Existence of periodic solutions for a class of $(\phi_{1}, \phi_{2})$-Laplacian discrete Hamiltonian systems

1.   Introduction and main results

In this paper, we investigate the existence of periodic solutions for the following system involving classical $(\phi_{1}, \phi_{2})$-Laplacian:

where $\Delta$ is the forward difference operator, $n\in \mathbb{Z}$, $F:\mathbb{Z}\times \mathbb{R}^{N}\times \mathbb{R}^{N}\rightarrow \mathbb{R}$ and $\phi_m$, $m = 1, 2$ satisfy the following condition:

$(\mathcal {A}0)$ $\phi_{m}$ is a homeomorphism from ${\mathbb R}^{N}$ onto ${\mathbb R}^{N}$ such that $\phi_{m}(0) = 0$, $\phi_{m} = \nabla\Phi_{m},$ where $\Phi_{m}\in C^{1}({\mathbb R}^{N}, [0, +\infty))$ strictly convex and $\Phi_{m}(0) = 0$, $m = 1, 2$.

$(\mathcal {F})$ $F(n, x_{1}, x_{2})$ is continuously differentiable in $(x_{1}, x_{2})$, there exist $a_1, a_2\in C(\mathbb{R}^{+}, \mathbb{R}^{+})$, $b:\mathbb{Z}[1, T]\rightarrow \mathbb{R}^{+}$ such that

Remark 1.1. Assumption $(\mathcal {A}0)$ given in ^[1] is used to characterize the classical homeomorphisms. Moreover, if $\Phi_{m}:\mathbb R^{N}\rightarrow \mathbb R$ is coercive ($i.e.$, $\Phi_{m}(x)\rightarrow +\infty$ as $|x|\rightarrow \infty$), then there exists $\delta_{m} = \min_{|x| = 1}\Phi_{m}(x) > 0, m = 1, 2$ such that

In recent years, critical point theory (see ^[2,3,4,5,6]) plays an important role in studying the Hamiltonian systems, nonlinear differential equations, nonlinear difference system, etc (see ^{[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32]}). In ^[1] and ^[7], by virtue of critical point theorem, Mawhin investigated the existence and multiplicity of periodic solutions for the following nonlinear difference systems with $\phi$-Laplacian:

where $\phi$ is a homeomorphism from $X\subset\mathbb{R}^{N}$ onto $Y\subset \mathbb{R}^{N}$ and the following three different types of homeomorphisms were discussed:

(1) classical homeomorphism: if $\phi:\mathbb{R}^N\rightarrow \mathbb{R}^{N}$;

(2) bounded homeomorphism: if $\phi:\mathbb{R}^{N}\rightarrow B_{a}\; (a < +\infty)$;

(3) singular homeomorphism: if $\phi:B_{a}\subset \mathbb{R}^{N}\rightarrow \mathbb{R}^{N}$,

where $B_{a}$ is a ball with its center at origin and radius $a$.

Inspired by ^[1,7], in ^[20], by using some abstract critical point theorems, the authors obtained some multiplicity results of periodic solutions for difference systems involving $(\phi_{1}, \phi_{2})$-Laplacian. In ^[21], by using Clark theorem, the authors obtained that a class of nonlinear difference systems involving classical $(\phi_{1}, \phi_{2})$-Laplacian has at least $m$ distinct pairs of homoclinic solutions. In ^[22], by using an important three critial point theorem, the authors obtained that a class of $(\phi_{1}, \phi_{2})$-Laplacian system has at least three T-periodic solutions. In ^[23], by using the least action principle and saddle point theorem, the authors obtained that system with classical and bounded $(\phi_{1}, \phi_{2})$-Laplacian has at least one periodic solution when $F$ has $(p, q)$-sublinear growth. In ^[24], the authors assumed that the nonlinear term $F$ in system satisfies a corresponding sub-linear growth condition in Orlicz-Sobolev space, by using the least action principle, they obtained that nonlinear and non-homogeneous elliptic system involving $(\phi_{1}, \phi_{2})$-Laplacian has at least a nontrivial solution, and by using the genus theory, obtained that system has infinitely many solutions under an additional symmetric condition.

As $\Phi_{1}(x) = \Phi_{2}(x) = \frac{1}{2}|x|^2$ and $F(n, x, y)\equiv F(n, x)$, system (1.1) reduces to the following second order discrete Hamiltonian system:

In ^[10], Guan and Yang investigated the existence of periodic solutions for system (1.4). By using the variational minimizing method and the saddle point theorem, they obtained that system (1.4) has at least one $T$-periodic solution.

