The theory of nonlinear difference equations has been widely used to study discrete models in many fields, such as statistics, neural network, computer science, electrical circuit analysis, optimal control, biological models, data classification, and so on.
The existence results on periodic solutions were usually obtained by analytic techniques or various fixed point theorems. In [1, 2], Guo and Yu developed a new method to study the existence and multiplicity of periodic and subharmonic solutions of the second order difference equation via variational methods. In 2005, Zhou et al. [3] applied the same approach for subharmonic solutions of a class of subquadratic Hamiltonian systems. Here we also point out the contribution of Mawhin [4, 5] in the study of second order nonlinear difference systems with φ-Laplacian and periodic potential by using critical point theory.
During the past decade, periodic solutions, subharmonic solutions, and homoclinic orbits for second order discrete Hamiltonian systems have captured special attention, and some solvability conditions have been given under distinct hypotheses on potential function [6–13].
Especially, Yan et al. [12] considered the second order discrete Hamiltonian system
$$ \Delta^{2}x(k-1)+\nabla F\bigl(k,x(k)\bigr)=0,\quad t\in \mathbb{Z}, $$
(1.1)
where \(\mathbb{Z}\) is the set of all integers, \(\Delta x(k)=x(k+1)-x(k)\) is the forward difference, \(\Delta^{2}x(k)=\Delta (\Delta x(k))\), \(F: \mathbb{Z}\times\mathbb{R}^{N}\rightarrow\mathbb{R}\), and \(\nabla F(k,x)\) denotes the gradient of \(F(k,x)\) in x.
In [12], the authors obtained some existence results for system (1.1) with partially periodic potentials and sublinear nonlinearity.
Theorem A
([12])
Suppose that
F
satisfies the following conditions:
-
(F1)
There exists an integer
\(r\in[0,N]\)
such that
\(F(k,x)\)
is
\(T_{i}\)-periodic in
\(x_{i}\), \(1\leq i\leq r\), where
\(x_{i}\)
is the
ith component of
\(x=(x_{1},x_{2},\ldots,x_{N})^{T}\in\mathbb{R}^{N}\).
-
(F2)
There exist constants
\(M_{1}>0\), \(M_{2}>0\), and
\(0\leq \alpha <1\)
such that
$$\bigl\vert \nabla F(k,x) \bigr\vert \leq M_{1} \vert x \vert ^{\alpha}+M_{2} $$
for all
\((k,x)\in\mathbb{Z} [1,T]\times\mathbb{R}^{N}\), where
\(\mathbb{Z} [a,b]: =\mathbb{Z}\cap[a,b]\)
for every
\(a,b\in\mathbb{Z}\)
with
\(a\leq b\).
-
(F3)
\(\lim_{|x|\rightarrow\infty}|x|^{-2\alpha}\sum_{k=1}^{T} F(k,x)=+\infty\), \(x\in\{0\}\times\mathbb{R}^{N-r}\).
Then problem (1.1) possesses at least
\(r+1\)
distinct
T-periodic solutions.
Recently, Jiang et al. [13] extended Theorem A, and they proved the same results under more general coercive condition:
-
(F4)
\(\liminf_{|x|\rightarrow\infty}|x|^{-2\alpha}\sum_{k=1}^{T} F(k,x)>L\), \(x\in\{0\}\times\mathbb{R}^{N-r}\), where L is a positive constant.
Theorem B
([13])
Suppose that
F
satisfies (F1), (F2), and (F4). Then problem (1.1) possesses at least
\(r+1\)
distinct
T-periodic solutions.
In [10, 13], when F satisfies (F1) and \(\nabla F(k,x)\) is growing linearly, that is, there exist constants \(M_{1}>0\) and \(M_{2}>0\), such that
$$\bigl\vert \nabla F(k,x) \bigr\vert \leq M_{1} \vert x \vert +M_{2} $$
for all \((k,x)\in\mathbb{Z} [1,T]\times\mathbb{R}^{N}\), the authors considered the multiple periodic solutions for system (1.1) and got some interesting results.
In recent years, many scholars were interested in difference equations involving the discrete variable exponent Laplacian operator. For instance, the case of homoclinic solutions of a class of \(p(k)\)-Laplacian difference systems was first considered by Chen et al. [14]. The existence of nontrivial homoclinic solutions was obtained by using the mountain pass theorem and the symmetric mountain pass theorem.
