6)

where $x={({\mathrm{△}}^{n-1}{u}_1,{\mathrm{△}}^{n-1}{u}_2,\dots ,{\mathrm{△}}^{n-1}{u}_{mT})}^{\mathrm{tr}}$.

Take $\sigma =a_1\underline{r}{d}_{1,s}^s{\lambda }_{min}^{\frac{ns}{2}}{\rho }^s-2^{\frac{s}{2}}{a}_2{d}_{2,s}^s{\rho }^s$. Then $\sigma \ge 0$ and

$$J(u)\ge \sigma ,u\in {\tilde{E}}_m\cap \partial B_{\rho }.$$

(4.7)

Therefore, ${\alpha }_0={sup}_{u\in E_m}J(u)\ge \sigma >0$. From (4.7), we have also proved that J satisfiesthe condition (J₁) of the Linking Theorem.

For all $u\in {\overline{E}}_m$, we have

$$J(u)=-\sum _{k=1}^{mT}F(k,u_{k+1},u_k)\le 0.$$

Thus, the critical point $\overline{u}$ of J corresponding to the critical value${\alpha }_0$ is a nontrivial mT-periodic solution of (1.1). Inthe following, we will verify the condition (J₂).

Take $e\in \partial B_1\cap {\tilde{E}}_m$, for any $z\in {\overline{E}}_m$ and $r>0$, let $u=re+z$. Then

$$\begin{array}{rl}J(u)& =\sum _{k=1}^{mT}{r}_{k-1}\mathrm{\Phi }\left({\mathrm{△}}^n{u}_{k-1}\right)-\sum _{k=1}^{mT}F(k,u_{k+1},u_k)\\ =\sum _{k=1}^{mT}{r}_{k-1}\mathrm{\Phi }\left(r{\mathrm{△}}^n{e}_{k-1}\right)-\sum _{k=1}^{mT}F(k,u_{k+1},u_k)\\ \le \sum _{k=1}^{mT}{r}_{k-1}{b}_1{\left|r{\mathrm{△}}^n{e}_{k-1}\right|}^p+mT{c}_1^{\prime }-\sum _{k=1}^{mT}{b}_2{\left(\sqrt{u_{k+1}^2+u_k^2}\right)}^p+mT{c}_2^{\prime }\\ \le b_1\overline{r}{r}^p{d}_{2,p}^p{\left(\sum _{k=1}^{mT}{\left|{\mathrm{△}}^n{e}_{k-1}\right|}^2\right)}^{\frac{p}{2}}-b_2{\left[{\left(\sum _{k=1}^{mT}{\left(\sqrt{u_{k+1}^2+u_k^2}\right)}^p\right)}^{\frac{1}{p}}\right]}^p+mT(c_1^{\prime }+c_2^{\prime })\\ \le b_1\overline{r}{r}^p{d}_{2,p}^p{\left(y^{\mathrm{tr}}My\right)}^{\frac{p}{2}}-b_2{d}_{1,p}^p{\left(2{\parallel u\parallel }^2\right)}^{\frac{p}{2}}+mT(c_1^{\prime }+c_2^{\prime })\\ \le b_1\overline{r}{r}^p{d}_{2,p}^p{\lambda }_{max}^{\frac{p}{2}}{\parallel y\parallel }^p-2^{\frac{p}{2}}{b}_2{d}_{1,p}^p(r^p+{\parallel z\parallel }^p)+mT(c_1^{\prime }+c_2^{\prime })\\ \le (b_1\overline{r}{d}_{2,p}^p{\lambda }_{max}^{\frac{np}{2}}-2^{\frac{p}{2}}{b}_2{d}_{1,p}^p)r^p-2^{\frac{p}{2}}{b}_2{d}_{1,p}^p{\parallel z\parallel }^p+mT(c_1^{\prime }+c_2^{\prime }),\end{array}$$

(4.8)

where $y={({\mathrm{△}}^{n-1}{e}_1,{\mathrm{△}}^{n-1}{e}_2,\dots ,{\mathrm{△}}^{n-1}{e}_{mT})}^{\mathrm{tr}}$.