Theorem A. (see ^[10]) Suppose that $F(n, x) = F_1(n, x)+F_2(x)$ satisfies

$(H_{1})$ $F(n, x)\in C^1(\mathbb R^N, \mathbb R)$ for any $n\in \mathbb Z$, $F(n+T, x) = F(n, x)$ for all $(n, x)\in \mathbb Z\times \mathbb R^N$, $T$ is a positive integer;

$(H_{2})$ there exist $f, g:\mathbb Z[1, T]\rightarrow \mathbb R^+$ and $\alpha\in[0, 1)$ such that

$(H_{3})$ there exist constants $r > 0$ and $\gamma\in[0, 2)$ such that

$(H_{4})$ ${\lim \inf}_{|x|\rightarrow +\infty}|x|^{-2\alpha} \sum\limits_{n = 1}^T F(n, x) > \dfrac{1}{8\sin^2\dfrac{\pi}{T}} \sum\limits_{n = 1}^T f^2(n).$

Then system (1.4) has at least one $T$-periodic solution.

Theorem B. (see ^[10]) Suppose that $F(n, x) = F_1(n, x)+F_2(x)$ satisfies $(H_{1})$, $(H_{2})$, $(H_{3})$ and the following conditions:

$(H_{5})$ there exist $\delta\in[0, 2)$ and $C > 0$ such that

$(H_{6})$ ${\lim \sup}_{|x|\rightarrow +\infty}|x|^{-2\alpha} \sum\limits_{n = 1}^T F(n, x) < -\dfrac{3}{8\sin^2\dfrac{\pi}{T}} \sum\limits_{n = 1}^T f^2(n).$

Then system (1.4) possesses at least one $T$-periodic solution.

Motivated by ^{[10,20,21,22,23]}, in this paper, we consider the existence of periodic solutions for system (1.1) and obtain the following main results.

Theorem 1.1. Suppose that $F(n, x_1, x_2) = F_1(n, x_1, x_2)+F_2(x_1, x_2)$ and the following conditions hold:

$(\mathcal {A}1)$ there exist constants $d_{1} > \frac{1}{p}$, $d_{2} > \frac{1}{q}$, $p > 1$ and $q > 1$ such that

$(F_{0})$ $F(n, x_1, x_2)\in C^1(\mathbb{R}^{N}\times\mathbb{R}^{N}, \mathbb{R})$ for every $n\in\mathbb{Z}$, and $F(n+T, x_{1}, x_{2}) = F(n, x_{1}, x_{2})$ for all $(n, x_{1}, x_{2})\in \mathbb{Z}\times\mathbb{R}^{N}\times\mathbb{R}^{N}$;

$(F_{1})$ there exist $f_i, h_i:\mathbb{Z}[1, T]\rightarrow \mathbb{R}^{+}, i = 1, 2$ and $\alpha_1\in[0, p-1)$, $\alpha_2\in[0, q-1)$ such that

for all $(n, x_1, x_2)\in \mathbb Z[1, T]\times \mathbb{R}^{N}\times \mathbb{R}^{N}$;

$(F_{2})$ there exist constants $r_i\in (0, +\infty), i = 1, 2$, $\gamma_1\in [0, p)$ and $\gamma_2\in [0, q)$ such that

and

for all $(x_1, x_2)$, $(y_1, y_2)\in\mathbb{R}^{N}\times\mathbb{R}^{N}$;

$(F_{3})$

where

Then system (1.1) possesses at least one $T$-periodic solution.

Theorem 1.2. Suppose that $F(n, x_1, x_2) = F_1(n, x_1, x_2)+F_2(x_1, x_2)$, $(\mathcal {A}1)$, $(F_{0})$, $(F_{1})$, $(F_{3})$ and the following conditions hold:

$(F_{4})$ there exist constants $r_1\in\big[0, \frac{pd_1-1}{C(p, p')}\big)$, $r_2\in[0, +\infty)$, $r_3\in\big[0, \frac{qd_2-1}{C(q, q')}\big)$, $r_4\in[0, +\infty)$, $\alpha_0\in[0, p)$ and $\beta_0\in[0, q)$ such that

and

for all $(x_1, x_2)$, $(y_1, y_2)\in\mathbb{R}^{N}\times\mathbb{R}^{N}$.

Then system (1.1) possesses at least one $T$-periodic solution.

By Theorem 1.2, it is easy to obtain the following corollary.