In [15], when \(N=1\), Bereanu et al. considered the existence of periodic or Neumann boundary value problems for the discrete \(p(k)\)-Laplacian equations of this type
$$-\Delta\bigl( \bigl\vert \Delta x(k-1) \bigr\vert ^{p(k)-2} \Delta x(k-1) \bigr)=f(k), $$
through the use of Rabinowitz saddle point theorem, where \(p(k): \mathbb {Z} [0,T]\rightarrow(1,+\infty)\) and the nonlinear term \(f(k): \mathbb {Z} [0,T]\rightarrow\mathbb{R}\) is continuous and bounded.
In this paper, we further investigate the existence and multiplicity of periodic solution for the nonautonomous discrete \(p(k)\)-Laplacian system
$$ -\Delta\bigl( \bigl\vert \Delta x(k-1) \bigr\vert ^{p(k)-2} \Delta x(k-1) \bigr) = \nabla F\bigl(k,x(k)\bigr), \quad k\in\mathbb{Z}, $$
(1.2)
where the variable exponent \(p(k): \mathbb{Z} [0,T]\rightarrow (1,+\infty )\) satisfies \(p(0)=p(T)\), T is a positive integer, and \(\Delta x(k)=x(k+1)-x(k)\) is the forward difference operator, F: \(\mathbb{Z}\times\mathbb{R}^{N}\mapsto \mathbb{R}\) is continuously differentiable in x for every \(k\in \mathbb{Z}\) and T-periodic in k for all \(x\in\mathbb{R}^{N}\).
We may think of (1.2) as being a discrete analogue of the following \(p(t)\)-Laplacian system:
$$-\frac{d}{dt}\bigl( \bigl\vert \dot{x}(t) \bigr\vert ^{p(t)-2} \dot{x}(t)\bigr)=\nabla F\bigl(t,x(t)\bigr), $$
where \(\Phi(x)=-\frac{d}{dt}(|\dot{x}(t)|^{p(t)-2}\dot{x}(t))\) is said to be \(p(t)\)-Laplacian.
During the last fifteen years, differential and partial differential equations with variable exponent growth conditions have become increasingly popular. This type of problems has very strong background, the \(p(t)\)-Laplacian systems provide a natural description of the physical phenomena with “pointwise different properties” which first arose from the nonlinear elasticity theory, see [16]. In [17], the authors proposed a framework for image restoration based on a nonhomogeneous \(p(t)\)-Laplacian operator.
In addition, problem (1.2) is also very interesting from a purely mathematical point of view. When the variable exponent \(p(k)\equiv2\), discrete \(p(k)\)-Laplacian system (1.2) becomes the second order discrete Hamiltonian system (1.1), problem (1.2) represents the extension to the variable exponent space setting. The \(p(k)\)-Laplacian operator possesses more complicated nonlinearity than the constant case, for example, it is inhomogeneous, which provokes some mathematical difficulties. We point out that commonly known methods and techniques for studying constant exponent equations fail in the setting of problems involving variable exponents, thus our problem (1.2) is more difficult and more delicate.
Inspired by the above-mentioned papers, the objective of this article is to use a control function \(\omega(|x|)\) instead of \(|x|^{\alpha}\) in conditions (F2), (F3), and (F4). By using the theory of variable exponent Sobolev spaces and the generalized saddle point theorem in [18], we will prove the existence of multiple periodic solutions for (1.2) for a new and large range of nonlinear terms.
Now, we state the assumptions on function F:
-
(F5)
There exist constants \(K_{0}>0\), \(K_{1}>0\), \(K_{2}>0\), \(\alpha\in [0,p^{-}-1)\) and a nonnegative function \(\omega\in C ([0,\infty ),[0,\infty))\) such that
- (\(\omega_{1}\)):
-
\(\omega(s)\leq\omega(t)\), \(\forall s\leq t\), \(s,t\in [0,\infty )\).
- (\(\omega_{2}\)):
-
\(\omega(s+t)\leq K_{0} (\omega(s)+\omega(t))\), \(\forall s,t\in[0,\infty)\).
- (\(\omega_{3}\)):
-
\(0\leq\omega(s)\leq K_{1} s^{\alpha}+K_{2}\), \(\forall s,t\in[0,\infty)\).
- (\(\omega_{4}\)):
-
\(\omega(s)\rightarrow\infty\), as \(s\rightarrow \infty\).