Since $\frac{b_1}{b_2}{(\frac{d_{2,p}}{d_{1,p}})}^p\frac{\overline{r}{\lambda }_{max}^{\frac{np}{2}}}{2^{\frac{p}{2}}}<1$, and $J(z)\le 0$ for $z\in {\overline{E}}_m$, thus there exists a constant $R_0>\rho$ such that $J{|}_{\partial Q}\le 0$ where $Q=({\overline{B}}_{R_0}\cap {\tilde{E}}_m)\oplus \{re|0<r<R_0\}$.

Case 3. Assume that (${\text{H}}_{1,s}$) and (${\text{H}}_{2,p}$) are satisfied. Similar to Case 1, by (4.6), we see that${\alpha }_0>0$. Similar to Case 2, by (4.5), we see that there exists aconstant $R_0>\rho$ such that $J{|}_{\partial Q}\le 0$ where $Q=({\overline{B}}_{R_0}\cap {\tilde{E}}_m)\oplus \{re|0<r<R_0\}$. We have also proved that J satisfies thecondition (J₁) and (J₂) of the Linking Theorem.

Case 4. Assume that (${\text{H}}_{2,s}$) and (${\text{H}}_{1,p}$) are satisfied. Similar to Case 1, by (4.3), we see that${\alpha }_0>0$. Similar to Case 2, by (4.8), we see that there exists aconstant $R_0>\rho$ such that $J{|}_{\partial Q}\le 0$ where $Q=({\overline{B}}_{R_0}\cap {\tilde{E}}_m)\oplus \{re|0<r<R_0\}$. We have also proved that J satisfies thecondition (J₁) and (J₂) of the Linking Theorem.

By one of the above four cases and the Linking Theorem, J possesses criticalvalue $\alpha \ge \sigma >0$. Moreover, α can be characterized as

$$\alpha =\underset{h\in \mathrm{\Gamma }}{inf}\underset{x\in Q}{sup}J(h(x)),$$

where $\mathrm{\Gamma }=\{h\in C(\overline{Q},E_m):h{|}_{\partial Q}=\mathrm{id}{|}_{\partial Q}\}$ and $\mathrm{id}{|}_{\partial Q}$ is the identity operator on ∂Q. Let$\tilde{u}\in E_m$ be a critical point associated to the critical valueα of J, i.e., $J(\tilde{u})=\alpha$. If $\tilde{u}\ne \overline{u}$, then The proof is complete. Otherwise,$\tilde{u}=\overline{u}$. Then ${\alpha }_0=J(\overline{u})=J(\tilde{u})=\alpha$, i.e., ${sup}_{u\in E_m}J(u)={inf}_{h\in \mathrm{\Gamma }}{sup}_{x\in Q}J(h(x))$. Choosing $h=\mathrm{id}$, we have ${sup}_{u\in Q}J(u)={\alpha }_0$. Take $-e\in \partial B_1\cap {\tilde{E}}_m$. Similarly, there exists a positive number$R_1>\rho$, $J{|}_{\partial Q_1}\le 0$, where $Q_1=({\overline{B}}_{R_1}\cap {\tilde{E}}_m)\oplus \{-re|0<r<R_1\}$. Again, by the Linking Theorem, J possesses acritical value ${\alpha }^{\prime }\ge \sigma >0$. Moreover, ${\alpha }^{\prime }$ can be characterized as

$${\alpha }^{\prime }=\underset{h\in {\mathrm{\Gamma }}_1}{inf}\underset{x\in Q_1}{sup}J(h(x)),$$

where ${\mathrm{\Gamma }}_1=\{h\in C({\overline{Q}}_1,E_m):h{|}_{\partial Q_1}=\mathrm{id}{|}_{\partial Q_1}\}$ and $\mathrm{id}{|}_{\partial Q_1}$ is the identity operator on $\partial Q_1$. If ${\alpha }^{\prime }\ne {\alpha }_0$, then the proof is finished. If ${\alpha }^{\prime }={\alpha }_0$, then ${sup}_{u\in Q_1}J(u)={\alpha }_0$. Due to the fact that $J{|}_{\partial Q}\le 0$ and $J{|}_{\partial Q_1}\le 0$, J attains its maximum at some points in theinterior of sets Q and $Q_1$. However, $Q\cap Q_1\subset {\overline{E}}_m$ and $J{|}_{{\overline{E}}_m}\le 0$. Therefore, there must be a point $u^{\prime }\in E_m$, $u^{\prime }\ne \tilde{u}$ and $J(u^{\prime })={\alpha }^{\prime }={\alpha }_0$.

