Multiple periodic solutions for a class of nonlinear difference systems with classical or bounded $({\varphi }_1,{\varphi }_2)$-Laplacian

Abstract

In this paper, we consider the multiplicity of periodic solutions for a class of difference systems involving the $({\varphi }_1,{\varphi }_2)$-Laplacian in the cases when the gradient of the nonlinearity has a sublinear growth. By using the variational method, some existence results are obtained. Our results generalize some recent results in (Mawhin in Discrete Contin. Dyn. Syst. 6:1065-1076, 2013).

1 Introduction and main results

Let ℝ denote the real numbers and ℤ the integers. Given $a<b$ in ℤ. Let $\mathbb{Z}[a,b]=\{a,a+1,\dots ,b\}$. Let $T>1$ and N be fixed positive integers.

In this paper, we investigate the multiplicity of periodic solutions for the following nonlinear difference systems:

$$\{\begin{array}{c}\mathrm{\Delta }{\varphi }_1(\mathrm{\Delta }{u}_1(t-1))={\mathrm{\nabla }}_{u_1}F(t,u_1(t),u_2(t))+h_1(t),\hfill \\ \mathrm{\Delta }{\varphi }_2(\mathrm{\Delta }{u}_2(t-1))={\mathrm{\nabla }}_{u_2}F(t,u_1(t),u_2(t))+h_2(t),\hfill \end{array}$$

(1.1)

where $F:\mathbb{Z}\times {\mathbb{R}}^N\times {\mathbb{R}}^N\to \mathbb{R}$ and ${\varphi }_m$, $m=1,2$, satisfy the following condition:

($\mathcal{A}0$) ${\varphi }_m$ is a homeomorphism from ${\mathbb{R}}^N$ onto $B_a\subset {\mathbb{R}}^N$ ($a\in (0,+\mathrm{\infty }]$), such that ${\varphi }_m(0)=0$, ${\varphi }_m=\mathrm{\nabla }{\mathrm{\Phi }}_m$, with ${\mathrm{\Phi }}_m\in C^1({\mathbb{R}}^N,[0,+\mathrm{\infty }])$ strictly convex and ${\mathrm{\Phi }}_m(0)=0$, $m=1,2$.

Remark 1.1 Assumption ($\mathcal{A}0$) is given in [1], which is used to characterize the classical homeomorphism and the bounded homeomorphism. ${\varphi }_m$ is called classical when $a=+\mathrm{\infty }$ and bounded when $a<+\mathrm{\infty }$. If furthermore ${\mathrm{\Phi }}_m:{\mathbb{R}}^N\to \mathbb{R}$ is coercive (i.e. ${\mathrm{\Phi }}_m(x)\to +\mathrm{\infty }$ as $|x|\to \mathrm{\infty }$), there exists ${\delta }_m>0$ such that

$${\mathrm{\Phi }}_m(x)\ge {\delta }_m(|x|-1),x\in {\mathbb{R}}^N,$$

(1.2)

where ${\delta }_m={min}_{|x|=1}{\mathrm{\Phi }}_m(x)$, $m=1,2$ (see [1]).

It is well known that the variational method has been an important tool to study the existence and multiplicity of solutions for various difference systems. Lots of contributions have been obtained (for example, see [1–13]). However, to the best of our knowledge, few people investigated system (1.1). Recently, in [1] and [14], by using the variational approach, Mawhin investigated the following second order nonlinear difference systems with ϕ-Laplacian:

$$\mathrm{\Delta }\varphi [\mathrm{\Delta }u(n-1)]={\mathrm{\nabla }}_{u}F[n,u(n)]+h(n)(n\in \mathbb{Z}),$$

(1.3)

where $\varphi =\mathrm{\nabla }\mathrm{\Phi }$, Φ strictly convex, is a homeomorphism of ${\mathbb{R}}^N$ onto the ball $B_a\subset {\mathbb{R}}^N$ or of $B_a$ onto ${\mathbb{R}}^N$. By using the variational approach, under different conditions, the author found that system (1.3) has at least one or $N+1$ geometrically distinct T-periodic solutions. It is interesting that Mawhin considered three kinds of ϕ: (1) $\varphi :{\mathbb{R}}^N\to {\mathbb{R}}^N$ is a classical homeomorphism, for example, $\varphi (x)={|x|}^{p-1}x$ for some $p>1$ and all $x\in {\mathbb{R}}^N$; (2) $\varphi :{\mathbb{R}}^N\to B_a$ ($a<+\mathrm{\infty }$) is a bounded homeomorphism, for example, $\varphi (x)=\frac{x}{\sqrt{1+{|x|}^2}}\in B_1$ for all $x\in {\mathbb{R}}^N$; (3) $\varphi :B_a\subset {\mathbb{R}}^N\to {\mathbb{R}}^N$ is a singular homeomorphism, for example, $\varphi (x)=\frac{x}{\sqrt{1-{|x|}^2}}$ for all $x\in B_1$.

For a classical and bounded homeomorphism, in [14], Mawhin obtained the following multiplicity results.

Theorem A (see [14], Theorem 4.1)

Assume that the following assumptions hold:

(HB) ϕ is a homeomorphism from ${\mathbb{R}}^N$ onto ${\mathbb{R}}^N$, such that $\varphi (0)=0$, $\varphi =\mathrm{\nabla }\mathrm{\Phi }$, with $\mathrm{\Phi }\in C^1({\mathbb{R}}^N,[0,+\mathrm{\infty }])$ strictly convex and $\mathrm{\Phi }(0)=0$.

(HF) $F\in C(\mathbb{Z}\times {\mathbb{R}}^N,\mathbb{R})$, $F(n,\cdot )\in C^1({\mathbb{R}}^N,\mathbb{R})$, and there exist an integer $T>0$ and real numbers ${\omega }_1>0,{\omega }_2>0,\dots ,{\omega }_N>0$ such that

$$F(t+T,u_1+{\omega }_1,u_2+{\omega }_2,\dots ,u_N+{\omega }_N)=F(t,u_1,u_2,\dots ,u_N)$$

for all $t\in \mathbb{R}$ and $u=(u_1,u_2,\dots ,u_N)\in {\mathbb{R}}^N$.

If there exist $\gamma >0$ and $p>1$ such that

$$|\mathrm{\Phi }(u)|\ge \gamma {|u|}^p(u\in {\mathbb{R}}^N).$$

Then, for any $h\in H_T$ such that $\frac{1}{T}{\sum }_{t=1}^{T}h(t)=0$ (the definition of $H_T$ can be seen in [14]), system (1.3) has at least $N+1$ geometrically distinct T-periodic solutions.

Theorem B (see [14], Theorem 4.2)

Assume that assumption (HF) and the following condition hold:

(HB)′ ϕ is a homeomorphism from ${\mathbb{R}}^N$ onto $B_a\subset {\mathbb{R}}^N$ ($a\in (0,+\mathrm{\infty })$), such that $\varphi (0)=0$, $\varphi =\mathrm{\nabla }\mathrm{\Phi }$, with $\mathrm{\Phi }\in C^1({\mathbb{R}}^N,[0,+\mathrm{\infty }])$ strictly convex and $\mathrm{\Phi }(0)=0$.

If $\mathrm{\Phi }:{\mathbb{R}}^N\to \mathbb{R}$ is coercive, $h\in H_T$ such that $\frac{1}{T}{\sum }_{t=1}^{T}h(t)=0$ and ${|H|}_{\mathrm{\infty }}<\delta$, system (1.3) has at least $N+1$ geometrically distinct T-periodic solutions, where $\delta >0$ is given by (5) in [14]and $H={(H(n))}_{n\in \mathbb{Z}}\in H_T$ is such that $\mathrm{\Delta }H(n)=h(n)$, $n\in \mathbb{Z}$.

Obviously, (HF) implies that F is periodic on all variables $u_1,\dots ,u_N$. Hence, a natural question is that what will occur if F is periodic on some of variables $u_1,\dots ,u_N$. For differential systems, in [15] and [16], the arguments on this question have been given. In [15], Tang and Wu considered the second order Hamiltonian system

$$\{\begin{array}{c}\ddot{u}(t)+\mathrm{\nabla }F(t,u(t))=e(t),\text{a.e. }t\in [0,T],\hfill \\ u(0)-u(T)=\dot{u}(0)-\dot{u}(T)=0,\hfill \end{array}$$

(1.4)

and in [16], Zhang and Tang generalized and improved the results in [15]. They considered the following ordinary p-Laplacian system:

$$\{\begin{array}{c}{({|u^{\prime }(t)|}^{p-2}{u}^{\prime }(t))}^{\prime }+\mathrm{\nabla }F(t,u(t))=e(t),\text{a.e. }t\in [0,T],\hfill \\ u(0)-u(T)=\dot{u}(0)-\dot{u}(T)=0.\hfill \end{array}$$

(1.5)

Inspired by [1, 14, 15] and [16], in this paper, we investigate system (1.1), which is different from (1.3), and consider the case that $F(t,x_1,x_2)$ is periodic on some of the variables $x_1^{(1)},\dots ,x_N^{(1)}$ and some of the variables $x_1^{(1)},\dots ,x_N^{(2)}$, where $x_1={(x_1^{(1)},\dots ,x_N^{(1)})}^{\tau }$ and $x_2={(x_1^{(1)},\dots ,x_N^{(2)})}^{\tau }$. We generalize Theorem A and Theorem B.

Next, in order to present our main results, we consider two decompositions ${\mathbb{R}}^N={\mathcal{R}}_1\oplus {\mathcal{S}}_1$ and ${\mathbb{R}}^N={\mathcal{R}}_2\oplus {\mathcal{S}}_2$ with

$$\begin{array}{r}{\mathcal{R}}_1=span〈e_{i_1},\dots ,e_{i_{r_1}}〉,{\mathcal{S}}_1=span〈e_{i_{r_1+1}},\dots ,e_{i_N}〉,\\ {\mathcal{R}}_2=span〈e_{j_1},\dots ,e_{j_{r_2}}〉,{\mathcal{S}}_2=span〈e_{j_{r_2+1}},\dots ,e_{j_N}〉,\end{array}$$

where $e_{i_k}$ and $e_{j_s}$ are the canonical basis of ${\mathbb{R}}^N$ for $1\le k\le N$, $1\le s\le N$, $1\le r_1\le N$, and $1\le r_2\le N$.

In this paper, we make the following assumptions:

($\mathcal{A}1$) Let $p>1$, $q>1$, ${\beta }_1\in [0,p)$, and ${\beta }_2\in [0,q)$. Assume that there exist positive constants ${\gamma }_1$, ${\gamma }_2$, ${\gamma }_3$, ${\gamma }_4$ such that

$${\mathrm{\Phi }}_1(x)\ge {\gamma }_1{|x|}^p-{\gamma }_2{|x|}^{{\beta }_1},{\mathrm{\Phi }}_2(y)\ge {\gamma }_3{|y|}^q-{\gamma }_4{|y|}^{{\beta }_2},\mathrm{\forall }x,y\in {\mathbb{R}}^N.$$

($\mathcal{A}2$) There exist positive constants $d_1$, $d_2$, $d_3$, $d_4$ with $d_1>\frac{1}{p}$ and $d_3>\frac{1}{q}$, ${\beta }_3\in [0,p)$, and ${\beta }_4\in [0,q)$ such that

$$({\varphi }_1(x),x)\ge d_1{|x|}^p-d_2{|x|}^{{\beta }_3},({\varphi }_2(x),x)\ge d_3{|x|}^q-d_4{|x|}^{{\beta }_4},\mathrm{\forall }x\in {\mathbb{R}}^N.$$

($\mathcal{A}3$) There exist constants $c_{m0}>0$, $k_{m1}>0$, $k_{m2}>0$, ${\alpha }_1\in [0,p-1)$, ${\alpha }_2\in [0,q-1)$, and two nonnegative functions $w_m\in C([0,+\mathrm{\infty }),[0,+\mathrm{\infty }))$, where $m=1,2$, with the properties:

• (i) $w_m(s)\le w_m(t)$ $\mathrm{\forall }s\le t$, $s,t\in [0,+\mathrm{\infty })$,

• (ii) $w_m(s+t)\le c_{m0}(w_m(s)+w_m(t))$ $\mathrm{\forall }s,t\in [0,+\mathrm{\infty })$,

• (iii) $0\le w_1(t)\le k_{11}{t}^{{\alpha }_1}+k_{12}$, $0\le w_2(t)\le k_{21}{t}^{{\alpha }_2}+k_{22}$, $\mathrm{\forall }t\in [0,+\mathrm{\infty })$,

• (iv) $w_m(t)\to +\mathrm{\infty }$, as $t\to +\mathrm{\infty }$.

($\mathcal{F}1$) $F:\mathbb{Z}\times {\mathbb{R}}^N\times {\mathbb{R}}^N⟶{\mathbb{R}}^N$, $(t,x_1,x_2)⟶F(t,x_1,x_2)$ is T-periodic in t for all $(x_1,x_2)\in {\mathbb{R}}^N\times {\mathbb{R}}^N$ and continuously differentiable in $(x_1,x_2)$ for every $t\in \mathbb{Z}[1,T]$, where $x_1={(x_1^{(1)},\dots ,x_N^{(1)})}^{\tau }$, $x_2={(x_1^{(2)},\dots ,x_N^{(2)})}^{\tau }$.

($\mathcal{F}2$) $F(t,x_1,x_2)$ is $T_{i_k}^{(1)}$-periodic in $x_{i_k}^{(1)}$, where $x_{i_k}^{(1)}$ is a component of vector $x_1$ and $T_{i_k}^{(1)}>0$, $1\le k\le r_1$, and $T_{j_s}^{(2)}$-periodic in $x_{j_s}^{(2)}$, where $x_{j_s}^{(2)}$ is a component of vector $x_2$ and $T_{j_s}^{(2)}>0$, $1\le s\le r_2$.