Moreover, there exist \(f,g: \mathbb{Z} [0,T]\rightarrow\mathbb {R}^{+}\) such that
$$\bigl\vert \nabla F(k,x) \bigr\vert \leq f(k) \omega\bigl( \vert x \vert \bigr)+g(k) $$
for all \((k,x)\in\mathbb{Z} [1,T]\times\mathbb{R}^{N}\).
-
(F6)
Let \(\frac{1}{q^{+}}+\frac{1}{p^{-}}=1\), and
$$\liminf_{|x|\rightarrow\infty}\frac{\sum_{k=1}^{T} F(k,x)}{\omega ^{q^{+}}(|x|)}>\frac{p^{-} ( 2K_{0}C_{0}\sum_{k=1}^{T}f(k) )^{q^{+}}}{q^{+}(p^{-}-1)}, $$
as \(x\in\{0\}\times\mathbb{R}^{N-r}\), where \(C_{0}\) is a positive constant.
Our main results are the following theorems.
Theorem 1.1
Suppose that
F
satisfies (F1), (F5), and (F6). Then problem (1.2) possesses at least
\(r+1\)
distinct
T-periodic solutions.
Remark 1.1
Obviously, Theorem 1.1 generalizes Theorem A, which corresponds to the spacial case \(p(k)=2\), \(f(k)=M_{1}\), \(g(k)=M_{2}\) and control function \(\omega(|x|)=|x|^{\alpha}\).
Comparing with the results in [6–11, 13–15], Theorem 1.1 is a different result even in the case \(p(k)=2\). For example, \(x=(x_{1},x_{2},\ldots,x_{N})^{T}\in\mathbb{R}^{N}\), let \(p(k)=2\) and
$$\begin{aligned} F(t,x)&=M_{1} \ln^{\frac{3}{2}} \Biggl[1+ \Biggl(r+1+\sum _{j=1}^{r}\sin ^{2}x_{j}+ \frac{1}{2}\sum_{j=r+1}^{N}x_{j}^{2} \Biggr) \Biggr] \\ &\quad {} +M_{2} \ln \Biggl[1+ \Biggl(r+1+\sum _{j=1}^{r}\sin^{2}x_{j}+ \frac {1}{2}\sum_{j=r+1}^{N}x_{j}^{2} \Biggr) \Biggr], \end{aligned}$$
where \(M_{1}\) and \(M_{2}\) are positive constants. Then F satisfies (F1) with \(T_{i}=\pi\), \(i=1,2,\ldots,r\). Choose
$$f(t)=\frac{3}{2} M_{1},\qquad g(t)=M_{2},\qquad K_{0}=2 $$
and control function
$$\omega\bigl( \vert x \vert \bigr)=\ln^{\frac{1}{2}} \bigl[1+\bigl(r+1+ \vert x \vert ^{2}\bigr) \bigr], $$
it is easy to see that all the conditions of Theorem 1.1 hold, but F is not covered by the results in [6–13].
Remark 1.2
When \(p(k)\equiv2\), (F5) was introduced in [19], which is an extension of the usual sublinear growth condition, that is, there exist \(\alpha\in[0,1)\) and \(f,g: \mathbb{Z} [0,T]\rightarrow \mathbb {R}^{+}\) such that
$$\bigl\vert \nabla F(k,x) \bigr\vert \leq f(k) \vert x \vert ^{\alpha}+g(k) $$
for all \((k,x)\in\mathbb{Z} [1,T]\times\mathbb{R}^{N}\). From (F5), we can see that the nonlinearity \(\nabla F(k,x)\) grows slightly slower than \(|x|^{\alpha}\). Comparing with the results in [19], the periodicity and coercivity conditions in our theorems are only in a part of variables of potentials, and
$$\liminf_{|x|\rightarrow\infty}\frac{\sum_{k=1}^{T} F(k,x)}{\omega ^{q^{+}}(|x|)} $$
has appropriate lower bound.
By Theorem 1.1, it is easy to obtain the following corollary.
Corollary 1.1
Suppose that (F1), (F5) hold and
$$\lim_{|x|\rightarrow\infty}\frac{\sum_{k=1}^{T} F(k,x)}{\omega ^{q^{+}}(|x|)}=+\infty, $$
as
\(x\in\{0\}\times\mathbb{R}^{N-r}\), where
\(\frac{1}{q^{+}}+\frac {1}{p^{-}}=1\). Then problem (1.2) possesses at least
\(r+1\)
distinct
T-periodic solutions.