Corollary 1.3. Suppose that $F(n, x_1, x_2) = F_1(n, x_1, x_2)+F_2(x_1, x_2)$, $(\mathcal {A}1)$, $(F_{0})$, $(F_{1})$, $(F_{3})$ and the following conditions hold:

$(F_{5})$ there exist constants $r_1\in\big[0, \frac{pd_1-1}{C(p, p')}\big)$, $r_2\in[0, +\infty)$, $r_3\in\big[0, \frac{qd_2-1}{C(q, q')}\big)$, $r_4\in[0, +\infty)$, $\alpha_0\in[0, p)$ and $\beta_0\in[0, q)$ such that

and

for all $(x_1, x_2)$, $(y_1, y_2)\in\mathbb{R}^{N}\times\mathbb{R}^{N}$.

Then system (1.1) possesses at least one $T$-periodic solution.

Remark 1.2. Theorem 1.1, Theorem 1.2 and Corollary 1.3 generalize Theorem A. In fact, when $\Phi_1(x) = \Phi_2(x) = \frac{1}{2}|x|^2$ and $F(n, x, y)\equiv F(n, x)$, System (1.1) reduces to system (1.4) and Theorem 1.1 become Theorem A. Moreover, Theorem 1.2 and Corollary 1.3 is still a new result, which shows that $(F_{2})$ can be weaken to $(F_{4})$.

Theorem 1.4. Suppose that $F(n, x_1, x_2) = F_1(n, x_1, x_2)+F_2(n, x_1, x_2)$, $F_1$ and $F_2$ satisfy $(\mathcal {F})$, $(\mathcal {A}1)$, $(F_{0})$, $(F_{1})$ and the following conditions hold:

$(F_{6})$

for all $(n, x_1, x_2)\in\mathbb{Z}\times\mathbb{R}^{N}\times\mathbb{R}^{N}$ with $|x_1| > 1$, $|x_2| > 1$;

$(F_{7})$ $F_2(n, ., .)$ is $(\lambda, \mu)$-subconvex with $\lambda = (\lambda_1, \lambda_2)$, $\mu = (\mu_1, \mu_2)$ and $\lambda_1 > \frac{1}{p}$, $\mu_1 < 2^{p-1}\lambda_1^p$, $\lambda_2 > \frac{1}{q}$, $\mu_2 < 2^{q-1}\lambda_2^q$, that is

$(F_{8})$

where $M = \max\bigg\{\frac{2^{q\alpha_1}}{q}[C(p, p')]^{\frac{q}{p}} \sum\limits_{n = 1}^T f_1(n)^q, \; \ \ \frac{2^{p\alpha_2}}{p}[C(q, q')]^{\frac{p}{q}} \sum\limits_{n = 1}^T f_2(n)^p\bigg\}.$

Then system (1.1) possesses at least one $T$-periodic solution.

2.   Preliminaries

First, we present some basic notations. We wse $(\cdot, \cdot)$ and $|\cdot|$ to denote the inner product and the Euclidean norm in $\mathbb{R}^{N}$. Let

For $1 < s < +\infty$ and $v\in H_{T}$, we define

For $v\in H_{T}$, set

Let $E = H_{T}\times H_{T}$. For $u = (u_{1}, u_{2})^{\tau}\in E$, define

For any $u\in H_T$, it can be expressed as $u(n) = \overline{u}+\widetilde{u}(n)$, where $\overline{u} = \frac{1}{T} \sum\limits_{n = 1}^{T}u(n)$, and $\sum\limits_{n = 1}^{T}\widetilde{u}(n) = 0$.

Lemma 2.1. (see ^[16])Let $\widetilde{u}(n) = (\widetilde u_{1}(n), \widetilde u_{2}(n))^{\tau}$, then

where $1/s+1/s' = 1$ and $s, s' > 1$.

Lemma 2.2. (see ^[20])For any $u = (u_{1}, u_{2})^{\tau}, \; v = (v_{1}, v_{2})^{\tau}\in E$, the following two equalities hold:

Define

then

and then it is easy to obtain that critical point of $\mathcal{J}$ in $E$ is $T$-periodic solution of system (1.1).

Assume that $E$ is a real Banach space and for $\varphi\in C^{1}(E, \mathbb{R})$, we say that $\varphi$ satisfies the Palais-Smale(PS) condition if any sequence $(u_{n})\subset E$ for which $\varphi(u_{n})$ is bounded and $\varphi'(u_{n})\rightarrow 0$ as $n\rightarrow \infty$ possesses a convergent subsequence.

Lemma 2.3. (see ^[6])Assume that $X$ is a real Banach space, $\varphi \in C^1(X, \mathbb R)$ is bounded from below and satisfies the (PS) condition, then $c = \inf_{u\in X}\varphi(u)$ is a critical value of $\varphi$.