($\mathcal{F}3$) There exist $f_m,g_m:\mathbb{Z}[1,T]\to \mathbb{R}$, $m=1,2$, such that

$$\begin{array}{r}|{\mathrm{\nabla }}_{x_1}F(t,x_1,x_2)|\le f_1(t)w_1(|x_1|)+g_1(t),\\ |{\mathrm{\nabla }}_{x_2}F(t,x_1,x_2)|\le f_2(t)w_2(|x_2|)+g_2(t)\end{array}$$

for all $(x_1,x_2)\in {\mathbb{R}}^N\times {\mathbb{R}}^N$ and $t\in \mathbb{Z}[1,T]$.(ℰ)

$$\sum _{t=1}^T{h}_1(t)=\sum _{t=1}^T{h}_2(t)=0.$$

Remark 1.2 A condition similar to ($\mathcal{A}3$) and ($\mathcal{F}3$) was given first in [17] for the second order Hamiltonian systems

$$\{\begin{array}{c}\ddot{u}(t)=\mathrm{\nabla }F(t,u(t)),\hfill \\ u(0)-u(T)=\dot{u}(0)-\dot{u}(T).\hfill \end{array}$$

(1.6)

The condition presented some advantages over the following subquadratic condition: there exist $\alpha \in [0,1)$ and $f,g\in L^1([0,T];{\mathbb{R}}^N)$ such that

$$|\mathrm{\nabla }F(t,x)|\le f(t){|x|}^{\alpha }+g(t).$$

We refer readers to [17] for more details.

Moreover, assume that $p^{\prime }>1$ and $q^{\prime }>1$ satisfying $1/p+1/p^{\prime }=1$ and $1/q+1/q^{\prime }=1$. Let

$$C\left(p^{\prime }\right)=min\{\frac{{(T-1)}^{(p^{\prime }+1)/p^{\prime }}}{T},{\left(\frac{{(T+1)}^{p^{\prime }+1}-2}{T^{p^{\prime }}(p^{\prime }+1)}\right)}^{1/p^{\prime }}\},$$

(1.7)

$$C\left(q^{\prime }\right)=min\{\frac{{(T-1)}^{(q^{\prime }+1)/q^{\prime }}}{T},{\left(\frac{{(T+1)}^{q^{\prime }+1}-2}{T^{q^{\prime }}(q^{\prime }+1)}\right)}^{1/q^{\prime }}\},$$

(1.8)

$$C(p,p^{\prime })=min\{\frac{{(T-1)}^{2p-1}}{T^{p-1}},\frac{T^{p-1}\mathrm{\Theta }(p^{\prime },p)}{{(p^{\prime }+1)}^{p/p^{\prime }}}\},$$

(1.9)

$$\begin{array}{r}C(q,q^{\prime })=min\{\frac{{(T-1)}^{2q-1}}{T^{q-1}},\frac{T^{q-1}\mathrm{\Theta }(q^{\prime },q)}{{(q^{\prime }+1)}^{q/q^{\prime }}}\},\\ \mathrm{\Theta }(p^{\prime },p)=\sum _{t=1}^T{[{\left(\frac{t}{T}\right)}^{p^{\prime }+1}+{(1-\frac{t}{T}+\frac{1}{T})}^{p^{\prime }+1}-\frac{2}{T^{p^{\prime }+1}}]}^{p/p^{\prime }},\\ \mathrm{\Theta }(q^{\prime },q)=\sum _{t=1}^T{[{\left(\frac{t}{T}\right)}^{q^{\prime }+1}+{(1-\frac{t}{T}+\frac{1}{T})}^{q^{\prime }+1}-\frac{2}{T^{q^{\prime }+1}}]}^{q/q^{\prime }}.\end{array}$$

(1.10)

Next, we present our main results.

(I) For classical homeomorphism

Theorem 1.1 Assume that ($\mathcal{A}0$) with $a=+\mathrm{\infty }$, ($\mathcal{A}1$), ($\mathcal{A}3$), ($\mathcal{F}1$)-($\mathcal{F}3$), and (ℰ) hold. Assume that F satisfies the following condition:

($\mathcal{F}4$) For $(x_1,x_2)\in {\mathcal{S}}_1\times {\mathcal{S}}_2$,

$$\begin{array}{rcl}\underset{|x_1|+|x_2|\to +\mathrm{\infty }}{lim}\frac{{\sum }_{t=1}^{T}F(t,x_1,x_2)}{w_1^{p^{\prime }}(|x_1|)+w_2^{q^{\prime }}(|x_2|)}& >& max\{\frac{{[c_{10}C(p^{\prime })]}^{p^{\prime }}}{{[p{\gamma }_1]}^{p^{\prime }-1}{p}^{\prime }}{(\sum _{t=1}^T{f}_1(t))}^{p^{\prime }},\\ \frac{{[c_{20}C(q^{\prime })]}^{q^{\prime }}}{{[q{\gamma }_3]}^{q^{\prime }-1}{q}^{\prime }}{(\sum _{t=1}^T{f}_2(t))}^{q^{\prime }}\}.\end{array}$$

Then system (1.1) has at least $r_1+r_2+1$ geometrically distinct solutions in ℋ, where the definition of ℋ is given in Section  2 below.

Theorem 1.2 Assume that ($\mathcal{A}0$) with $a=+\mathrm{\infty }$, ($\mathcal{A}1$), ($\mathcal{A}2$), ($\mathcal{A}3$), ($\mathcal{F}1$)-($\mathcal{F}3$), and (ℰ) hold. Assume that F satisfies the following condition:

($\mathcal{A}1$)′ Let ${\theta }_1\in [0,p)$ and ${\theta }_2\in [0,q)$. Assume that there exist positive constants ${\zeta }_1$, ${\zeta }_2$, ${\zeta }_3$, ${\zeta }_4$ such that

$${\mathrm{\Phi }}_1(x)\le {\zeta }_1{|x|}^p+{\zeta }_2{|x|}^{{\theta }_1},{\mathrm{\Phi }}_2(y)\le {\zeta }_3{|y|}^q+{\zeta }_4{|y|}^{{\theta }_2},\mathrm{\forall }x,y\in {\mathbb{R}}^N;$$

($\mathcal{F}4$)′ For $(x_1,x_2)\in {\mathcal{S}}_1\times {\mathcal{S}}_2$,

$$\begin{array}{r}\underset{|x_1|+|x_2|\to +\mathrm{\infty }}{lim}\frac{{\sum }_{t=1}^{T}F(t,x_1,x_2)}{w_1^{p^{\prime }}(|x_1|)+w_2^{q^{\prime }}(|x_2|)}\\ <-max\{\frac{{[C(p^{\prime })c_{10}]}^{p^{\prime }}}{p^{\prime }}[\frac{1+p{\zeta }_1}{d_{1}p-1}+\frac{1+q{\zeta }_3}{d_{3}q-1}+1]{(\sum _{t=1}^T{f}_1(t))}^{p^{\prime }},\\ \frac{{[C(q^{\prime })c_{20}]}^{q^{\prime }}}{q^{\prime }}[\frac{1+p{\zeta }_1}{d_{1}p-1}+\frac{1+q{\zeta }_3}{d_{3}q-1}+1]{(\sum _{t=1}^T{f}_2(t))}^{q^{\prime }}\}.\end{array}$$

Then system (1.1) has at least $r_1+r_2+1$ geometrically distinct solutions in ℋ.

(II) For bounded homeomorphism

Theorem 1.3 Assume that ($\mathcal{A}0$) with $a<+\mathrm{\infty }$, ${\mathrm{\Phi }}_m:{\mathbb{R}}^N\to \mathbb{R}$ are coercive, $m=1,2$, ($\mathcal{F}1$), ($\mathcal{F}2$), and (ℰ) hold. Assume that F satisfies the following conditions:

($\mathcal{F}5$) There exists a nonnegative $b_m:\mathbb{Z}[1,T]\to {\mathbb{R}}^{+}$, $m=1,2$, such that

$$\begin{array}{r}|{\mathrm{\nabla }}_{x_1}F(t,x_1,x_2)|\le b_1(t),\\ |{\mathrm{\nabla }}_{x_2}F(t,x_1,x_2)|\le b_2(t)\end{array}$$

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

($\mathcal{F}6$) For $(x_1,x_2)\in {\mathcal{S}}_1\times {\mathcal{S}}_2$,

$$\underset{|x_1|+|x_2|\to +\mathrm{\infty }}{lim}\sum _{t=1}^{T}F(t,x_1,x_2)=+\mathrm{\infty };$$

($\mathcal{F}7$)

$$\begin{array}{r}\sum _{t=1}^T{b}_1(t)+\sum _{t=1}^T|h_1(t)|<\frac{{\delta }_1}{C(p^{\prime })},\\ \sum _{t=1}^T{b}_2(t)+\sum _{t=1}^T|h_2(t)|<\frac{{\delta }_2}{C(q^{\prime })},\end{array}$$

where ${\delta }_m$, $m=1,2$ are given in (1.2). Then system (1.1) has at least $r_1+r_2+1$ geometrically distinct solutions in ℋ.

Theorem 1.4 Assume that ($\mathcal{A}0$) with $a<+\mathrm{\infty }$, ${\mathrm{\Phi }}_m:{\mathbb{R}}^N\to \mathbb{R}$ are coercive, $m=1,2$, ($\mathcal{F}1$), ($\mathcal{F}2$), ($\mathcal{F}5$), and (ℰ) hold. If F satisfies the following conditions:

($\mathcal{F}6$)′ For $(x_1,x_2)\in {\mathcal{S}}_1\times {\mathcal{S}}_2$,

$$\underset{|x_1|+|x_2|\to +\mathrm{\infty }}{lim}\sum _{t=1}^{T}F(t,x_1,x_2)=-\mathrm{\infty };$$

($\mathcal{F}7$)′

$$\begin{array}{r}C\left(p^{\prime }\right)\sum _{t=1}^T{b}_1(t)+C\left(p^{\prime }\right)\sum _{t=1}^T|h_1(t)|+{(C(p,p^{\prime })+1)}^{1/p}<{\delta }_1,\\ C\left(q^{\prime }\right)\sum _{t=1}^T{b}_2(t)+C\left(q^{\prime }\right)\sum _{t=1}^T|h_2(t)|+{(C(q,q^{\prime })+1)}^{1/q}<{\delta }_2,\end{array}$$

where ${\delta }_m$, $m=1,2$ are given in (1.2), then system (1.1) has at least $r_1+r_2+1$ geometrically distinct solutions in ℋ.

2 Preliminaries

First, we present some basic notations. We use $|\cdot |$ to denote the usual Euclidean norm in ${\mathbb{R}}^N$. Define

$$\begin{array}{rcl}\mathcal{V}& =& \{u={(u_1,u_2)}^{\tau }=\{u(t)\}|u(t)={(u_1(t),u_2(t))}^{\tau }\in {\mathbb{R}}^{2N},\\ u_m=\{u_m(t)\},u_m(t)\in {\mathbb{R}}^N,m=1,2,t\in \mathbb{Z}\}.\end{array}$$

ℋ is defined as a subspace of $\mathcal{V}$ by

$$\mathcal{H}=\{u=\{u(t)\}\in \mathcal{V}|u(t+T)=u(t),t\in \mathbb{Z}\}.$$

Define

$${\mathcal{H}}_m=\{u_m=\{u_m(t)\}|u_m(t+T)=u_m(t),u_m(t)\in {\mathbb{R}}^N,t\in \mathbb{Z}\},m=1,2.$$

Then $\mathcal{H}={\mathcal{H}}_1\times {\mathcal{H}}_2$. For $u_m\in {\mathcal{H}}_m$, set

$${\parallel u_m\parallel }_r={\left(\sum _{t=1}^T{|u_m(t)|}^r\right)}^{1/r}\text{and}{\parallel u_m\parallel }_{\mathrm{\infty }}=\underset{t\in \mathbb{Z}[1,T]}{max}|u_m(t)|,m=1,2,r>1.$$

Obviously, we have

$${\parallel u_m\parallel }_{\mathrm{\infty }}\le {\parallel u_m\parallel }_2,m=1,2.$$

(2.1)

For $1<p,q<+\mathrm{\infty }$, on ${\mathcal{H}}_1$, we define

$${\parallel u_1\parallel }_p={(\sum _{t=1}^T{|\mathrm{\Delta }{u}_1(t)|}^p+\sum _{t=1}^T{|u_1(t)|}^p)}^{1/p}$$

and, on ${\mathcal{H}}_2$, we define

$${\parallel u_2\parallel }_q={(\sum _{t=1}^T{|\mathrm{\Delta }{u}_2(t)|}^q+\sum _{t=1}^T{|u_2(t)|}^q)}^{1/q}.$$

For $u={(u_1,u_2)}^{\tau }\in \mathcal{H}$, we define

$$\parallel u\parallel ={\parallel u_1\parallel }_p+{\parallel u_2\parallel }_q.$$

Let

$$\mathcal{W}=\{u={(u_1,u_2)}^{\tau }\in \mathcal{H}|u_m(1)=\cdots =u_m(T)=\frac{1}{T}\sum _{t=1}^T{u}_m(t),m=1,2\}$$

and

$$\tilde{\mathcal{H}}=\{u={(u_1,u_2)}^{\tau }\in \mathcal{H}|\sum _{t=1}^T{u}_m(t)=0,m=1,2\}.$$

Then ℋ can be decomposed into the direct sum $\mathcal{H}=\mathcal{W}\oplus \tilde{\mathcal{H}}$. So, for any $u\in \mathcal{H}$, u can be expressed in the form $u=\tilde{u}+\overline{u}$, where $\tilde{u}={({\tilde{u}}_1,{\tilde{u}}_2)}^{\tau }\in \mathcal{V}$ and $\overline{u}={({\overline{u}}_1,{\overline{u}}_2)}^{\tau }\in \mathcal{W}$. Obviously, $u_m={\tilde{u}}_m+{\overline{u}}_m$, $m=1,2$.

For $u={(u_1,u_2)}^{\tau }\in \tilde{\mathcal{H}}$, let

$${\parallel \mathrm{\Delta }{u}_m\parallel }_r={\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_m(t)|}^r\right)}^{1/r},$$

where $m=1,2$, $r>1$. It is easy to verify that

$$\parallel \mathrm{\Delta }u\parallel ={\parallel \mathrm{\Delta }{u}_1\parallel }_p+{\parallel \mathrm{\Delta }{u}_2\parallel }_q$$

is also a norm on $\tilde{\mathcal{H}}$. Since $\tilde{\mathcal{H}}$ is finite-dimensional, the norm $\parallel \mathrm{\Delta }u\parallel$ is equivalent to the norm $\parallel u\parallel$ in ℋ if $u\in \tilde{\mathcal{H}}$.

