Multiple Periodic Solutions for Odd Perturbations of the Discrete Relativistic Operator

Abstract

We obtain the existence of multiple pairs of periodic solutions for difference equations of type $-\Delta \left(\frac{\Delta u(n - 1)}{\sqrt{1 - {|\Delta u(n - 1)|}^2}}\right)=\lambda g\left(u\left(n\right)\right)(n\in \mathbb{Z}),$ where $g:\mathbb{R}\to \mathbb{R}$ is a continuous odd function with anticoercive primitive, and $\lambda >0$ is a real parameter. The approach is variational and relies on the critical point theory for convex, lower semicontinuous perturbations of $C^1$-functionals.

1. Introduction

In this note, we are concerned with the multiplicity of solutions for difference equations with relativistic operator of type

$$-\Delta \left[\varphi (\Delta u(n-1\left)\right)\right]=\lambda g\left(u\left(n\right)\right),u\left(n\right)=u(n+T)(n\in \mathbb{Z}),$$

(1)

where $\Delta u\left(n\right)=u(n+1)-u\left(n\right)$ is the usual forward difference operator, $\lambda >0$ is a real parameter, $g:\mathbb{R}\to \mathbb{R}$ is a continuous odd function, and

$${\displaystyle \varphi \left(y\right)=\frac{y}{\sqrt{1-y^2}}(y\in (-1,1)).}$$

In recent years, special attention has been paid to the existence and multiplicity of T-periodic solutions for problems with a discrete relativistic operator. Thus, for instance, in [1,2], variational arguments were employed to prove the solvability of systems of difference equations having the form

$$\Delta \left[{\varphi }_N(\Delta u(n-1))\right]={\nabla }_{u}V(n,u\left(n\right))+h\left(n\right)(n\in \mathbb{Z}),$$

(2)

under various hypotheses upon V and h (coerciveness, growth restriction, convexity or periodicity conditions); here, ${\varphi }_N$ is the N-dimensional variant of $\varphi$, i.e.,

$${\displaystyle {\varphi }_N\left(y\right)=\frac{y}{\sqrt{1-{\left|y\right|}^2}}(y\in {\mathbb{R}}^N,\left|y\right|<1).}$$

The existence of at least $N+1$ geometrically distinct T-periodic solutions of (2) was proved in [3], under the assumptions that h is T-periodic, ${\sum }_{j=1}^{T}h\left(j\right)=0$, and the mapping $V(n,x)$ is T-periodic in n and ${\omega }_i$-periodic $({\omega }_i>0)$ with respect to each $x_i$$(i=1,\dots ,N)$. For the proof, using an idea from the differential case [4], the singular problem (2) was reduced to an equivalent non-singular one to which classical Ljusternik–Schnirelmann category methods can be applied. In addition, under some similar assumptions on V and h, were obtained in [5] using Morse theory, conditions under which system (2) has at least $2^N$ geometrically distinct T-periodic solutions.

The motivation of the present study mainly comes from paper [6], where for problems involving Fisher-Kolmogorov nonlinearities of type

$$-\Delta \left[\varphi (\Delta u(n-1\left)\right)\right]={\lambda u\left(n\right)(1-|u\left(n\right)|}^q),u\left(n\right)=u(n+T)(n\in \mathbb{Z}),$$

(3)

with $q>0$ fixed and $\lambda >0$ a real parameter, it was proved that if $\lambda >8mT$ for some $m\in \mathbb{N}$ with $2\le m\le T$, then problem (3) has at least m distinct pairs of nontrivial solutions. We also refer the interested reader to [6] for a discussion concerning the origin and steps in the study of this type of nonlinearity. In this respect, we shall see in Example 1 below that a sharper result holds true, namely,

(i)

If ${\displaystyle \lambda >8{sin}^2\frac{m\pi }{T}}$ with$0\le m\le \left\{\begin{array}{c}(T-1)/2ifTisodd\hfill \\ (T-2)/2ifTiseven\hfill \end{array}\right.,$then problem (3) has at least $2m+1$ distinct pairs of nontrivial solutions.

(ii)

If T is even and $\lambda >8$, then (3) has at least T distinct pairs of nontrivial solutions.

Moreover, we prove in Theorem 2 that the above statements (i) and (ii) still remain valid for a larger class of periodic problems.