3.   proofs

Proof of Theorem 1.1. It follows from $(F_1)$, Hölder inequality, Young inequality and Lemma 2.1 that

and

Then by $(F_2)$, Hölder inequality and Lemma 2.1, we have

and

Then by $(\mathcal {A}1)$, (3.1), (3.2), (3.3), (3.4), we have

where $M = \max\bigg\{\frac{2^{q\alpha_1}}{q}[C(p, p')]^{\frac{q}{p}} \sum\limits_{n = 1}^T f_1(n)^q, \frac{2^{p\alpha_2}}{p}[C(q, q')]^{\frac{p}{q}} \sum\limits_{n = 1}^T f_2(n)^p\bigg\}$. Note that $(F_3)$, $\alpha_1\in[0, p-1)$, $\alpha_2\in[0, q-1)$, $\gamma_1\in [0, p)$ and $\gamma_2\in [0, q)$. Hence, together with (3.5), implies that

Hence $\mathcal{J}$ is bounded from below and (PS) condition holds. Then by Lemma 2.3, it is easy to know that $\mathcal{J}$ has at least one critical point $u_0$ such that

Thus the proof is complete.

Proof of Theorem 1.2. It follows from $(F_{4})$, Hölder inequality, Young inequality and Lemma 2.1 that

and

Then by $(\mathcal {A}1)$, (3.1), (3.2), (3.7), (3.8), we have

where $M = \max\bigg\{\frac{2^{q\alpha_1}}{q}[C(p, p')]^{\frac{q}{p}} \sum\limits_{n = 1}^T f_1(n)^q, \frac{2^{p\alpha_2}}{p}[C(q, q')]^{\frac{p}{q}} \sum\limits_{n = 1}^T f_2(n)^p\bigg\}$. Note that $(F_3)$, $r_1\in\big[0, \frac{pd_1-1}{C(p, p')}\big)$, $r_3\in\big[0, \frac{qd_2-1}{C(q, q')}\big)$, $\alpha_0\in[0, p)$ and $\beta_0\in[0, q)$. Hence, together with (3.9), implies that

Hence $\mathcal{J}$ is bounded from below and (PS) condition holds. Then by Lemma 2.3, it is easy to know that $\mathcal{J}$ has at least one critical point $u_0$ such that

Thus the proof is complete.

Proof of Theorem 1.3. Let $\beta_i = \log_{2\lambda_i}(2\mu_i)$, $i = 1, 2$ then $\beta_1 < p$, $\beta_2 < q$. For $|x_i| > 1, i = 1, 2$ there exist positive integers $l$, $m$ such that

then

by $(\mathcal {F})$, $(F_{6})$ and $(F_{7})$, we have

for a.e. $n\in[0, T]$ and $|x_1|, |x_2| > 1$, where $a_0 = \max_{0\leq s\leq 1} a_i(s), i = 1, 2$. Then, for a.e. $n\in[0, T]$ and all $|x_1|, |x_2|$, we have

Then by $(\mathcal {A}1)$, (3.1), (3.2), (3.12), we have

where $C_1 = (2\mu_1)^{K-l+1}[\sum\limits_{n = 1}^T b(n)]^{\frac{p-\beta_1}{p}}[C(p, p')]^{\frac{p-\beta_1}{p}}$, $C_2 = (2\mu_1)^{K-m+1}[\sum\limits_{n = 1}^T b(n)]^{\frac{q-\beta_2}{q}}[C(q, q')]^{\frac{q-\beta_2}{q}}$, $M = \max\bigg\{\frac{2^{q\alpha_1}}{q}[C(p, p')]^{\frac{q}{p}} \sum\limits_{n = 1}^T f_1(n)^q, \frac{2^{p\alpha_2}}{p}[C(q, q')]^{\frac{p}{q}} \sum\limits_{n = 1}^T f_2(n)^p\bigg\}$. Hence, together with (3.13), implies that

Hence $\mathcal{J}$ is bounded from below and (PS) condition holds. Then by Lemma 2.3, it is easy to know that $\mathcal{J}$ has at least one critical point $u_0$ such that

Thus the proof is complete.

Acknowledgments

The project is supported by the NSF of Hunan Province, China (No: 2022JJ30463), Research Foundation of Education Bureau of Hunan Province, China (Nos. 19C1465, 21C0649, 22A0540), Huaihua University Scientific Research Project, China (Nos. HHUY2019-3, HHUY2018-12, HHUY2022-11) and the Huaihua University Double First-Class Initiative Applied Characteristic Discipline of Control Science and Engineering.

Conflict of interest

The authors declare that they have no competing interests.