Lemma 2.1 (see [12])

Let $u=(u_1,u_2)\in \tilde{\mathcal{H}}$. Then

$$\underset{t\in \mathbb{Z}[1,T]}{max}|u_m(t)|\le C\left(p^{\prime }\right){\left(\sum _{s=1}^T{|\mathrm{\Delta }{u}_m(s)|}^p\right)}^{1/p},m=1,2,$$

(2.2)

$$\underset{t\in \mathbb{Z}[1,T]}{max}|u_m(t)|\le C\left(q^{\prime }\right){\left(\sum _{s=1}^T{|\mathrm{\Delta }{u}_m(s)|}^q\right)}^{1/q},m=1,2,$$

(2.3)

and

$$\sum _{t=1}^T{|u_m(t)|}^p\le C(p,p^{\prime })\sum _{s=1}^T{|\mathrm{\Delta }{u}_m(s)|}^p,m=1,2,$$

(2.4)

$$\sum _{t=1}^T{|u_m(t)|}^q\le C(q,q^{\prime })\sum _{s=1}^T{|\mathrm{\Delta }{u}_m(s)|}^q,m=1,2,$$

(2.5)

where $C(p^{\prime })$, $C(q^{\prime })$, $C(p,p^{\prime })$, and $C(q,q^{\prime })$ are defined by (1.7)-(1.10).

Lemma 2.2 (see [16])

Let $a>0$, $b,c\ge 0$, $\epsilon >0$.

• (i) If $\alpha \in (0,1]$, then ${(a+b+c)}^{\alpha }\le a^{\alpha }+b^{\alpha }+c^{\alpha }$;

• (ii) if $\alpha \in (1,+\mathrm{\infty })$, then there exists $B(\epsilon )>1$ such that

  $${(a+b+c)}^{\alpha }\le (1+\epsilon )a^{\alpha }+B(\epsilon )b^{\alpha }+B(\epsilon )c^{\alpha }.$$

Lemma 2.3 For any $u=(u_1,u_2),v=(v_1,v_2)\in \mathcal{H}$, the following two equalities hold:

$$-\sum _{t=1}^T(\mathrm{\Delta }{\varphi }_1(\mathrm{\Delta }{u}_1(t-1)),v_1(t))=\sum _{t=1}^T(\mathrm{\Delta }{\varphi }_1(\mathrm{\Delta }{u}_1(t)),\mathrm{\Delta }{v}_1(t)),$$

(2.6)

$$-\sum _{t=1}^T(\mathrm{\Delta }{\varphi }_2(\mathrm{\Delta }{u}_2(t-1)),v_2(t))=\sum _{t=1}^T(\mathrm{\Delta }{\varphi }_2(\mathrm{\Delta }{u}_2(t)),\mathrm{\Delta }{v}_2(t)).$$

(2.7)

Proof In fact, since $u_1(t)=u_1(t+T)$ and $v_1(t)=v_1(t+T)$ for all $t\in \mathbb{Z}$, we have

$$\begin{array}{r}-\sum _{t=1}^T(\mathrm{\Delta }{\varphi }_1(\mathrm{\Delta }{u}_1(t-1)),v_1(t))\\ =-\sum _{t=1}^T({\varphi }_1(\mathrm{\Delta }{u}_1(t)),v_1(t))+\sum _{t=1}^T({\varphi }_1(\mathrm{\Delta }{u}_1(t-1)),v_1(t))\\ =-\sum _{t=1}^T({\varphi }_1(\mathrm{\Delta }{u}_1(t)),v_1(t))+\sum _{t=1}^{T-1}({\varphi }_1(\mathrm{\Delta }{u}_1(t)),v_1(t+1))+({\varphi }_1(\mathrm{\Delta }{u}_1(0)),v_1(1))\\ =\sum _{t=1}^T({\varphi }_1(\mathrm{\Delta }{u}_1(t)),\mathrm{\Delta }{v}_1(t))+({\varphi }_1(\mathrm{\Delta }{u}_1(0)),v_1(1))-({\varphi }_1(\mathrm{\Delta }{u}_1(T)),v_1(T+1))\\ =\sum _{t=1}^T({\varphi }_1(\mathrm{\Delta }{u}_1(t)),\mathrm{\Delta }{v}_1(t)).\end{array}$$

Hence, (2.6) holds. Similarly, it is easy to obtain (2.7). The proof is complete. □

Lemma 2.4 Let $L:\mathbb{Z}[1,T]\times {\mathbb{R}}^N\times {\mathbb{R}}^N\times {\mathbb{R}}^N\times {\mathbb{R}}^N⟶\mathbb{R}$, $(t,x_1,x_2,y_1,y_2)⟶L(t,x_1,x_2,y_1,y_2)$ and assume that L is continuously differential in $(x_1,x_2,y_1,y_2)$ for all $t\in \mathbb{Z}[1,T]$. Then the function $\phi :\mathcal{H}\to \mathbb{R}$ defined by

$$\phi (u)=\phi (u_1,u_2)=\sum _{t=1}^{T}L(t,u_1(t),u_2(t),\mathrm{\Delta }{u}_1(t),\mathrm{\Delta }{u}_2(t))$$

is continuously differentiable on ℋ and

$$\begin{array}{rl}〈{\phi }^{\prime }(u),v〉=& 〈{\phi }^{\prime }(u_1,u_2),(v_1,v_2)〉\\ =& \sum _{t=1}^T[(D_{x_1}L(t,u_1(t),u_2(t),\mathrm{\Delta }{u}_1(t),\mathrm{\Delta }{u}_2(t)),v_1(t))\\ +(D_{y_1}L(t,u_1(t),u_2(t),\mathrm{\Delta }{u}_1(t),\mathrm{\Delta }{u}_2(t)),\mathrm{\Delta }{v}_1(t))\\ +(D_{x_2}L(t,u_1(t),u_2(t),\mathrm{\Delta }{u}_1(t),\mathrm{\Delta }{u}_2(t)),v_2(t))\\ +(D_{y_2}L(t,u_1(t),u_2(t),\mathrm{\Delta }{u}_1(t),\mathrm{\Delta }{u}_2(t)),\mathrm{\Delta }{v}_2(t))],\end{array}$$

where $u,v\in \mathcal{H}$.

Proof Define $G:[-1,1]\times \mathbb{Z}[1,T]\to \mathbb{R}$, $(\lambda ,t)\to G(\lambda ,t)$ by

$$G(\lambda ,t)=L(t,u_1(t)+\lambda v_1(t),u_2(t)+\lambda v_2(t),\mathrm{\Delta }{u}_1(t)+\lambda \mathrm{\Delta }{v}_1(t),\mathrm{\Delta }{u}_2(t)+\lambda \mathrm{\Delta }{v}_2(t)).$$

Since L is continuously differential in $(x_1,x_2,y_1,y_2)$ for all $t\in \mathbb{Z}[1,T]$, $G(\lambda ,t)$ is differential in λ and

$$\begin{array}{r}{G}^{\prime }(\lambda ,t)\\ =(D_{x_1}L(t,u_1(t)+\lambda v_1(t),u_2(t)+\lambda v_2(t),\mathrm{\Delta }{u}_1(t)+\lambda \mathrm{\Delta }{v}_1(t),\mathrm{\Delta }{u}_2(t)+\lambda \mathrm{\Delta }{v}_2(t)),v_1(t))\\ +\left(D_{x_2}L\right(t,u_1(t)+\lambda v_1(t),u_2(t)+\lambda v_2(t),\\ \phantom{}\mathrm{\Delta }{u}_1(t)+\lambda \mathrm{\Delta }{v}_1(t),\mathrm{\Delta }{u}_2(t)+\lambda \mathrm{\Delta }{v}_2(t)),v_2(t))\\ +\left(D_{y_1}L\right(t,u_1(t)+\lambda v_1(t),u_2(t)+\lambda v_2(t),\\ \phantom{}\mathrm{\Delta }{u}_1(t)+\lambda \mathrm{\Delta }{v}_1(t),\mathrm{\Delta }{u}_2(t)+\lambda \mathrm{\Delta }{v}_2(t)),\mathrm{\Delta }{v}_1(t))\\ +\left(D_{y_2}L\right(t,u_1(t)+\lambda v_1(t),u_2(t)+\lambda v_2(t),\\ \phantom{}\mathrm{\Delta }{u}_1(t)+\lambda \mathrm{\Delta }{v}_1(t),\mathrm{\Delta }{u}_2(t)+\lambda \mathrm{\Delta }{v}_2(t)),\mathrm{\Delta }{v}_2(t)).\end{array}$$

Hence,

$$\begin{array}{rl}〈{\phi }^{\prime }(u),v〉=& \underset{\lambda \to 0}{lim}\frac{\phi (u+\lambda v)-\phi (u)}{\lambda }\\ =& \underset{\lambda \to 0}{lim}\frac{{\sum }_{t=1}^{T}G(\lambda ,t)-{\sum }_{t=1}^{T}G(0,t)}{\lambda }\\ =& \sum _{t=1}^T\underset{\lambda \to 0}{lim}\frac{G(\lambda ,t)-G(0,t)}{\lambda }\\ =& \sum _{t=1}^T{G}^{\prime }(0,t)\\ =& \sum _{t=1}^T[(D_{x_1}L(t,u_1(t),u_2(t),\mathrm{\Delta }{u}_1(t),\mathrm{\Delta }{u}_2(t)),v_1(t))\\ +(D_{y_1}L(t,u_1(t),u_2(t),\mathrm{\Delta }{u}_1(t),\mathrm{\Delta }{u}_2(t)),\mathrm{\Delta }{v}_1(t))\\ +(D_{x_2}L(t,u_1(t),u_2(t),\mathrm{\Delta }{u}_1(t),\mathrm{\Delta }{u}_2(t)),v_2(t))\\ +(D_{y_2}L(t,u_1(t),u_2(t),\mathrm{\Delta }{u}_1(t),\mathrm{\Delta }{u}_2(t)),\mathrm{\Delta }{v}_2(t))].\end{array}$$

The proof is complete. □

Let

$$L(t,x_1,x_2,y_1,y_2)={\mathrm{\Phi }}_1(y_1)+{\mathrm{\Phi }}_2(y_2)+F(t,x_1,x_2)+(h_1(t),x_1)+(h_2(t),x_2).$$

Then

$$\begin{array}{rcl}\phi (u)& =& \phi (u_1,u_2)\\ =& \sum _{t=1}^T[{\mathrm{\Phi }}_1(\mathrm{\Delta }{u}_1(t))+{\mathrm{\Phi }}_2(\mathrm{\Delta }{u}_2(t))+F(t,u_1(t),u_2(t))\\ +(h_1(t),u_1(t))+(h_2(t),u_2(t))].\end{array}$$

(2.8)

It follows from ($\mathcal{A}0$), ($\mathcal{F}1$), and Lemma 2.4 that

$$\begin{array}{rl}〈{\phi }^{\prime }(u),v〉=& 〈{\phi }^{\prime }(u_1,u_2),(v_1,v_2)〉\\ =& \sum _{t=1}^T[({\varphi }_1(\mathrm{\Delta }{u}_1(t)),\mathrm{\Delta }{v}_1(t))+({\varphi }_2(\mathrm{\Delta }{u}_2(t)),\mathrm{\Delta }{v}_2(t))\\ +({\mathrm{\nabla }}_{u_1}F(t,u_1(t),u_2(t)),v_1(t))+({\mathrm{\nabla }}_{u_2}F(t,u_1(t),u_2(t)),v_2(t))\\ +(h_1(t),v_1(t))+(h_2(t),v_2(t))],\mathrm{\forall }u,v\in \mathcal{H}.\end{array}$$

(2.9)

By Lemma 2.3, it is easy to see that the critical points of φ in ℋ are periodic solutions of system (1.1).

Next, we recall a definition. Let G be a discrete subgroup of a Banach space X and let $\pi :X\to X/G$ be the canonical surjection. A subset A of X is G-invariant if ${\pi }^{-1}(\pi (A))=A$. A function f defined on X is G-invariant if $f(u+g)=f(u)$ for every $u\in X$ and every $g\in G$ (see [18]).

Definition 2.1 (see [18], Definition 4.2)

A G-invariant differentiable functional $\phi :X\to \mathbb{R}$ satisfies the (PS) _G condition, if for every sequence $\{u_k\}$ in X such that $\phi (u_k)$ is bounded and ${\phi }^{\prime }(u_k)\to 0$, the sequence $\{\pi (u_k)\}$ contains a convergent subsequence.

We will use the following two lemmas to obtain the critical points of φ.

Lemma 2.5 (see [18], Theorem 4.12)

Let $\phi \in C^1(x,\mathbb{R})$ be a G-invariant functional satisfying the (PS) _G condition. If φ is bounded from below and if the dimension N of the space generated by G is finite, then φ has at least $N+1$ critical orbits.

Lemma 2.6 (see [19])

Let X be a Banach space and have a decomposition: $X=Y+Z$ where Y and Z are two subspaces of X with $dimY<+\mathrm{\infty }$. Let V be a finite-dimensional, compact $C^2$-manifold without boundary. Let $f:X\times V\to \mathbb{R}$ be a $C^1$-function and satisfy the (PS) condition. Suppose that f satisfies

$$\underset{u\in Z\times V}{inf}f(u)\ge a,\underset{u\in S\times V}{sup}f(u)\le b<a,$$

where $S=\partial D$, $D=\{u\in Y|\parallel u\parallel \le R\}$, R, a, and b are constants. Then the function f has at least $\mathit{cuplength}(V)+1$ critical points.