As in [6], our approach to problem (1) is variational and combines a Clark-type abstract result for convex, lower semicontinuous perturbations of $C^1$-functionals, based on Krasnoselskii’s genus. However, our technique here brings the novelty that it exploits the interference of the geometry of the energy functional with fine spectral properties of the operator $-{\Delta }^2$; recall that

$${\Delta }^{2}u(n-1):=\Delta (\Delta u(n-1))=u(n+1)-2u\left(n\right)+u(n-1).$$

It is worth noting that in paper [7] analogous multiplicity results are obtained in the differential case for potential systems involving parametric odd perturbations of the relativistic operator. In addition, we mention the recent paper [8], where the authors obtain the existence and multiplicity of sign-changing solutions for a slightly modified parametric problem of type (1) using bifurcation techniques.

We conclude this introductory part by briefly recalling some topics in the frame of Szulkin’s critical point theory [9], which is needed in the sequel. Let $(Y,\parallel ·\parallel )$ be a real Banach space and $\mathcal{I}:Y\to (-\infty ,+\infty ]$ be a functional having the following structure:

$$\begin{array}{c}\hfill \mathcal{I}=\mathcal{F}+\psi ,\end{array}$$

(4)

where $\mathcal{F}\in C^1(Y,\mathbb{R})$ and $\psi :Y\to (-\infty ,+\infty ]$ is proper, convex and lower semicontinuous. A point $u\in D\left(\psi \right)$ is said to be a critical point of $\mathcal{I}$ if it satisfies the inequality

$$⟨{\mathcal{F}}^{\prime }\left(u\right),v-u⟩+\psi \left(v\right)-\psi \left(u\right)\ge 0\forall v\in D\left(\psi \right).$$

A sequence $\left\{u_n\right\}\subset D\left(\psi \right)$ is called a (PS)-sequence if $\mathcal{I}\left(u_n\right)\to c\in \mathbb{R}$ and

$$⟨{\mathcal{F}}^{\prime }\left(u_n\right),v-u_n⟩+\psi \left(v\right)-\psi \left(u_n\right)\ge -{\epsilon }_n\parallel v-u_n\parallel \forall v\in D\left(\psi \right),$$

where ${\epsilon }_n\to 0$. The functional $\mathcal{I}$ is said to satisfy the (PS) condition if any (PS)-sequence has a convergent subsequence in Y.

Let $\Sigma$ be the collection of all symmetric subsets of $Y\backslash \left\{0\right\}$ which are closed in Y. The genus of a nonempty set $A\in \Sigma$ is defined as being the smallest integer k with the property that there exists an odd continuous mapping $h:A\to {\mathbb{R}}^k\backslash \left\{0\right\}$; in this case, we write $\gamma \left(A\right)=k$. If such an integer does not exist, then $\gamma \left(A\right):=+\infty$. Notice that if $A\in \Sigma$ is homeomorphic to $S^{k-1}$ ($k-1$ dimension unit sphere in the Euclidean space ${\mathbb{R}}^k$) by an odd homeomorphism, then $\gamma \left(A\right)=k$ ([10], Corollary 5.5). For other properties and more details on the notion of genus, we refer the reader to [10,11]. The following theorem is an immediate consequence of ([9], Theorem 4.3).

Theorem 1. 

Let $\mathcal{I}$ be of type (4) with $\mathcal{F}$ and ψ even. In addition, suppose that $\mathcal{I}$ is bounded from below, satisfies the (PS) condition and $\mathcal{I}\left(0\right)=0$. If there exists a nonempty compact symmetric subset $A\subset Y\backslash \left\{0\right\}$ with $\gamma \left(A\right)\ge k$, such that

$$\underset{v\in A}{sup}\mathcal{I}\left(v\right)<0,$$

then the functional $\mathcal{I}$ has at least k distinct pairs of nontrivial critical points.