The proof of Theorem 3.1 is complete. □

Proof of Corollary 3.1 By (${\mathrm{\Phi }}_1^{\prime }$), there exists constant ${ϵ}_1>0$ such that $\mathrm{\Phi }(u)\ge \frac{d}{2}{|u|}^{\mu }$, for $|u|\le {ϵ}_1$. Hence (${\mathrm{\Phi }}_1^{\prime }$) implies (${\mathrm{\Phi }}_1$). By (${\mathrm{\Phi }}_2^{\prime }$), there exist constants ${\delta }_1>0$, $b_1>0$ and $c_1\ge 0$ such that $\mathrm{\Phi }(u)\le b_1{|u|}^{\nu }+c_1$, for $|u|\ge {\delta }_1$. So (${\mathrm{\Phi }}_2^{\prime }$) implies (${\mathrm{\Phi }}_2$).

By (${\mathrm{F}}_2^{\prime }$), there exist constants ${ϵ}_2>0$ and $a_2>0$ such that

$$F(k,v_1,v_2)\le a_2{\left(\sqrt{v_1^2+v_2^2}\right)}^{\theta },\sqrt{v_1^2+v_2^2}\le {ϵ}_2.$$

So (${\mathrm{F}}_2^{\prime }$) implies (F₂).

By (${\mathrm{F}}_3^{\prime }$), there exist constants ${\delta }_2>0$, $b_2>0$ and $c_2>0$ such that

$$F(k,v_1,v_2)\ge b_2{\left(\sqrt{v_1^2+v_2^2}\right)}^{\vartheta }-c_2,\sqrt{v_1^2+v_2^2}\ge {\delta }_2.$$

So (${\mathrm{F}}_3^{\prime }$) implies (F₃). Since $\vartheta >\nu$, (${\mathrm{F}}_3^{\prime }$) implies (${\text{H}}_{2,p}$).

If $\theta >\mu$, then (${\mathrm{F}}_2^{\prime }$) implies (${\text{H}}_{2,s}$). If $\theta =\mu =s$, then by (${\mathrm{F}}_2^{\prime }$), there exist constants ${ϵ}_2^{\prime }>0$ and $a_2^{\prime }=a_1{(\frac{d_{1,s}}{d_{2,s}})}^s\frac{\underline{r}{\lambda }_{min}^{\frac{ns}{2}}}{2^{\frac{s+2}{2}}}$ such that

$$F(k,v_1,v_2)\le a_2^{\prime }{\left(\sqrt{v_1^2+v_2^2}\right)}^s,\sqrt{v_1^2+v_2^2}\le {ϵ}_2^{\prime }.$$

we have $\frac{a_1}{a_2^{\prime }}{(\frac{d_{1,s}}{d_{2,s}})}^s\frac{\underline{r}{\lambda }_{min}^{\frac{ns}{2}}}{2^{\frac{s}{2}}}=2>1$. So, if $\theta =\mu =s$, then (${\mathrm{F}}_2^{\prime }$) implies (${\text{H}}_{1,s}$).

So, by Theorem 3.1, Corollary 3.1 holds. □

5 Example

As an application of Theorem 3.1, we give an example to illustrate our result.

Example 5.1 For a given positive integer T, consider the following2n th-order difference equation:

$${\mathrm{△}}^n\left(\frac{{\mathrm{△}}^n{u}_{k-1}}{\sqrt{1+{|{\mathrm{△}}^n{u}_{k-1}|}^2}}\right)={(-1)}^{n}f(k,u_{k+1},u_k,u_{k-1}),n\in Z(1),k\in Z,$$

(5.1)

where

$$f(k,v_1,v_2,v_3)=v_2((2+cos\frac{2\pi k}{T})(v_1^2+v_2^2)+(2+cos\frac{2\pi (k-1)}{T})(v_2^2+v_3^2)).$$

Let

$$F(k,v_1,v_2)=\frac{2+cos\frac{2\pi k}{T}}{4}{(v_1^2+v_2^2)}^2.$$

It is easy to verify that all the assumptions of Theorem 3.1 are satisfied. So, for anygiven positive integer m, (5.1) has at least three mT-periodicsolutions.