Let

$${\hat{u}}_m(t)=P_m{\overline{u}}_m+Q_m{\overline{u}}_m+{\tilde{u}}_m,m=1,2,$$

where

$$\begin{array}{r}{P}_1{\overline{u}}_1=\sum _{k=r_1+1}^N({\overline{u}}_1,e_{i_k})e_{i_k},Q_1{\overline{u}}_1=\sum _{k=1}^{r_1}[({\overline{u}}_1,e_{i_k})-m_{i_k}{T}_{i_k}]e_{i_k},\\ P_2{\overline{u}}_2=\sum _{s=r_2+1}^N({\overline{u}}_2,e_{j_s})e_{j_s},Q_2{\overline{u}}_2=\sum _{s=1}^{r_2}[({\overline{u}}_2,e_{j_s})-m_{j_s}{T}_{j_s}]e_{j_s},\end{array}$$

and $m_{i_k}$, $m_{j_s}$ are the unique integers such that

$$\begin{array}{r}0\le ({\overline{u}}_1,e_{i_k})-m_{i_k}{T}_{i_k}<T_{i_k},1\le k\le r_1,\\ 0\le ({\overline{u}}_2,e_{j_s})-m_{j_s}{T}_{j_s}<T_{j_s},1\le s\le r_2.\end{array}$$

Let

$$\begin{array}{rcl}G& =& \{g=\left(\begin{array}{c}{g}_1\\ g_2\end{array}\right)\in {\mathbf{R}}^N\times {\mathbf{R}}^N|\left(\begin{array}{c}{g}_1\\ g_2\end{array}\right)=\left(\begin{array}{c}{\sum }_{k=1}^{r_1}{m}_{i_k}{T}_{i_k}{e}_{i_k}\\ {\sum }_{s=1}^{r_2}{m}_{j_s}{T}_{j_s}{e}_{j_s}\end{array}\right),\\ \phantom{\left(\begin{array}{c}{g}_1\\ g_2\end{array}\right)}{m}_{i_k}\text{ and }{m}_{j_s}\text{ are integers},1\le k\le r_1,1\le s\le r_2\}.\end{array}$$

(2.10)

Let $Z=\tilde{\mathcal{H}}$, $Y={\mathcal{S}}_1\times {\mathcal{S}}_2$, $X=Y+Z$, and V is the quotient space $({\mathcal{R}}_1\times {\mathcal{R}}_2)/G$ which is isomorphic to the torus $T^{r_1+r_2}$. Now define $\mathrm{\Psi }:X\times T^{r_1+r_2}\to \mathbb{R}$ by

$$\mathrm{\Psi }\left((y+z(t),v)\right)=\phi (y+v+z(t)),\mathrm{\forall }(y,z,v)\in Y\times Z\times T^{r_1+r_2},$$

(2.11)

that is,

$$\mathrm{\Psi }\left((y+z(t),v)\right)=\phi (y_1+v_1+z_1(t),y_2+v_2+z_2(t)),$$

(2.12)

where

$$\begin{array}{r}y={(y_1,y_2)}^{\tau }\in Y,v={(v_1,v_2)}^{\tau }\in V,z=z(t)={(z_1(t),z_2(t))}^{\tau }\in Z,\\ y+v+z(t)={(y_1+v_1+z_1(t),y_2+v_2+z_2(t))}^{\tau }.\end{array}$$

It is easy to verify that Ψ is continuously differentiable and that

$$\begin{array}{r}〈{\mathrm{\Psi }}^{\prime }\left((y^{[1]}+z^{[1]}(t),v^{[1]})\right),(y^{[2]}+z^{[2]}(t),v^{[2]})〉\\ =〈{\phi }^{\prime }(y^{[1]}+v^{[1]}+z^{[1]}(t)),y^{[2]}+v^{[2]}+z^{[2]}(t)〉\\ =〈{\phi }^{\prime }(y_1^{[1]}+v_1^{[1]}+z_1^{[1]}(t),y_2^{[1]}+v_2^{[1]}+z_2^{[1]}(t)),\\ (y_1^{[2]}+v_1^{[2]}+z_1^{[2]}(t),y_2^{[2]}+v_2^{[2]}+z_2^{[2]}(t))〉,\\ \mathrm{\forall }(y^{[m]},z^{[m]},v^{[m]})\in Y\times Z\times T^{r_1+r_2},m=1,2.\end{array}$$

(2.13)

Then ($\mathcal{F}2$) implies that

$$F(t+T,x_1+g_1,x_2+g_2)=F(t,x_1,x_2),\mathrm{\forall }t\in \mathbb{Z}\text{ and }\mathrm{\forall }g\in G.$$

Hence, we have

$$\begin{array}{rl}F(t,u_1(t),u_2(t))& =F(t,{\hat{u}}_1(t)+\sum _{k=1}^{r_1}{m}_{i_k}{T}_{i_k}{e}_{i_k},{\hat{u}}_2(t)+\sum _{s=1}^{r_2}{m}_{j_s}{T}_{j_s}{e}_{j_s})\\ =F(t,{\hat{u}}_1(t),{\hat{u}}_2(t)),\end{array}$$

(2.14)

$$\begin{array}{rl}\mathrm{\nabla }F(t,u_1(t),u_2(t))& =\mathrm{\nabla }F(t,{\hat{u}}_1(t)+\sum _{k=1}^{r_1}{m}_{i_k}{T}_{i_k}{e}_{i_k},{\hat{u}}_2(t)+\sum _{s=1}^{r_2}{m}_{j_s}{T}_{j_s}{e}_{j_s})\\ =\mathrm{\nabla }F(t,{\hat{u}}_1(t),{\hat{u}}_2(t))\end{array}$$

(2.15)

and, by (ℰ), we have

$$\begin{array}{rl}\sum _{t=1}^T(h_1(t),u_1(t))& =\sum _{t=1}^T(h_1(t),{\hat{u}}_1(t)+\sum _{k=1}^{r_1}{m}_{i_k}{T}_{i_k}{e}_{i_k})\\ =\sum _{t=1}^T(h_1(t),{\hat{u}}_1(t)),\end{array}$$

(2.16)

$$\begin{array}{rl}\sum _{t=1}^T(h_2(t),u_2(t))& =\sum _{t=1}^T(h_2(t),{\hat{u}}_2(t)+\sum _{k=1}^{r_1}{m}_{i_k}{T}_{i_k}{e}_{i_k})\\ =\sum _{t=1}^T(h_2(t),{\hat{u}}_2(t)).\end{array}$$

(2.17)

Hence $\phi (u)=\phi (\hat{u})$ and ${\phi }^{\prime }(u)={\phi }^{\prime }(\hat{u})$.

3 Proofs

For the sake of convenience, we denote by $C_{ij}$ and $D_{ij}$, $i=1,2$, $j=0,1,\dots ,9$ below the various positive constants, by $C_{ij}(\epsilon )$ and $D_{ij}(\epsilon )$, $i=1,2$, $j=0,1,\dots ,9$ below the various positive constants depending on ε and

$$\begin{array}{r}{M}_{11}=\sum _{t=1}^T{f}_1(t),M_{12}=\sum _{t=1}^T{g}_1(t),M_{13}={\left(\sum _{k=1}^{r_1}{T}_{i_k}^2\right)}^{1/2},\\ M_{14}=\sum _{t=1}^T|h_1(t)|,M_{15}=\sum _{t=1}^T{b}_1(t),\\ M_{21}=\sum _{t=1}^T{f}_2(t),M_{22}=\sum _{t=1}^T{g}_2(t),M_{23}={\left(\sum _{s=1}^{r_2}{T}_{j_s}^2\right)}^{1/2},\\ M_{24}=\sum _{t=1}^T|h_2(t)|,M_{25}=\sum _{t=1}^T{b}_2(t).\end{array}$$

Proof of Theorem 1.1 It follows from ($\mathcal{F}4$) that there exist $a_1>\frac{C(p^{\prime })}{p{\gamma }_1}$ and $a_2>\frac{C(q^{\prime })}{q{\gamma }_3}$ such that

$$\underset{|x_1|+|x_2|\to \mathrm{\infty }}{lim}\frac{F(t,x_1,x_2)}{w_1^{p^{\prime }}(|x_1|)+w_2^{q^{\prime }}(|x_2|)}>max\{\frac{c_{10}^{p^{\prime }}{M}_{11}^{p^{\prime }}{a}_1^{p^{\prime }/p}C(p^{\prime })}{p^{\prime }},\frac{c_{20}^{q^{\prime }}{M}_{21}^{q^{\prime }}{a}_2^{q^{\prime }/q}C(q^{\prime })}{q^{\prime }}\},$$

(3.1)

for $(x_1,x_2)\in {\mathcal{S}}_1\times {\mathcal{S}}_2$. It follows from ($\mathcal{A}3$), ($\mathcal{F}3$), Lemma 2.1, and Lemma 2.2 that