2. Variational Approach and Preliminaries

To introduce the variational formulation for problem (1), let $H_T$ be the space of all T-periodic $\mathbb{Z}$-sequences in $\mathbb{R}$, i.e., of mappings $u:\mathbb{Z}\to \mathbb{R}$, such that $u\left(n\right)=u(n+T)$ for all $n\in \mathbb{Z}$. On $H_T$, we consider the following inner product and corresponding norm:

$$\left(u\right|v):=\sum _{j=1}^{T}u\left(j\right)v\left(j\right),\parallel u\parallel ={\left(\sum _{j=1}^T{\left|u\left(j\right)\right|}^2\right)}^{1/2},$$

which makes it a Hilbert space. In addition, for each $u\in H_T$, we set

$$\overline{u}:=\frac{1}{T}\sum _{j=1}^{T}u\left(j\right),\tilde{u}:=u-\overline{u}.$$

It is not difficult to check that

$$|\tilde{u}\left(i\right)|\le T^{\frac{1}{2}}{\left(\sum _{j=1}^T{|\Delta u\left(j\right)|}^2\right)}^{1/2}(i\in \{1,\dots ,T\}).$$

(5)

Now, let the closed convex subset K of $H_T$ be defined by

$$K:=\{u\in H_T:|\Delta u{|}_{\infty }\le 1\},$$

where ${|\Delta u|}_{\infty }:={max}_{i=1,\dots ,T}|\Delta u\left(i\right)|.$ Then, from (5), one has

$$|\overline{u}| - T\le |u\left(i\right)| \le |\overline{u}| + T(i\in \{1,\dots ,T\}),$$

(6)

for all $u\in K$. We introduce the even functions

$$\Psi \left(u\right)=\left\{\begin{array}{c}{\displaystyle \sum _{j=1}^T\Phi [\Delta u\left(j\right)],\mathrm{if} u\in K,}\hfill \\ \\ +\infty ,\mathrm{otherwise},\hfill \end{array}\right.$$

where $\Phi \left(y\right)=1-\sqrt{1-y^2}(y\in [-1,1])$ and

$${\mathcal{G}}_{\lambda }\left(u\right)=-\lambda \sum _{j=1}^{T}G\left(u\left(j\right)\right)(u\in H_T),$$

with G the primitive

$$G\left(x\right)={\int }_0^{x}g\left(\tau \right)d\tau (x\in \mathbb{R}).$$

It is not difficult to see that $\Psi$ is convex and lower semicontinuos, while ${\mathcal{G}}_{\lambda }$ is of class $C^1$, its derivative being given by

$$⟨{\mathcal{G}}_{\lambda }^{\prime }\left(u\right),v⟩=-\lambda \sum _{j=1}^{T}g\left(u\left(j\right)\right)v\left(j\right)(u,v\in H_T).$$

Then, the functional $I_{\lambda }:H_T\to (-\infty ,+\infty ]$ associated to (1) is

$$I_{\lambda }=\Psi +{\mathcal{G}}_{\lambda }$$

and it is clear that it has the structure required by Szulkin’s critical point theory. A solution of problem (1) is an element $u\in H_T$ such that $|\Delta u(n\left)\right|<1$, for all $n\in \mathbb{Z}$, which satisfies the equation in (1). The following result reduces the search of solutions of problem (1) to finding critical points of $I_{\lambda }$.

Proposition 1. 

Any critical point of $I_{\lambda }$ is a solution of problem (1).

Proof. 

Let $e\in H_T$. By virtue of Lemmas 5 and 6 in [1], the problem

$$\Delta \left[\varphi (\Delta u(n-1\left)\right)\right]=\overline{u}+e\left(n\right),u\left(n\right)=u(n+T)(n\in \mathbb{Z})$$

has a unique solution $u_e$, which is also the unique solution of the variational inequality

$$\sum _{j=1}^T\left\{\Phi [\Delta v\left(j\right)]-\Phi [\Delta u\left(j\right)]+\overline{u}(\overline{v}-\overline{u})+e\left(j\right)(v\left(j\right)-u\left(j\right))\right\}\ge 0,\forall v\in K$$

(7)

([6], Proposition 3.1). Next, let $w\in K$ be a critical point of $I_{\lambda }$. Then, for any $v\in K,$ one has

$$\sum _{j=1}^T\left\{\Phi [\Delta v(j\left)\right]-\Phi [\Delta w(j\left)\right]-\lambda g\left(w\right(j\left)\right)\left(v\right(j)-w(j\left)\right)\right\}\ge 0,$$

which can be written as

$$\sum _{j=1}^T\{\Phi [\Delta v\left(j\right)]-\Phi [\Delta w\left(j\right)]+\overline{w}(v\left(j\right)-w\left(j\right))\}-\sum _{j=1}^T\left[\lambda g\left(w\left(j\right)\right)+\overline{w}\right](v\left(j\right)-w\left(j\right))\ge 0.$$

Hence, w is a solution of the variational inequality

$$\sum _{j=1}^T\{\Phi [\Delta v\left(j\right)]-\Phi [\Delta w\left(j\right)]+\overline{w}(\overline{v}-\overline{w})+e_w\left(j\right)(v\left(j\right)-w\left(j\right))\}\ge 0,\forall v\in K,$$

(8)

with $e_w\in H_T$ being given by $e_w\left(n\right)=-\lambda g\left(w\left(n\right)\right)-\overline{w}$$(n\in \mathbb{Z})$.