$$\begin{array}{r}\sum _{t=1}^T|F(t,{\hat{u}}_1(t),{\hat{u}}_2(t))-F(t,P_1{\overline{u}}_1,P_2{\overline{u}}_2)|\\ \le \sum _{t=1}^T|F(t,{\hat{u}}_1(t),{\hat{u}}_2(t))-F(t,P_1{\overline{u}}_1,{\hat{u}}_2(t))|\\ +\sum _{t=1}^T|F(t,P_1{\overline{u}}_1,{\hat{u}}_2(t))-F(t,P_1{\overline{u}}_1,P_2{\overline{u}}_2)|\\ \le \sum _{t=1}^T\left|{\int }_0^1({\mathrm{\nabla }}_{x_1}F(t,P_1{\overline{u}}_1+s(Q_1{\overline{u}}_1+{\tilde{u}}_1(t)),{\hat{u}}_2(t)),Q_1{\overline{u}}_1+{\tilde{u}}_1(t))ds\right|\\ +\sum _{t=1}^T\left|{\int }_0^1({\mathrm{\nabla }}_{x_2}F(t,P_1{\overline{u}}_1,P_2{\overline{u}}_2+s(Q_2{\overline{u}}_2+{\tilde{u}}_2(t))),Q_2{\overline{u}}_2+{\tilde{u}}_2(t))ds\right|\\ \le (|Q_1{\overline{u}}_1|+{\parallel {\tilde{u}}_1\parallel }_{\mathrm{\infty }})\sum _{t=1}^T{\int }_0^1\left|{\mathrm{\nabla }}_{x_1}F(t,P_1{\overline{u}}_1+s(Q_1{\overline{u}}_1+{\tilde{u}}_1(t)),{\hat{u}}_2(t))\right|ds\\ +(|Q_2{\overline{u}}_2|+{\parallel {\tilde{u}}_2\parallel }_{\mathrm{\infty }})\sum _{t=1}^T{\int }_0^1\left|{\mathrm{\nabla }}_{x_2}F(t,P_1{\overline{u}}_1,P_2{\overline{u}}_2+s(Q_2{\overline{u}}_2+{\tilde{u}}_2(t)))\right|ds\\ \le (M_{13}+{\parallel {\tilde{u}}_1\parallel }_{\mathrm{\infty }})\sum _{t=1}^T{\int }_0^1[f_1(t)w_1\left(|P_1{\overline{u}}_1+s(Q_1{\overline{u}}_1+{\tilde{u}}_1(t))|\right)+g_1(t)]ds\\ +(M_{23}+{\parallel {\tilde{u}}_2\parallel }_{\mathrm{\infty }})\sum _{t=1}^T{\int }_0^1[f_2(t)w_2\left(|P_2{\overline{u}}_2+s(Q_2{\overline{u}}_2+{\tilde{u}}_2(t))|\right)+g_2(t)]ds\\ \le (M_{13}+{\parallel {\tilde{u}}_1\parallel }_{\mathrm{\infty }})\sum _{t=1}^T{\int }_0^1[c_{10}{f}_1(t)w_1(|P_1{\overline{u}}_1|)+c_{10}{f}_1(t)w_1\left(s|Q_1{\overline{u}}_1+{\tilde{u}}_1(t)|\right)]ds\\ +(M_{23}+{\parallel {\tilde{u}}_2\parallel }_{\mathrm{\infty }})\sum _{t=1}^T{\int }_0^1[c_{20}{f}_2(t)w_2(|P_2{\overline{u}}_2|)+c_{20}{f}_2(t)w_2\left(s|Q_2{\overline{u}}_2+{\tilde{u}}_2(t)|\right)]ds\\ +(M_{13}+{\parallel {\tilde{u}}_1\parallel }_{\mathrm{\infty }})\sum _{t=1}^T{\int }_0^1{g}_1(t)ds+(M_{23}+{\parallel {\tilde{u}}_2\parallel }_{\mathrm{\infty }})\sum _{t=1}^T{\int }_0^1{g}_2(t)ds\\ \le (M_{13}+{\parallel {\tilde{u}}_1\parallel }_{\mathrm{\infty }})w_1(|P_1{\overline{u}}_1|)\sum _{t=1}^T{c}_{10}{f}_1(t)+(M_{23}+{\parallel {\tilde{u}}_2\parallel }_{\mathrm{\infty }})w_2(|P_2{\overline{u}}_2|)\sum _{t=1}^T{c}_{20}{f}_2(t)\\ +(M_{13}+{\parallel {\tilde{u}}_1\parallel }_{\mathrm{\infty }})\sum _{t=1}^T{\int }_0^1[c_{10}{f}_1(t)k_{11}{\left|s(Q_1{\overline{u}}_1+{\tilde{u}}_1(t))\right|}^{{\alpha }_1}+f_1(t)c_{10}{k}_{12}+g_1(t)]ds\\ +(M_{23}+{\parallel {\tilde{u}}_2\parallel }_{\mathrm{\infty }})\sum _{t=1}^T{\int }_0^1[c_{20}{f}_2(t)k_{21}{\left|s(Q_2{\overline{u}}_2+{\tilde{u}}_2(t))\right|}^{{\alpha }_2}+f_2(t)c_{20}{k}_{22}+g_2(t)]ds\\ \le (M_{13}+{\parallel {\tilde{u}}_1\parallel }_{\mathrm{\infty }})w_1(|P_1{\overline{u}}_1|)\sum _{t=1}^T{c}_{10}{f}_1(t)\\ +\frac{1+{\epsilon }_1}{{\alpha }_1+1}(M_{13}+{\parallel {\tilde{u}}_1\parallel }_{\mathrm{\infty }})\sum _{t=1}^T{f}_1(t)c_{10}{k}_{11}{|Q_1{\overline{u}}_1|}^{{\alpha }_1}\\ +\frac{B({\epsilon }_1)}{{\alpha }_1+1}(M_{13}+{\parallel {\tilde{u}}_1\parallel }_{\mathrm{\infty }})\sum _{t=1}^T{f}_1(t)c_{10}{k}_{11}{|{\tilde{u}}_1(t)|}^{{\alpha }_1}\\ +(M_{13}+{\parallel {\tilde{u}}_1\parallel }_{\mathrm{\infty }})\sum _{t=1}^T[f_1(t)c_{10}{k}_{12}+g_1(t)]\\ +(M_{23}+{\parallel {\tilde{u}}_2\parallel }_{\mathrm{\infty }})w_2(|P_2{\overline{u}}_2|)\sum _{t=1}^T{c}_{20}{f}_2(t)\\ +\frac{1+{\epsilon }_2}{{\alpha }_2+1}(M_{23}+{\parallel {\tilde{u}}_2\parallel }_{\mathrm{\infty }})\sum _{t=1}^T{f}_2(t)c_{20}{k}_{21}{|Q_2{\overline{u}}_2|}^{{\alpha }_2}\\ +\frac{B({\epsilon }_2)}{{\alpha }_2+1}(M_{23}+{\parallel {\tilde{u}}_2\parallel }_{\mathrm{\infty }})\sum _{t=1}^T{f}_2(t)c_{20}{k}_{21}{|{\tilde{u}}_2(t)|}^{{\alpha }_2}\\ +(M_{23}+{\parallel {\tilde{u}}_2\parallel }_{\mathrm{\infty }})\sum _{t=1}^T[f_2(t)c_{20}{k}_{22}+g_2(t)]\\ \le M_{11}{c}_{10}{w}_1(|P_1{\overline{u}}_1|){\parallel {\tilde{u}}_1\parallel }_{\mathrm{\infty }}+M_{11}{M}_{13}{c}_{10}{w}_1(|P_1{\overline{u}}_1|)\\ +\frac{(1+{\epsilon }_1)M_{13}^{{\alpha }_1+1}{c}_{10}{k}_{11}{M}_{11}}{{\alpha }_1+1}+\frac{(1+{\epsilon }_1)M_{13}^{{\alpha }_1}{c}_{10}{k}_{11}{M}_{11}}{{\alpha }_1+1}{\parallel {\tilde{u}}_1\parallel }_{\mathrm{\infty }}\\ +\frac{B({\epsilon }_1)M_{13}{c}_{10}{k}_{11}{M}_{11}}{{\alpha }_1+1}{\parallel {\tilde{u}}_1\parallel }_{\mathrm{\infty }}^{{\alpha }_1}+\frac{B({\epsilon }_1)c_{10}{k}_{11}{M}_{11}}{{\alpha }_1+1}{\parallel {\tilde{u}}_1\parallel }_{\mathrm{\infty }}^{{\alpha }_1+1}\\ +M_{13}\sum _{t=1}^T[f_1(t)c_{10}{k}_{12}+g_1(t)]+{\parallel {\tilde{u}}_1\parallel }_{\mathrm{\infty }}\sum _{t=1}^T[f_1(t)c_{10}{k}_{12}+g_1(t)]\\ +M_{21}{c}_{20}{w}_2(|P_2{\overline{u}}_2|){\parallel {\tilde{u}}_2\parallel }_{\mathrm{\infty }}+M_{21}{M}_{23}{c}_{20}{w}_2(|P_2{\overline{u}}_2|)\\ +\frac{(1+{\epsilon }_2)M_{23}^{{\alpha }_2+1}{c}_{20}{k}_{21}{M}_{21}}{{\alpha }_2+1}+\frac{(1+{\epsilon }_2)M_{23}^{{\alpha }_2}{c}_{20}{k}_{21}{M}_{21}}{{\alpha }_2+1}{\parallel {\tilde{u}}_2\parallel }_{\mathrm{\infty }}\\ +\frac{B({\epsilon }_2)M_{23}{c}_{20}{k}_{21}{M}_{21}}{{\alpha }_2+1}{\parallel {\tilde{u}}_2\parallel }_{\mathrm{\infty }}^{{\alpha }_2}+\frac{B({\epsilon }_2)c_{20}{k}_{21}{M}_{21}}{{\alpha }_2+1}{\parallel {\tilde{u}}_2\parallel }_{\mathrm{\infty }}^{{\alpha }_2+1}\\ +M_{23}\sum _{t=1}^T[c_{20}{f}_2(t)k_{22}+g_2(t)]+{\parallel {\tilde{u}}_2\parallel }_{\mathrm{\infty }}\sum _{t=1}^T[c_{20}{f}_2(t)k_{22}+g_2(t)]\\ \le \frac{1}{p{a}_1}{\left(\frac{1}{C(p^{\prime })}\right)}^{p/p^{\prime }}{\parallel {\tilde{u}}_1\parallel }_{\mathrm{\infty }}^p+\frac{M_{11}^{p^{\prime }}{c}_{10}^{p^{\prime }}{a}_1^{p^{\prime }/p}C(p^{\prime })}{p^{\prime }}{w}_1^{p^{\prime }}(|P_1{\overline{u}}_1|)\\ +C_{11}({\epsilon }_1){\parallel {\tilde{u}}_1\parallel }_{\mathrm{\infty }}^{{\alpha }_1+1}+C_{12}({\epsilon }_1){\parallel {\tilde{u}}_1\parallel }_{\mathrm{\infty }}^{{\alpha }_1}+C_{13}({\epsilon }_1){\parallel {\tilde{u}}_1\parallel }_{\mathrm{\infty }}+C_{14}\\ +\frac{1}{q{a}_2}{\left(\frac{1}{C(q^{\prime })}\right)}^{q/q^{\prime }}{\parallel {\tilde{u}}_2\parallel }_{\mathrm{\infty }}^q+\frac{M_{21}^{q^{\prime }}{c}_{20}^{q^{\prime }}{a}_2^{q^{\prime }/q}C(q^{\prime })}{q^{\prime }}{w}_2^{q^{\prime }}(|P_2{\overline{u}}_2|)\\ +C_{21}({\epsilon }_2){\parallel {\tilde{u}}_2\parallel }_{\mathrm{\infty }}^{{\alpha }_2+1}+C_{22}({\epsilon }_2){\parallel {\tilde{u}}_2\parallel }_{\mathrm{\infty }}^{{\alpha }_2}+C_{23}({\epsilon }_2){\parallel {\tilde{u}}_2\parallel }_{\mathrm{\infty }}+C_{24}\\ +M_{11}{M}_{13}{c}_{10}{w}_1(|P_1{\overline{u}}_1|)+M_{21}{M}_{23}{c}_{20}{w}_2(|P_2{\overline{u}}_2|)\\ \le \frac{C(p^{\prime })}{p{a}_1}\sum _{t=1}^T{|\mathrm{\Delta }{u}_1(t)|}^p+\frac{M_{11}^{p^{\prime }}{c}_{10}^{p^{\prime }}{a}_1^{p^{\prime }/p}C(p^{\prime })}{p^{\prime }}{w}_1^{p^{\prime }}(|P_1{\overline{u}}_1|)\\ +C_{11}({\epsilon }_1){\left[C\left(p^{\prime }\right)\right]}^{{\alpha }_1+1}{\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_1(t)|}^p\right)}^{\frac{{\alpha }_1+1}{p}}+C_{14}\\ +C_{13}({\epsilon }_1)C\left(p^{\prime }\right){\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_1(t)|}^p\right)}^{\frac{1}{p}}+C_{12}({\epsilon }_1){\left[C\left(p^{\prime }\right)\right]}^{{\alpha }_1}{\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_1(t)|}^p\right)}^{\frac{{\alpha }_1}{p}}\\ +\frac{C(q^{\prime })}{q{a}_2}\sum _{t=1}^T{|\mathrm{\Delta }{u}_2(t)|}^q+\frac{M_{21}^{q^{\prime }}{c}_{20}^{q^{\prime }}{a}_2^{q^{\prime }/q}C(q^{\prime })}{q^{\prime }}{w}_2^{q^{\prime }}(|P_2{\overline{u}}_2|)\\ +C_{21}({\epsilon }_2){\left[C\left(q^{\prime }\right)\right]}^{{\alpha }_2+1}{\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_2(t)|}^q\right)}^{\frac{{\alpha }_2+1}{q}}+C_{24}\\ +C_{23}({\epsilon }_2)C\left(q^{\prime }\right){\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_2(t)|}^q\right)}^{\frac{1}{q}}+C_{22}({\epsilon }_2){\left[C\left(q^{\prime }\right)\right]}^{{\alpha }_2}{\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_2(t)|}^q\right)}^{\frac{{\alpha }_2}{q}}\\ +M_{11}{M}_{13}{c}_{10}{w}_1(|P_1{\overline{u}}_1|)+M_{21}{M}_{23}{c}_{20}{w}_2(|P_2{\overline{u}}_2|).\end{array}$$

(3.2)

By ($\mathcal{A}1$), (3.2), and Lemma 2.1, we have

$$\begin{array}{rl}\phi (u)=& \phi ({\hat{u}}_1,{\hat{u}}_2)\\ =& \sum _{t=1}^T[{\mathrm{\Phi }}_1(\mathrm{\Delta }{u}_1(t))+{\mathrm{\Phi }}_2(\mathrm{\Delta }{u}_2(t))+F(t,{\hat{u}}_1(t),{\hat{u}}_2(t))\\ +(h_1(t),{\hat{u}}_1(t))+(h_2(t),{\hat{u}}_2(t))]\\ =& \sum _{t=1}^T[{\mathrm{\Phi }}_1(\mathrm{\Delta }{u}_1(t))+{\mathrm{\Phi }}_2(\mathrm{\Delta }{u}_2(t))+F(t,{\hat{u}}_1(t),{\hat{u}}_2(t))\\ -F(t,P_1{\overline{u}}_1,P_2{\overline{u}}_2)+F(t,P_1{\overline{u}}_1,P_2{\overline{u}}_2)+(h_1(t),{\hat{u}}_1(t))+(h_2(t),{\hat{u}}_2(t))]\\ \ge & \sum _{t=1}^T({\gamma }_1{|\mathrm{\Delta }{u}_1(t)|}^p+{\gamma }_3{|\mathrm{\Delta }{u}_2(t)|}^q-{\gamma }_2{|\mathrm{\Delta }{u}_1(t)|}^{{\beta }_1}-{\gamma }_4{|\mathrm{\Delta }{u}_2(t)|}^{{\beta }_2})\\ -\frac{C(p^{\prime })}{p{a}_1}\sum _{t=1}^T{|\mathrm{\Delta }{u}_1(t)|}^p-\frac{M_{11}^{p^{\prime }}{c}_{10}^{p^{\prime }}{a}_1^{p^{\prime }/p}C(p^{\prime })}{p^{\prime }}{w}_1^{p^{\prime }}(|P_1{\overline{u}}_1|)\\ -C_{11}({\epsilon }_1){\left[C\left(p^{\prime }\right)\right]}^{{\alpha }_1+1}{\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_1(t)|}^p\right)}^{\frac{{\alpha }_1+1}{p}}-C_{14}-{\parallel {\tilde{u}}_1\parallel }_{\mathrm{\infty }}\sum _{t=1}^T|h_1(t)|\\ -C_{13}({\epsilon }_1)C\left(p^{\prime }\right){\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_1(t)|}^p\right)}^{\frac{1}{p}}-C_{12}({\epsilon }_1){\left[C\left(p^{\prime }\right)\right]}^{{\alpha }_1}{\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_1(t)|}^p\right)}^{\frac{{\alpha }_1}{p}}\\ -\frac{C(q^{\prime })}{q{a}_2}\sum _{t=1}^T{|\mathrm{\Delta }{u}_2(t)|}^q-\frac{M_{21}^{q^{\prime }}{c}_{20}^{q^{\prime }}{a}_2^{q^{\prime }/q}C(q^{\prime })}{q^{\prime }}{w}_2^{q^{\prime }}(|P_2{\overline{u}}_2|)\\ -C_{21}({\epsilon }_2){\left[C\left(q^{\prime }\right)\right]}^{{\alpha }_2+1}{\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_2(t)|}^q\right)}^{\frac{{\alpha }_2+1}{q}}-C_{24}-{\parallel {\tilde{u}}_2\parallel }_{\mathrm{\infty }}\sum _{t=1}^T|h_2(t)|\\ -C_{23}({\epsilon }_2)C\left(q^{\prime }\right){\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_2(t)|}^q\right)}^{\frac{1}{q}}-C_{22}({\epsilon }_2){\left[C\left(q^{\prime }\right)\right]}^{{\alpha }_2}{\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_2(t)|}^q\right)}^{\frac{{\alpha }_2}{q}}\\ -M_{11}{M}_{13}{c}_{10}{w}_1(|P_1{\overline{u}}_1|)-M_{21}{M}_{23}{c}_{20}{w}_2(|P_2{\overline{u}}_2|)\\ \ge & ({\gamma }_1-\frac{C(p^{\prime })}{p{a}_1})\sum _{t=1}^T{|\mathrm{\Delta }{u}_1(t)|}^p-M_{11}{M}_{13}{c}_{10}{w}_1(|P_1{\overline{u}}_1|)\\ -C_{11}({\epsilon }_1){\left[C\left(p^{\prime }\right)\right]}^{{\alpha }_1+1}{\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_1(t)|}^p\right)}^{\frac{{\alpha }_1+1}{p}}\\ -C_{12}({\epsilon }_1){\left[C\left(p^{\prime }\right)\right]}^{{\alpha }_1}{\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_1(t)|}^p\right)}^{\frac{{\alpha }_1}{p}}\\ -C_{13}({\epsilon }_1)C\left(p^{\prime }\right){\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_1(t)|}^p\right)}^{\frac{1}{p}}-{\gamma }_2{T}^{1-\frac{{\beta }_1}{p}}{\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_1(t)|}^p\right)}^{\frac{{\beta }_1}{p}}\\ -C_{14}+({\gamma }_3-\frac{C(q^{\prime })}{q{a}_2})\sum _{t=1}^T{|\mathrm{\Delta }{u}_2(t)|}^q-M_{21}{M}_{23}{c}_{20}{w}_2(|P_2{\overline{u}}_2|)\\ -C_{21}({\epsilon }_2){\left[C\left(q^{\prime }\right)\right]}^{{\alpha }_2+1}{\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_2(t)|}^q\right)}^{\frac{{\alpha }_2+1}{q}}\\ -C_{22}({\epsilon }_2){\left[C\left(q^{\prime }\right)\right]}^{{\alpha }_2}{\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_2(t)|}^q\right)}^{\frac{{\alpha }_2}{q}}\\ -C_{23}({\epsilon }_2)C\left(q^{\prime }\right){\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_2(t)|}^q\right)}^{\frac{1}{q}}-{\gamma }_4{T}^{1-\frac{{\beta }_2}{q}}{\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_2(t)|}^q\right)}^{\frac{{\beta }_2}{q}}\\ -C_{24}+[w_1^{p^{\prime }}(|P_1{\overline{u}}_1|)+w_2^{q^{\prime }}(|P_2{\overline{u}}_2|)][\frac{F(t,P_1{\overline{u}}_1,P_2{\overline{u}}_2)}{w_1^{p^{\prime }}(|P_1{\overline{u}}_1|)+w_2^{q^{\prime }}(|P_2{\overline{u}}_2|)}\\ -max\{\frac{M_{11}^{p^{\prime }}{c}_{10}^{p^{\prime }}{a}_1^{p^{\prime }/p}C(p^{\prime })}{p^{\prime }},\frac{M_{21}^{q^{\prime }}{c}_{20}^{q^{\prime }}{a}_2^{q^{\prime }/q}C(q^{\prime })}{q^{\prime }}\}]\\ -C\left(p^{\prime }\right){\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_1(t)|}^p\right)}^{\frac{1}{p}}\sum _{t=1}^T|h_1(t)|\\ -C\left(q^{\prime }\right){\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_2(t)|}^q\right)}^{\frac{1}{q}}\sum _{t=1}^T|h_2(t)|.\end{array}$$