Therefore, by (8) and the uniqueness of the solution of (7), we obtain that, in fact, w solves problem (1). □

Proposition 2. 

If G is anticoercive, i.e.,

$$\underset{\left|x\right|\to +\infty }{lim}G\left(x\right)=-\infty ,$$

(9)

then $I_{\lambda }$ is bounded from below and satisfies the (PS) condition.

Proof. 

From (9) we have that $-G,$ hence ${\mathcal{G}}_{\lambda }$, are bounded from below on $\mathbb{R}$, respectively on $H_T$. This, together with the fact that $\Psi$ is bounded from below, ensure that the same is true for $I_{\lambda }$.

To see that $I_{\lambda }$ satisfies the (PS) condition, let $\left\{u_n\right\}\subset K$ be a (PS)-sequence. Assuming by contradiction that $\left\{\right|{\overline{u}}_n\left|\right\}$ is not bounded, we may suppose, going, if necessary, to a subsequence, that $|{\overline{u}}_n|\to +\infty$. Then, by virtue of (6) and (9), we deduce that $I_{\lambda }\left(u_n\right)\to -\infty$, contradicting the fact that $\left\{I_{\lambda }\left(u_n\right)\right\}$ is convergent. Consequently, $\left\{\right|{\overline{u}}_n\left|\right\}$ is bounded. This, together with $|{\tilde{u}}_n| \le T$ shows that $\left\{u_n\right\}$ is bounded in the finite-dimensional space $H_T$; hence, it contains a convergent subsequence. □

Remark 1. 

Notice that until here in this section, no parity assumptions on the continuous function $g:\mathbb{R}\to \mathbb{R}$ must be required.

We end this section by reviewing some spectral properties of the operator $-{\Delta }^2$, which is needed in the sequel. A real number $\lambda \in \mathbb{R}$ is said to be an eigenvalue of $-{\Delta }^2$ on $H_T$, if there is some $u\in H_T\backslash \{0_{H_{{}_T}}\}$ such that

$$-{\Delta }^{2}u(n-1)=\lambda u\left(n\right),(n\in \mathbb{Z})$$

(10)

and in this case, u is called eigensequence corresponding to the eigenvalue $\lambda$. On account of the periodicity of u, relation (10) is equivalent to the system

$$\left\{\begin{array}{c}-u\left(2\right)+2u\left(1\right)-u\left(T\right)=\lambda u\left(1\right)\hfill \\ -u\left(3\right)+2u\left(2\right)-u\left(1\right)=\lambda u\left(2\right)\hfill \\ ⋮\hfill \\ -u\left(T\right)+2u(T-1)-u(T-2)=\lambda u(T-1)\hfill \\ -u\left(1\right)+2u\left(T\right)-u(T-1)=\lambda u\left(T\right).\hfill \end{array}\right.$$

(11)

If we consider the particular circulant matrix

$$M_T:=\left(\begin{array}{ccccccc}2& -1& 0& \cdots & 0& 0& -1\\ -1& 2& -1& \cdots & 0& 0& 0\\ ⋮\\ 0& 0& 0& \cdots & -1& 2& -1\\ -1& 0& 0& \cdots & 0& -1& 2\end{array}\right)$$

then, having in view (11), the eigenvalues of $-{\Delta }^2$ are precisely the characteristic roots of $M_T.$ In addition, if $y=(y_1,\dots ,y_T)\in {\mathbb{R}}^T\backslash \left\{0_{{\mathbb{R}}^T}\right\}$ is an eigenvector corresponding to a characteristic root $\lambda$, then its extension $u^y\in H_T$, defined by $u^y\left(i\right)=y_i$ for $i=\overline{1,T}$, is an eigensequence corresponding to the eigenvalue $\lambda .$ This means that an orthonormal basis of eigensequences $u^1,\dots ,u^T$ can be constructed from an orthonormal basis of eigenvectors $x^1,\dots ,x^T$ of $M_T$ by extending $x^i$ in $H_T$ ($i=\overline{1,T}$) as above.

From ([12], p. 38), we know that the characteristic roots of $M_T$, hence the eigenvalues of $-{\Delta }^2$, are $4{sin}^{2}i\pi /T$ ($i=\overline{0,T-1}$). We can label them according to the parity of T as follows:

$\underline{T\mathit{odd}}:$

$${\lambda }_0=0,{\lambda }_{2k-1}={\lambda }_{2k}=4{sin}^2\frac{k\pi }{T},k=1,\dots ,\frac{T-1}{2};$$

$\underline{T\mathit{even}}:$

$${\lambda }_0=0,{\lambda }_{2k-1}={\lambda }_{2k}=4{sin}^2\frac{k\pi }{T},k=1,\dots ,\frac{T-2}{2},{\lambda }_{T-1}=4.$$

In both cases, we consider an orthonormal basis $e^0,\dots ,e^{T-1}$ in $H_T$, such that $e^i$ is an eigensequence corresponding to ${\lambda }_i$ ($i=\overline{0,T-1}$). Observe that, by multiplying equality (10) by arbitrary $v\in H_T$ and using summation by parts formula, one obtains that if $u\in H_T$ and $\lambda \in \mathbb{R}$ satisfy (10), then

$$\sum _{j=1}^T\Delta u\left(j\right)\Delta v\left(j\right)=\lambda \left(u\right|v).$$

This yields

$$\sum _{j=1}^T\Delta e^i\left(j\right)\Delta e^k\left(j\right)={\lambda }_k{\delta }_{ik}(i,k\in \{0,\dots ,T-1\}),$$

(12)

where ${\delta }_{ik}$ stands for the Kronecker delta function.

3. Main Result

Our main result is given in the following.

Theorem 2. 

Assume that $g:\mathbb{R}\to \mathbb{R}$ is a continuous odd function and that G satisfies (9) together with

$$\underset{x\to 0}{lim\; inf}\frac{2G\left(x\right)}{x^2}\ge 1.$$

(13)

Then, the following hold true:

(i)

If

$$\lambda >8{sin}^2\frac{m\pi }{T}(=2{\lambda }_{2m})with 0\le m\le \left\{\begin{array}{c}(T-1)/2if T is odd\hfill \\ (T-2)/2if T is even\hfill \end{array}\right.,$$

(14)

then problem (1) has at least $2m+1$ distinct pairs of nontrivial solutions.

(ii)

If T is even and

$$\lambda >8(=2{\lambda }_{T-1}),$$

(15)

then (1) has at least T distinct pairs of nontrivial solutions.

Proof. 

We show $\left(i\right)$ in the odd case because the even case follows by exactly the same arguments, and under assumption (15), a quite similar strategy works by simply replacing “$2m$” with “$T-1$”.

Thus, let $0\le m\le (T-1)/2.$ On account of Theorem 1 and Propositions 1 and 2, we have to prove that there exists a nonempty compact symmetric subset $A_m\subset H_T\backslash \left\{0\right\}$ with $\gamma \left(A_m\right)\ge 2m+1$, such that

$$\underset{v\in A_m}{sup}{I}_{\lambda }\left(v\right)<0.$$

(16)

Since $\lambda >2{\lambda }_{2m}$, we can choose $\epsilon \in (0,1)$, so that $\lambda >2{\lambda }_{2m}/(1-\epsilon )$. Then, by virtue of (13), there exists $\delta >0$ such that

$$2G\left(x\right)\ge (1-\epsilon )x^2\mathrm{as}\left|x\right|\le \delta .$$

(17)

Next, we introduce the set

$$A_m:=\left\{\sum _{k=0}^{2m}{\alpha }_k{e}^k:{\alpha }_0^2+\cdots +{\alpha }_{2m}^2={\rho }^2\right\},$$

where $\rho$ is a positive number, which is chosen ${\displaystyle \le min\left\{\frac{1}{2\sqrt{2m+1}},\delta \right\}}$.

Then, it is not difficult to see that the odd mapping $H:A_m\to S^{2m}$ defined by

$$H\left(\sum _{k=0}^{2m}{\alpha }_k{e}^k\right)=\left(\frac{{\alpha }_0}{\rho },\frac{{\alpha }_1}{\rho }\dots ,\frac{{\alpha }_{2m}}{\rho }\right)$$

is a homeomorphism between $A_m$ and $S^{2m}$; therefore, $\gamma \left(A_m\right)=2m+1$.