(3.3)

It follows from (3.1), (3.3), $a_1>\frac{C(p^{\prime })}{p{\gamma }_1}$, and $a_2>\frac{C(q^{\prime })}{q{\gamma }_3}$ that φ is bounded from below. Let G be a discrete subgroup of ℋ defined by (2.10) and let $\pi :\mathcal{H}\to \mathcal{H}/G$ be the canonical surjection. By (2.14)-(2.17), it is easy to verify that φ is G-invariant. In what follows, we show that the functional φ satisfies the (PS) _G condition, that is, for every sequence $\{u_m\}$ in ℋ such that $\{\phi (u_n)\}$ is bounded and ${\phi }^{\prime }(u_n)\to 0$, the sequence $\{\pi (u_n)\}$ has a convergent subsequence. In fact, the boundedness of $\phi (u_n)$, (3.1), (3.3), and the facts that $a_1>\frac{C(p^{\prime })}{p{\gamma }_1}$ and $a_2>\frac{C(q^{\prime })}{q{\gamma }_3}$ imply that $(P{\overline{u}}_n)$ and ${\sum }_{t=1}^T{|\mathrm{\Delta }{u}_n(t)|}^2$ are bounded. Furthermore, by Lemma 2.1, we know that $({\tilde{u}}_n)$ is also bounded. Hence $\{{\hat{u}}_n\}$ is bounded in ℋ. Since $dim\mathcal{H}<\mathrm{\infty }$, we know that $\{{\hat{u}}_n\}$ has a convergent subsequence. Since $\pi (u_n)=\pi ({\hat{u}}_n)$, $\{\pi (u_n)\}$ also has a convergent subsequence. Thus, by Lemma 2.5, we know that φ has $r_1+r_2+1$ critical orbits. Hence, system (1.1) has at least $r_1+r_2+1$ geometrically distinct solutions in ℋ. The proof is complete. □

Proof of Theorem 1.2 First, we prove that Ψ defined by (2.11) satisfies the (PS) condition. Assume that ${\{(y^{[n]}+z^{[n]},v^{[n]})\}}_{n=1}^{\mathrm{\infty }}\subset X\times T^{r_1+r_2}$ is (PS) sequence for Ψ, that is, $\{\mathrm{\Psi }((y^{[n]}+z^{[n]},v^{[n]}))\}$ is bounded and ${\mathrm{\Psi }}^{\prime }((y^{[n]}+z^{[n]},v^{[n]}))\to 0$, where $y^{[n]}={(y_1^{[n]},y_2^{[n]})}^{\tau }\in Y$, $z^{[n]}=z^{[n]}(t)={(z_1^{[n]}(t),z_2^{[n]}(t))}^{\tau }\in Z$, $v^{[n]}={(v_1^{[n]},v_2^{[n]})}^{\tau }\in T^{r_1+r_2}$ for $n=1,2,\dots$ . Let

$$u^{[n]}=y^{[n]}+v^{[n]}+z^{[n]}={(y_1^{[n]}+v_1^{[n]}+z_1^{[n]},y_2^{[n]}+v_2^{[n]}+z_2^{[n]})}^{\tau },n=1,2\dots .$$

Then it is easy to see that

$$y_m^{[n]}=P_m{\overline{u}}_m^{[n]},v_m^{[n]}=Q_m{\overline{u}}_m^{[n]},z_m^{[n]}(t)={\tilde{u}}_m^{[n]}(t),m=1,2,n=1,2\dots .$$

By (2.12) and (2.13), we find that $\{\phi (u_1^{[n]},u_2^{[n]})\}$ is bounded and ${\phi }^{\prime }(u_1^{[n]},u_2^{[n]})\to 0$. Then there exists a positive constant $D_0$ such that

$$\left|\phi (u_1^{[n]},u_2^{[n]})\right|\le D_0,\parallel {\phi }^{\prime }(u_1^{[n]},u_2^{[n]})\parallel \le D_0,\mathrm{\forall }n\in \mathbb{R}.$$

(3.4)

By ($\mathcal{F}4$)′, there exist $a_3>C(p^{\prime })$ and $a_4>C(q^{\prime })$ such that

$$\begin{array}{r}\underset{|x_1|+|x_2|\to \mathrm{\infty }}{lim}\frac{F(t,x_1,x_2)}{w_1^{p^{\prime }}(|x_1|)+w_2^{q^{\prime }}(|x_2|)}\\ <-max\{\frac{{[a_3{M}_{11}{c}_{10}]}^{p^{\prime }}}{p^{\prime }}[\frac{1+p{\zeta }_1}{d_{1}p-1}+\frac{1+q{\zeta }_3}{d_{3}q-1}+1],\\ \frac{{[a_4{M}_{21}{c}_{20}]}^{q^{\prime }}}{q^{\prime }}[\frac{1+p{\zeta }_1}{d_{1}p-1}+\frac{1+q{\zeta }_3}{d_{3}q-1}+1]\}.\end{array}$$

(3.5)

It follows from ($\mathcal{F}3$), Lemma 2.1, and Young’s inequality that, for all $(u_1,u_2)\in \mathcal{H}$,

$$\begin{array}{r}|\sum _{t=1}^T({\mathrm{\nabla }}_{x_1}F(t,{\hat{u}}_1(t),{\hat{u}}_2(t)),{\tilde{u}}_1(t))+\sum _{t=1}^T({\mathrm{\nabla }}_{x_2}F(t,{\hat{u}}_1(t),{\hat{u}}_2(t)),{\tilde{u}}_2(t))|\\ \le \left|\sum _{t=1}^T({\mathrm{\nabla }}_{x_1}F(t,P_1{\overline{u}}_1+Q_1{\overline{u}}_1+{\tilde{u}}_1(t),{\hat{u}}_2(t)),{\tilde{u}}_1(t))\right|\\ +\left|\sum _{t=1}^T({\mathrm{\nabla }}_{x_2}F(t,{\hat{u}}_1(t),P_2{\overline{u}}_2+Q_2{\overline{u}}_2+{\tilde{u}}_2(t)),{\tilde{u}}_2(t))\right|\\ \le \sum _{t=1}^T{f}_1(t)w_1\left(|P_1{\overline{u}}_1+Q_1{\overline{u}}_1+{\tilde{u}}_1(t)|\right)|{\tilde{u}}_1(t)|+\sum _{t=1}^T{g}_1(t)|{\tilde{u}}_1(t)|\\ +\sum _{t=1}^T{f}_2(t)w_2\left(|P_2{\overline{u}}_2+Q_2{\overline{u}}_2+{\tilde{u}}_2(t)|\right)|{\tilde{u}}_2(t)|+\sum _{t=1}^T{g}_2(t)|{\tilde{u}}_2(t)|\\ \le {\parallel {\tilde{u}}_1\parallel }_{\mathrm{\infty }}{c}_{10}(w_1(|P_1{\overline{u}}_1|)+w_1(|Q_1{\overline{u}}_1+{\tilde{u}}_1|))\sum _{t=1}^T{f}_1(t)+{\parallel {\tilde{u}}_1\parallel }_{\mathrm{\infty }}\sum _{t=1}^T{g}_1(t)\\ +{\parallel {\tilde{u}}_2\parallel }_{\mathrm{\infty }}{c}_{20}(w_2(|P_2{\overline{u}}_2|)+w_2(|Q_2{\overline{u}}_2+{\tilde{u}}_2|))\sum _{t=1}^T{f}_2(t)+{\parallel {\tilde{u}}_2\parallel }_{\mathrm{\infty }}\sum _{t=1}^T{g}_2(t)\\ \le \frac{1}{p{a}_3^p}{\parallel {\tilde{u}}_1\parallel }_{\mathrm{\infty }}^p+\frac{{[a_3{M}_{11}{c}_{10}]}^{p^{\prime }}}{p^{\prime }}{w}_1^{p^{\prime }}(|P_1{\overline{u}}_1|)\\ +{\parallel {\tilde{u}}_1\parallel }_{\mathrm{\infty }}{c}_{10}(k_{11}{|Q_1{\overline{u}}_1+{\tilde{u}}_1(t)|}^{{\alpha }_1}+k_{12})\sum _{t=1}^T{f}_1(t)\\ +\frac{1}{q{a}_4^q}{\parallel {\tilde{u}}_2\parallel }_{\mathrm{\infty }}^q+\frac{{[a_4{M}_{21}{c}_{20}]}^{q^{\prime }}}{q^{\prime }}{w}_2^{q^{\prime }}(|P_2{\overline{u}}_2|)\\ +{\parallel {\tilde{u}}_2\parallel }_{\mathrm{\infty }}{c}_{20}(k_{21}{|Q_2{\overline{u}}_2+{\tilde{u}}_2(t)|}^{{\alpha }_2}+k_{22})\sum _{t=1}^T{f}_2(t)\\ +{\parallel {\tilde{u}}_1\parallel }_{\mathrm{\infty }}\sum _{t=1}^T{g}_1(t)+{\parallel {\tilde{u}}_2\parallel }_{\mathrm{\infty }}\sum _{t=1}^T{g}_2(t)\\ \le \frac{1}{p{a}_3^p}{\parallel {\tilde{u}}_1\parallel }_{\mathrm{\infty }}^p+\frac{{[a_3{M}_{11}{c}_{10}]}^{p^{\prime }}}{p^{\prime }}{w}_1^{p^{\prime }}(|P_1{\overline{u}}_1|)\\ +M_{11}{c}_{10}{k}_{11}(1+{\epsilon }_1){|Q_1{\overline{u}}_1|}^{{\alpha }_1}{\parallel {\tilde{u}}_1\parallel }_{\mathrm{\infty }}+{\parallel {\tilde{u}}_1\parallel }_{\mathrm{\infty }}^{{\alpha }_1+1}{M}_{11}{c}_{10}{k}_{11}{B}_1({\epsilon }_1)\\ +{\parallel {\tilde{u}}_1\parallel }_{\mathrm{\infty }}{M}_{11}{c}_{10}{k}_{12}\\ +\frac{1}{q{a}_4^q}{\parallel {\tilde{u}}_2\parallel }_{\mathrm{\infty }}^q+\frac{{[a_4{M}_{21}{c}_{20}]}^{q^{\prime }}}{q^{\prime }}{w}_2^{q^{\prime }}(|P_2{\overline{u}}_2|)\\ +M_{21}{c}_{20}{k}_{21}(1+{\epsilon }_2){|Q_2{\overline{u}}_2|}^{{\alpha }_2}{\parallel {\tilde{u}}_2\parallel }_{\mathrm{\infty }}+{\parallel {\tilde{u}}_2\parallel }_{\mathrm{\infty }}^{{\alpha }_2+1}{M}_{21}{c}_{20}{k}_{21}{B}_2({\epsilon }_2)\\ +{\parallel {\tilde{u}}_2\parallel }_{\mathrm{\infty }}{M}_{21}{c}_{20}{k}_{22}\\ \le \frac{{[C(p^{\prime })]}^p}{p{a}_3^p}\sum _{t=1}^T{|\mathrm{\Delta }{u}_1(t)|}^p+\frac{{[a_3{M}_{11}{c}_{10}]}^{p^{\prime }}}{p^{\prime }}{w}_1^{p^{\prime }}(|P_1{\overline{u}}_1|)\\ +C_{15}({\epsilon }_1){\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_1(t)|}^p\right)}^{1/p}+C_{16}({\epsilon }_1){\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_1(t)|}^p\right)}^{({\alpha }_1+1)/p}\\ +\frac{{[C(q^{\prime })]}^q}{q{a}_4^q}\sum _{t=1}^T{|\mathrm{\Delta }{u}_2(t)|}^q+\frac{{[a_4{M}_{21}{c}_{20}]}^{q^{\prime }}}{q^{\prime }}{w}_2^{q^{\prime }}(|P_2{\overline{u}}_2|)\\ +C_{25}({\epsilon }_2){\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_2(t)|}^q\right)}^{1/q}+C_{26}({\epsilon }_2){\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_2(t)|}^q\right)}^{({\alpha }_2+1)/q}.\end{array}$$

(3.6)