We have that $A_m\subset K.$ Indeed, let $v={\sum }_{k=0}^{2m}{\alpha }_k{e}^k\in A_m$. Then, for all $j\in \{1,\dots ,T\}$, we obtain

$$\begin{array}{ccc}\hfill |\Delta v(j\left)\right|& \le & \sum _{k=0}^{2m}\left|{\alpha }_k{e}^k(j+1)\right|+\sum _{k=0}^{2m}\left|{\alpha }_k{e}^k\left(j\right)\right|\le 2\sum _{k=0}^{2m}\left|{\alpha }_k\right|\hfill \\ & \le & 2\sqrt{2m+1}{\left(\sum _{k=0}^{2m}{\alpha }_k^2\right)}^{1/2}=2\rho \sqrt{2m+1}\hfill \end{array}$$

(18)

and since $\rho \le 1/\left(2\sqrt{2m+1}\right)$, one has ${\displaystyle {|\Delta v|}_{\infty }\le 1}$, which shows that $v\in K$. On the other hand, using (12), we obtain

$$\begin{array}{ccc}\hfill \sum _{j=1}^T{|\Delta v\left(j\right)|}^2& =& \sum _{j=1}^T{\left|\Delta \left(\sum _{k=0}^{2m}{\alpha }_k{e}^k\left(j\right)\right)\right|}^2=\sum _{j=1}^T{\left(\sum _{k=0}^{2m}{\alpha }_k\Delta e^k\left(j\right)\right)}^2\hfill \\ & =& \sum _{j=1}^T\left(\sum _{k=0}^{2m}{\alpha }_k^2{(\Delta e^k\left(j\right))}^2+\sum _{\begin{array}{c}i,k=0\\ i\ne k\end{array}}^{2m}{\alpha }_i{\alpha }_k\Delta e^k\left(j\right)\Delta e^i\left(j\right)\right)\hfill \\ & =& \sum _{k=0}^{2m}{\alpha }_k^2\sum _{j=1}^T{(\Delta e^k\left(j\right))}^2+\sum _{\begin{array}{c}i,k=0\\ i\ne k\end{array}}^{2m}{\alpha }_i{\alpha }_k\sum _{j=1}^T\Delta e^k\left(j\right)\Delta e^i\left(j\right)\hfill \\ & =& \sum _{k=0}^{2m}{\lambda }_k{\alpha }_k^2\le {\lambda }_{2m}\sum _{k=0}^{2m}{\alpha }_k^2={\lambda }_{2m}{\rho }^2.\hfill \end{array}$$

(19)

In addition, it is clear that

$$\sum _{j=1}^T{\left|v\left(j\right)\right|}^2={\parallel v\parallel }^2=\left(v\right|v)=\sum _{k=0}^{2m}{\alpha }_k^2={\rho }^2.$$

(20)

Then, from (17), (19), (20) and $\left|v\right(j\left)\right|\le \rho \le \delta$ ($j\in \{1,\dots ,T\}$), it follows that

$$\begin{array}{ccc}\hfill I_{\lambda }\left(v\right)& =& \Psi \left(v\right)+{\mathcal{G}}_{\lambda }\left(v\right)\le \sum _{j=1}^T{|\Delta v\left(j\right)|}^2-\frac{\lambda }{2}(1-\epsilon )\sum _{j=1}^T{\left|v\left(j\right)\right|}^2\hfill \\ & \le & {\rho }^2{\lambda }_{2m}-\frac{\lambda }{2}(1-\epsilon ){\rho }^2={\rho }^2\frac{2{\lambda }_{2m}-\lambda (1-\epsilon )}{2}<0.\hfill \end{array}$$

Therefore, (16) holds true and the proof of $\left(i\right)$ is complete. □

Example 1. 

If (14) holds true, then problem (3) has at least $2m+1$ distinct pairs of nontrivial solutions. In addition, if T is even, under assumption (15), problem (3) has at least T distinct pairs of nontrivial solutions. Notice that besides the trivial solution, problem (3) always has the pair of constant solutions $u\equiv \pm 1$, and these are the only constant nontrivial solutions of (3). Therefore, problem (3) has at least $2m$ (resp. $T-1$) distinct pairs of nonconstant solutions if hypothesis (14) is satisfied (resp. (15) holds true).