Hence, by ($\mathcal{A}2$) and (3.6), we have

$$\begin{array}{r}{D}_0({\parallel {\tilde{u}}_1^{[n]}\parallel }_p+{\parallel {\tilde{u}}_2^{[n]}\parallel }_q)\\ \ge \left|〈{\phi }^{\prime }(u_1^{[n]},u_2^{[n]}),({\tilde{u}}_1^{[n]},{\tilde{u}}_2^{[n]})〉\right|\\ =\left|\sum _{t=1}^T\right[({\varphi }_1(\mathrm{\Delta }{u}_1^{[n]}(t)),\mathrm{\Delta }{u}_1^{[n]}(t))+({\varphi }_2(\mathrm{\Delta }{u}_2^{[n]}(t)),\mathrm{\Delta }{u}_2^{[n]}(t))\\ +({\mathrm{\nabla }}_{x_1}F(t,u_1^{[n]}(t),u_2^{[n]}(t)),{\tilde{u}}_1^{[n]}(t))+({\mathrm{\nabla }}_{x_2}F(t,u_1^{[n]}(t),u_2^{[n]}(t)),{\tilde{u}}_2^{[n]}(t))\\ +(h_1(t),{\tilde{u}}_1^{[n]}(t))+(h_2(t),{\tilde{u}}_2^{[n]}(t))\left]\right|\\ \ge d_1\sum _{t=1}^T{|\mathrm{\Delta }{u}_1^{[n]}(t)|}^p-d_2\sum _{t=1}^T{|\mathrm{\Delta }{u}_1^{[n]}(t)|}^{{\beta }_3}+d_3\sum _{t=1}^T{|\mathrm{\Delta }{u}_2^{[n]}(t)|}^p\\ -d_4\sum _{t=1}^T{|\mathrm{\Delta }{u}_2^{[n]}(t)|}^{{\beta }_4}-\frac{{[C(p^{\prime })]}^p}{p{a}_3^p}\sum _{t=1}^T{|\mathrm{\Delta }{u}_1^{[n]}(t)|}^p-\frac{{[a_3{M}_{11}{c}_{10}]}^{p^{\prime }}}{p^{\prime }}{w}_1^{p^{\prime }}\left(\left|P_1{\overline{u}}_1^{[n]}\right|\right)\\ -C_{15}({\epsilon }_1){\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_1^{[n]}(t)|}^p\right)}^{1/p}-C_{16}({\epsilon }_1){\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_1^{[n]}(t)|}^p\right)}^{({\alpha }_1+1)/p}\\ -\frac{{[C(q^{\prime })]}^q}{q{a}_4^q}\sum _{t=1}^T{|\mathrm{\Delta }{u}_2^{[n]}(t)|}^q-\frac{{[a_4{M}_{21}{c}_{20}]}^{q^{\prime }}}{q^{\prime }}{w}_2^{q^{\prime }}\left(\left|P_2{\overline{u}}_2^{[n]}\right|\right)\\ -C_{25}({\epsilon }_2){\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_2^{[n]}(t)|}^q\right)}^{1/q}-C_{26}({\epsilon }_2){\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_2^{[n]}(t)|}^q\right)}^{({\alpha }_2+1)/q}\\ -M_{14}C\left(p^{\prime }\right){\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_1^{[n]}(t)|}^p\right)}^{1/p}-M_{24}C\left(q^{\prime }\right){\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_2^{[n]}(t)|}^q\right)}^{1/q}\\ \ge (d_1-\frac{{[C(p^{\prime })]}^p}{p{a}_3^p})\sum _{t=1}^T{|\mathrm{\Delta }{u}_1^{[n]}(t)|}^p-d_2{T}^{1-\frac{{\beta }_3}{p}}{\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_1^{[n]}(t)|}^p\right)}^{{\beta }_3/p}\\ +(d_3-\frac{{[C(q^{\prime })]}^q}{q{a}_4^q})\sum _{t=1}^T{|\mathrm{\Delta }{u}_2^{[n]}(t)|}^p-d_4{T}^{1-\frac{{\beta }_4}{q}}{\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_2^{[n]}(t)|}^q\right)}^{{\beta }_4/q}\\ -\frac{{[a_3{M}_{11}{c}_{10}]}^{p^{\prime }}}{p^{\prime }}{w}_1^{p^{\prime }}\left(\left|P_1{\overline{u}}_1^{[n]}\right|\right)\\ -C_{15}({\epsilon }_1){\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_1^{[n]}(t)|}^p\right)}^{1/p}-C_{16}({\epsilon }_1){\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_1^{[n]}(t)|}^p\right)}^{({\alpha }_1+1)/p}\\ -\frac{{[a_4{M}_{21}{c}_{20}]}^{q^{\prime }}}{q^{\prime }}{w}_2^{q^{\prime }}\left(\left|P_2{\overline{u}}_2^{[n]}\right|\right)\\ -C_{25}({\epsilon }_2){\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_2^{[n]}(t)|}^q\right)}^{1/q}-C_{26}({\epsilon }_2){\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_2^{[n]}(t)|}^q\right)}^{({\alpha }_2+1)/q}\\ -M_{14}C\left(p^{\prime }\right){\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_1^{[n]}(t)|}^p\right)}^{1/p}-M_{24}C\left(q^{\prime }\right){\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_2^{[n]}(t)|}^q\right)}^{1/q}\end{array}$$

(3.7)

for all $n\in \mathbb{R}$. Moreover, by Lemma 2.1, we have

$$\begin{array}{r}{D}_0({\parallel {\tilde{u}}_1^{[n]}\parallel }_p+{\parallel {\tilde{u}}_2^{[n]}\parallel }_q)\\ \le D_0{(C(p,p^{\prime })+1)}^{1/p}{\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_1^{[n]}(t)|}^p\right)}^{1/p}\\ +D_0{(C(q,q^{\prime })+1)}^{1/q}{\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_2^{[n]}(t)|}^q\right)}^{1/q}.\end{array}$$

(3.8)

Then (3.7) and (3.8) imply that

$$\begin{array}{r}\frac{{[a_3{M}_{11}{c}_{10}]}^{p^{\prime }}}{p^{\prime }}{w}_1^{p^{\prime }}\left(\left|P_1{\overline{u}}_1^{[n]}\right|\right)+\frac{{[a_4{M}_{21}{c}_{20}]}^{q^{\prime }}}{q^{\prime }}{w}_2^{q^{\prime }}\left(\left|P_2{\overline{u}}_2^{[n]}\right|\right)\\ \ge (d_1-\frac{1}{p})\sum _{t=1}^T{|\mathrm{\Delta }{u}_1^{[n]}(t)|}^p+(d_3-\frac{1}{q})\sum _{t=1}^T{|\mathrm{\Delta }{u}_2^{[n]}(t)|}^q+D_1,\end{array}$$

(3.9)

where

$$\begin{array}{rcl}{D}_1& =& \underset{s\in [0,+\mathrm{\infty })}{min}\{(\frac{1}{p}-\frac{{[C(p^{\prime })]}^p}{p{a}_3^p})s^p-d_2{T}^{1-\frac{{\beta }_3}{p}}{s}^{{\beta }_3}-C_{16}({\epsilon }_1)s^{{\alpha }_1+1}\\ -(C_{15}({\epsilon }_1)+M_{14}C\left(p^{\prime }\right))s\}\\ +\underset{s\in [0,+\mathrm{\infty })}{min}\{(\frac{1}{q}-\frac{{[C(q^{\prime })]}^q}{q{a}_4^q})s^p-d_4{T}^{1-\frac{{\beta }_4}{q}}{s}^{{\beta }_4}-C_{26}({\epsilon }_2)s^{{\alpha }_2+1}\\ -(C_{25}({\epsilon }_2)+M_{24}C\left(q^{\prime }\right))s\}.\end{array}$$

Note that $a_3>C(p^{\prime })$, $a_4>C(q^{\prime })$, ${\alpha }_1,{\beta }_3\in [0,p)$, and ${\alpha }_2,{\beta }_3\in [0,q)$. Hence (3.9) implies that there exist positive constants $D_2$ and $D_3$ such that

$$\begin{array}{rl}\sum _{t=1}^T{|\mathrm{\Delta }{u}_1^{[n]}(t)|}^p\le & \frac{p{[a_3{M}_{11}{c}_{10}]}^{p^{\prime }}}{(d_{1}p-1)p^{\prime }}{w}_1^{p^{\prime }}\left(\left|P_1{\overline{u}}_1^{[n]}\right|\right)\\ +\frac{p{[a_4{M}_{21}{c}_{20}]}^{q^{\prime }}}{(d_{1}p-1)q^{\prime }}{w}_2^{q^{\prime }}\left(\left|P_2{\overline{u}}_2^{[n]}\right|\right)+D_2,\end{array}$$

(3.10)

$$\begin{array}{rl}\sum _{t=1}^T{|\mathrm{\Delta }{u}_2^{[n]}(t)|}^q\le & \frac{q{[a_3{M}_{11}{c}_{10}]}^{p^{\prime }}}{(d_{3}q-1)p^{\prime }}{w}_1^{p^{\prime }}\left(P_1{\overline{u}}_1^{[n]}\right)\\ +\frac{q{[a_4{M}_{21}{c}_{20}]}^{q^{\prime }}}{(d_{3}q-1)q^{\prime }}{w}_2^{q^{\prime }}\left(\left|P_2{\overline{u}}_2^{[n]}\right|\right)+D_3.\end{array}$$

(3.11)

Then it is easy to see that $-\mathrm{\infty }<D_1<0$. By (3.2), we know that

$$\begin{array}{r}\sum _{t=1}^T|F(t,u_1^{[n]}(t),u_2^{[n]}(t))-F(t,P_1{\overline{u}}_1^{[n]},P_2{\overline{u}}_2^{[n]})|\\ \le \frac{C(p^{\prime })}{p{a}_3}\sum _{t=1}^T{|\mathrm{\Delta }{u}_1^{[n]}(t)|}^p+\frac{M_{11}^{p^{\prime }}{c}_{10}^{p^{\prime }}{a}_3^{p^{\prime }/p}C(p^{\prime })}{p^{\prime }}{w}_1^{p^{\prime }}\left(\left|P_1{\overline{u}}_1^{[n]}\right|\right)+C_{14}\\ +C_{11}({\epsilon }_1){\left[C\left(p^{\prime }\right)\right]}^{{\alpha }_1+1}{\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_1^{[n]}(t)|}^p\right)}^{\frac{{\alpha }_1+1}{p}}+M_{11}{M}_{13}{c}_{10}{w}_1\left(\left|P_1{\overline{u}}_1^{[n]}\right|\right)\\ +C_{12}({\epsilon }_1){\left[C\left(p^{\prime }\right)\right]}^{{\alpha }_1}{\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_1^{[n]}(t)|}^p\right)}^{\frac{{\alpha }_1}{p}}+C_{13}({\epsilon }_1)C\left(p^{\prime }\right){\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_1^{[n]}(t)|}^p\right)}^{\frac{1}{p}}\\ +\frac{C(q^{\prime })}{q{a}_4}\sum _{t=1}^T{|\mathrm{\Delta }{u}_2^{[n]}(t)|}^q+\frac{M_{21}^{q^{\prime }}{c}_{20}^{q^{\prime }}{a}_4^{q^{\prime }/q}C(q^{\prime })}{q^{\prime }}{w}_2^{q^{\prime }}\left(\left|P_2{\overline{u}}_2^{[n]}\right|\right)+C_{24}\\ +M_{21}{M}_{23}{c}_{20}{w}_2\left(\left|P_2{\overline{u}}_2^{[n]}\right|\right)+C_{21}({\epsilon }_2){\left[C\left(q^{\prime }\right)\right]}^{{\alpha }_2+1}{\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_2^{[n]}(t)|}^q\right)}^{\frac{{\alpha }_2+1}{q}}\\ +C_{23}({\epsilon }_2)C\left(q^{\prime }\right){\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_2^{[n]}(t)|}^q\right)}^{\frac{1}{q}}\\ +C_{22}({\epsilon }_2){\left[C\left(q^{\prime }\right)\right]}^{{\alpha }_2}{\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_2^{[n]}(t)|}^q\right)}^{\frac{{\alpha }_2}{q}}.\end{array}$$

(3.12)