Consider the eigenvalue type problem

$$-\Delta \left[\varphi (\Delta u(n-1\left)\right)\right]=\lambda u\left(n\right)+h\left(u\left(n\right)\right),u\left(n\right)=u(n+T)(n\in \mathbb{Z})$$

(21)

and set $H\left(x\right)={\int }_0^{x}h\left(\tau \right)d\tau$$(x\in \mathbb{R}).$

Corollary 1. 

If the continuous function $h:\mathbb{R}\to \mathbb{R}$ is odd and

$$\underset{x\to 0}{lim\; inf}\frac{H\left(x\right)}{x^2}\ge 0,\underset{x\to +\infty }{lim}\frac{H\left(x\right)}{x^2}=-\infty ,$$

then the conclusions $\left(i\right)$ and $\left(ii\right)$ of Theorem 2 remain valid with (21) instead of (1).

Proof. 

Theorem 2 applies to the problem

$$-\Delta \left[\varphi (\Delta u(n-1\left)\right)\right]=\lambda \left(u\left(n\right)+\frac{h\left(u\right(n\left)\right)}{\lambda }\right),u\left(n\right)=u(n+T)(n\in \mathbb{Z}).$$

□

Theorem 2 can be employed to derive the multiplicity of nontrivial solutions of autonomous non-parametric problems having the form

$$-\Delta \left[\varphi (\Delta u(n-1\left)\right)\right]=f\left(u\left(n\right)\right),u\left(n\right)=u(n+T)(n\in \mathbb{Z}).$$

(22)

Setting $F\left(x\right)={\int }_0^{x}f\left(\tau \right)d\tau$ $(x\in \mathbb{R})$, we have the following.

Corollary 2. 

Assume that $f:\mathbb{R}\to \mathbb{R}$ is a continuous odd function and that

$$\underset{x\to +\infty }{lim}F\left(x\right)=-\infty .$$

(23)

Then, the following hold true:

(i)

If

$$\underset{x\to 0}{lim\; inf}\frac{F\left(x\right)}{x^2}>4{sin}^2\frac{m\pi }{T}with 0\le m\le \left\{\begin{array}{c}(T-1)/2if T is odd\hfill \\ (T-2)/2if T is even\hfill \end{array}\right.,$$

(24)

then problem (22) has at least $2m+1$ distinct pairs of nontrivial solutions.

(ii)

If T is even and

$$\underset{x\to 0}{lim\; inf}\frac{F\left(x\right)}{x^2}>4,$$

(25)

then (22) has at least T distinct pairs of nontrivial solutions.

Proof. 

From (24), there exists $\overline{\lambda }>0$ such that

$$\underset{x\to 0}{lim\; inf}\frac{2F\left(x\right)}{x^2}\ge \overline{\lambda }>8{sin}^2\frac{m\pi }{T}$$

and the result follows from Theorem 2 with $g\left(x\right)=f\left(x\right)/\overline{\lambda }$; a similar argument works when (25) is fulfilled. □

Example 2. 

Let $f_a:\mathbb{R}\to \mathbb{R}$ be given by

$$f_a\left(x\right)=2xsin{\left|x\right|}^{-\frac{1}{2}}-\frac{{x\left|x\right|}^{-\frac{1}{2}}cos{\left|x\right|}^{-\frac{1}{2}}}{2}+2ax-4x^3(x\in \mathbb{R}).$$

Then,

$$F_a\left(x\right)=x^2\left({sin\left|x\right|}^{-\frac{1}{2}}+a-x^2\right)(x\in \mathbb{R})$$

and by Corollary 2, we obtain that, if

$$a>1+4{sin}^2\frac{m\pi }{T}with 0\le m\le \left\{\begin{array}{c}(T-1)/2if T is odd\hfill \\ (T-2)/2if T is even\hfill \end{array}\right.,$$

then the equation

$$-\Delta \left[\varphi (\Delta u(n-1\left)\right)\right]=f_a\left(u\left(n\right)\right)(n\in \mathbb{Z})$$

(26)

has at least$2m+1$distinct pairs of nontrivial T-periodic solutions, while if T is even and$a>5$, then (26) has at least T distinct pairs of nontrivial T-periodic solutions.

Remark 2. 

A multiplicity result for odd perturbations of the discrete p-Laplacian operator is obtained in [13] using a Clark-type result in the frame of the classical critical point theory.