By ($\mathcal{A}1$)′, (3.10), (3.11), (3.12), and Lemma 2.1, we have

$$\begin{array}{r}\phi \left(u^{[n]}\right)\\ =\phi (u_1^{[n]},u_2^{[n]})\\ =\sum _{t=1}^T[{\mathrm{\Phi }}_1(\mathrm{\Delta }{u}_1^{[n]}(t))+{\mathrm{\Phi }}_2(\mathrm{\Delta }{u}_2^{[n]}(t))+F(t,u_1^{[n]}(t),u_2^{[n]}(t))\\ +(h_1(t),u_1^{[n]}(t))+(h_2(t),u_2^{[n]}(t))]\\ =\sum _{t=1}^T[{\mathrm{\Phi }}_1(\mathrm{\Delta }{u}_1^{[n]}(t))+{\mathrm{\Phi }}_2(\mathrm{\Delta }{u}_2^{[n]}(t))+F(t,u_1^{[n]}(t),u_2^{[n]}(t))-F(t,P_1{\overline{u}}_1^{[n]},P_2{\overline{u}}_2^{[n]})\\ +F(t,P_1{\overline{u}}_1^{[n]},P_2{\overline{u}}_2^{[n]})+(h_1(t),u_1^{[n]}(t))+(h_2(t),u_2^{[n]}(t))]\\ \le \sum _{t=1}^T({\zeta }_1{|\mathrm{\Delta }{u}_1^{[n]}(t)|}^p+{\zeta }_3{|\mathrm{\Delta }{u}_2^{[n]}(t)|}^q+{\zeta }_2{|\mathrm{\Delta }{u}_1^{[n]}(t)|}^{{\theta }_1}+{\zeta }_4{|\mathrm{\Delta }{u}_2^{[n]}(t)|}^{{\theta }_2})\\ +\frac{C(p^{\prime })}{p{a}_3}\sum _{t=1}^T{|\mathrm{\Delta }{u}_1^{[n]}(t)|}^p+\frac{M_{11}^{p^{\prime }}{c}_{10}^{p^{\prime }}{a}_3^{p^{\prime }/p}C(p^{\prime })}{p^{\prime }}{w}_1^{p^{\prime }}\left(\left|P_1{\overline{u}}_1^{[n]}\right|\right)\\ +M_{11}{M}_{13}{c}_{10}{w}_1\left(\left|P_1{\overline{u}}_1^{[n]}\right|\right)+C_{11}({\epsilon }_1){\left[C\left(p^{\prime }\right)\right]}^{{\alpha }_1+1}{\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_1^{[n]}(t)|}^p\right)}^{\frac{{\alpha }_1+1}{p}}\\ +C_{12}({\epsilon }_1){\left[C\left(p^{\prime }\right)\right]}^{{\alpha }_1}{\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_1^{[n]}(t)|}^p\right)}^{\frac{{\alpha }_1}{p}}+C_{13}({\epsilon }_1)C\left(p^{\prime }\right){\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_1^{[n]}(t)|}^p\right)}^{\frac{1}{p}}\\ +C_{14}+{\parallel {\tilde{u}}_1^{[n]}\parallel }_{\mathrm{\infty }}\sum _{t=1}^T|h_1(t)|\\ +\frac{C(q^{\prime })}{q{a}_4}\sum _{t=1}^T{|\mathrm{\Delta }{u}_2^{[n]}(t)|}^q+\frac{M_{21}^{q^{\prime }}{c}_{20}^{q^{\prime }}{a}_4^{q^{\prime }/q}C(q^{\prime })}{q^{\prime }}{w}_2^{q^{\prime }}\left(\left|P_2{\overline{u}}_2^{[n]}\right|\right)\\ +M_{21}{M}_{23}{c}_{20}{w}_2\left(\left|P_2{\overline{u}}_2^{[n]}\right|\right)+C_{21}({\epsilon }_2){\left[C\left(q^{\prime }\right)\right]}^{{\alpha }_2+1}{\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_2^{[n]}(t)|}^q\right)}^{\frac{{\alpha }_2+1}{q}}\\ +C_{22}({\epsilon }_2){\left[C\left(q^{\prime }\right)\right]}^{{\alpha }_2}{\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_2^{[n]}(t)|}^q\right)}^{\frac{{\alpha }_2}{q}}+C_{23}({\epsilon }_2)C\left(q^{\prime }\right){\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_2^{[n]}(t)|}^q\right)}^{\frac{1}{q}}\\ +C_{24}+{\parallel {\tilde{u}}_2^{[n]}\parallel }_{\mathrm{\infty }}\sum _{t=1}^T|h_2(t)|+\sum _{t=1}^{T}F(t,P_1{\overline{u}}_1^{[n]},P_2{\overline{u}}_2^{[n]})\\ \le (\frac{C(p^{\prime })}{p{a}_3}+{\zeta }_1)(\frac{p{[a_3{M}_{11}{c}_{10}]}^{p^{\prime }}}{(d_{1}p-1)p^{\prime }}{w}_1^{p^{\prime }}\left(\left|P_1{\overline{u}}_1^{[n]}\right|\right)+\frac{p{[a_4{M}_{21}{c}_{20}]}^{q^{\prime }}}{(d_{1}p-1)q^{\prime }}{w}_2^{q^{\prime }}\left(\left|P_2{\overline{u}}_2^{[n]}\right|\right))\\ +\frac{M_{11}^{p^{\prime }}{c}_{10}^{p^{\prime }}{a}_3^{p^{\prime }/p}C(p^{\prime })}{p^{\prime }}{w}_1^{p^{\prime }}\left(\left|P_1{\overline{u}}_1^{[n]}\right|\right)+M_{11}{M}_{13}{c}_{10}{w}_1\left(\left|P_1{\overline{u}}_1^{[n]}\right|\right)\\ +M_{21}{M}_{23}{c}_{20}{w}_2\left(\left|P_2{\overline{u}}_2^{[n]}\right|\right)+C_{11}({\epsilon }_1){\left[C\left(p^{\prime }\right)\right]}^{{\alpha }_1+1}{\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_1^{[n]}(t)|}^p\right)}^{\frac{{\alpha }_1+1}{p}}\\ +C_{13}({\epsilon }_1)C\left(p^{\prime }\right){\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_1^{[n]}(t)|}^p\right)}^{\frac{1}{p}}+{\zeta }_2{T}^{1-\frac{{\theta }_1}{p}}{\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_1^{[n]}(t)|}^p\right)}^{\frac{{\theta }_1}{p}}\\ +D_2(\frac{C(p^{\prime })}{p{a}_3}+{\zeta }_1)+C_{14}\\ +(\frac{C(q^{\prime })}{q{a}_4}+{\zeta }_3)(\frac{q{[a_3{M}_{11}{c}_{10}]}^{p^{\prime }}}{(d_{3}q-1)p^{\prime }}{w}_1^{p^{\prime }}\left(\left|P_1{\overline{u}}_1^{[n]}\right|\right)+\frac{q{[a_4{M}_{21}{c}_{20}]}^{q^{\prime }}}{(d_{3}q-1)q^{\prime }}{w}_2^{q^{\prime }}\left(\left|P_2{\overline{u}}_2^{[n]}\right|\right))\\ +\frac{M_{21}^{q^{\prime }}{c}_{20}^{q^{\prime }}{a}_4^{q^{\prime }/q}C(q^{\prime })}{q^{\prime }}{w}_2^{q^{\prime }}\left(\left|P_2{\overline{u}}_2^{[n]}\right|\right)+C_{22}({\epsilon }_2){\left[C\left(q^{\prime }\right)\right]}^{{\alpha }_2}{\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_2^{[n]}(t)|}^q\right)}^{\frac{{\alpha }_2}{q}}\\ +C_{21}({\epsilon }_2){\left[C\left(q^{\prime }\right)\right]}^{{\alpha }_2+1}{\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_2^{[n]}(t)|}^q\right)}^{\frac{{\alpha }_2+1}{q}}+M_{21}{M}_{23}{c}_{20}{w}_2\left(\left|P_2{\overline{u}}_2^{[n]}\right|\right)\\ +C_{23}({\epsilon }_2)C\left(q^{\prime }\right){\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_2^{[n]}(t)|}^q\right)}^{\frac{1}{q}}+{\zeta }_4{T}^{1-\frac{{\theta }_2}{p}}{\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_2^{[n]}(t)|}^q\right)}^{\frac{{\theta }_2}{q}}+C_{24}\\ +C\left(p^{\prime }\right){\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_1^{[n]}(t)|}^p\right)}^{\frac{1}{p}}\sum _{t=1}^T|h_1(t)|+C\left(q^{\prime }\right){\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_2^{[n]}(t)|}^q\right)}^{\frac{1}{q}}\sum _{t=1}^T|h_2(t)|\\ +\sum _{t=1}^{T}F(t,P_1{\overline{u}}_1^{[n]},P_2{\overline{u}}_2^{[n]})+D_3(\frac{C(q^{\prime })}{q{a}_4}+{\zeta }_3)\\ \le \frac{{[a_3{M}_{11}{c}_{10}]}^{p^{\prime }}}{p^{\prime }}\\ \times [\frac{C(p^{\prime })}{(d_{1}p-1)a_3}+\frac{p{\zeta }_1}{d_{1}p-1}+\frac{C(p^{\prime })}{a_3}+\frac{C(q^{\prime })}{(d_{3}q-1)a_4}+\frac{q{\zeta }_3}{d_{3}q-1}]w_1^{p^{\prime }}\left(\left|P_1{\overline{u}}_1^{[n]}\right|\right)\\ +\frac{{[a_4{M}_{21}{c}_{20}]}^{q^{\prime }}}{q^{\prime }}\\ \times [\frac{C(p^{\prime })}{(d_{1}p-1)a_3}+\frac{p{\zeta }_1}{d_{1}p-1}+\frac{C(q^{\prime })}{a_4}+\frac{C(q^{\prime })}{(d_{3}q-1)a_4}+\frac{q{\zeta }_3}{d_{3}q-1}]w_2^{q^{\prime }}\left(\left|P_2{\overline{u}}_2^{[n]}\right|\right)\\ +C_{17}(\epsilon )w_1^{\frac{p^{\prime }({\alpha }_1+1)}{p}}\left(\left|P_1{\overline{u}}_1^{[n]}\right|\right)+C_{27}(\epsilon )w_2^{\frac{q^{\prime }({\alpha }_2+1)}{q}}\left(\left|P_2{\overline{u}}_2^{[n]}\right|\right)\\ +C_{18}(\epsilon )w_1^{\frac{p^{\prime }{\alpha }_1}{p}}\left(\left|P_1{\overline{u}}_1^{[n]}\right|\right)+C_{28}(\epsilon )w_2^{\frac{q^{\prime }{\alpha }_2}{q}}\left(\left|P_2{\overline{u}}_2^{[n]}\right|\right)\\ +C_{19}{w}_1^{\frac{p^{\prime }{\theta }_1}{p}}\left(\left|P_1{\overline{u}}_1^{[n]}\right|\right)+C_{29}{w}_2^{\frac{q^{\prime }{\theta }_2}{q}}\left(\left|P_2{\overline{u}}_2^{[n]}\right|\right)\\ +D_{10}{w}_1^{\frac{p^{\prime }}{p}}\left(\left|P_1{\overline{u}}_1^{[n]}\right|\right)+D_{20}{w}_2^{\frac{q^{\prime }}{q}}\left(\left|P_2{\overline{u}}_2^{[n]}\right|\right)+\sum _{t=1}^{T}F(t,P_1{\overline{u}}_1^{[n]},P_2{\overline{u}}_2^{[n]})\\ \le \frac{{[a_3{M}_{11}{c}_{10}]}^{p^{\prime }}}{p^{\prime }}[\frac{1}{d_{1}p-1}+\frac{p{\zeta }_1}{d_{1}p-1}+1+\frac{1}{d_{3}q-1}+\frac{q{\zeta }_3}{d_{3}q-1}]w_1^{p^{\prime }}\left(\left|P_1{\overline{u}}_1^{[n]}\right|\right)\\ +\frac{{[a_4{M}_{21}{c}_{20}]}^{q^{\prime }}}{q^{\prime }}[\frac{1}{d_{1}p-1}+\frac{p{\zeta }_1}{d_{1}p-1}+1+\frac{1}{d_{3}q-1}+\frac{q{\zeta }_3}{d_{3}q-1}]w_2^{q^{\prime }}\left(\left|P_2{\overline{u}}_2^{[n]}\right|\right)\\ +C_{17}(\epsilon )w_1^{\frac{p^{\prime }({\alpha }_1+1)}{p}}\left(\left|P_1{\overline{u}}_1^{[n]}\right|\right)+C_{27}(\epsilon )w_2^{\frac{q^{\prime }({\alpha }_2+1)}{q}}\left(\left|P_2{\overline{u}}_2^{[n]}\right|\right)\\ +C_{18}(\epsilon )w_1^{\frac{p^{\prime }{\alpha }_1}{p}}\left(\left|P_1{\overline{u}}_1^{[n]}\right|\right)+C_{28}(\epsilon )w_2^{\frac{q^{\prime }{\alpha }_2}{q}}\left(\left|P_2{\overline{u}}_2^{[n]}\right|\right)+C_{19}{w}_1^{\frac{p^{\prime }{\theta }_1}{p}}\left(\left|P_1{\overline{u}}_1^{[n]}\right|\right)\\ +C_{29}{w}_2^{\frac{q^{\prime }{\theta }_2}{q}}\left(\left|P_2{\overline{u}}_2^{[n]}\right|\right)+D_{10}{w}_1^{\frac{p^{\prime }}{p}}\left(\left|P_1{\overline{u}}_1^{[n]}\right|\right)+D_{20}{w}_2^{\frac{q^{\prime }}{q}}\left(\left|P_2{\overline{u}}_2^{[n]}\right|\right)\\ +\sum _{t=1}^{T}F(t,P_1{\overline{u}}_1^{[n]},P_2{\overline{u}}_2^{[n]})\\ \le (w_1^{p^{\prime }}\left(\left|P_1{\overline{u}}_1^{[n]}\right|\right)+w_2^{q^{\prime }}\left(\left|P_2{\overline{u}}_2^{[n]}\right|\right))[max\{\frac{{[a_3{M}_{11}{c}_{10}]}^{p^{\prime }}}{p^{\prime }}[\frac{1+p{\zeta }_1}{d_{1}p-1}+\frac{1+q{\zeta }_3}{d_{3}q-1}+1],\\ \frac{{[a_4{M}_{21}{c}_{20}]}^{q^{\prime }}}{q^{\prime }}[\frac{1+p{\zeta }_1}{d_{1}p-1}+\frac{1+q{\zeta }_3}{d_{3}q-1}+1]\}+\frac{{\sum }_{t=1}^{T}F(t,P_1{\overline{u}}_1^{[n]},P_2{\overline{u}}_2^{[n]})}{w_1^{p^{\prime }}(|P_1{\overline{u}}_1^{[n]}|)+w_2^{q^{\prime }}(|P_2{\overline{u}}_2^{[n]}|)}]\\ +C_{17}(\epsilon )w_1^{\frac{p^{\prime }({\alpha }_1+1)}{p}}\left(\left|P_1{\overline{u}}_1^{[n]}\right|\right)+C_{18}(\epsilon )w_1^{\frac{p^{\prime }{\alpha }_1}{p}}\left(\left|P_1{\overline{u}}_1^{[n]}\right|\right)+C_{19}{w}_1^{\frac{p^{\prime }{\theta }_1}{p}}\left(\left|P_1{\overline{u}}_1^{[n]}\right|\right)\\ +C_{27}(\epsilon )w_2^{\frac{q^{\prime }({\alpha }_2+1)}{q}}\left(\left|P_2{\overline{u}}_2^{[n]}\right|\right)+C_{28}(\epsilon )w_2^{\frac{q^{\prime }{\alpha }_2}{q}}\left(\left|P_2{\overline{u}}_2^{[n]}\right|\right)+C_{29}{w}_2^{\frac{q^{\prime }{\theta }_2}{q}}\left(\left|P_2{\overline{u}}_2^{[n]}\right|\right)\\ +D_{10}{w}_1^{\frac{p^{\prime }}{p}}\left(\left|P_1{\overline{u}}_1^{[n]}\right|\right)+D_{20}{w}_2^{\frac{q^{\prime }}{q}}\left(\left|P_2{\overline{u}}_2^{[n]}\right|\right).\end{array}$$

(3.13)

Then (3.5) and ($\mathcal{A}3$) imply that $\{P_1{\overline{u}}_1^{[n]}\}$, $\{P_2{\overline{u}}_2^{[n]}\}$, $\{w_1(P_1{\overline{u}}_2^{[n]})\}$, and $\{w_2(P_2{\overline{u}}_2^{[n]})\}$ are bounded. Furthermore, (3.10), (3.11), and (3.8) imply that $\{{\tilde{u}}_1^{[n]}\}$ and $\{{\tilde{u}}_2^{[n]}\}$ are bounded. Then $\{u^{[n]}\}$ is bounded in ℋ. Since $dim\mathcal{H}<\mathrm{\infty }$, $\{u^{[n]}\}$ has a convergent subsequence. Hence, Ψ satisfies the (PS) condition.

In order to use Lemma 2.3, next we prove the following conclusions:

• (i) $inf\{\mathrm{\Psi }((z,v))|(z,v)\in Z\times T^{r_1+r_2}\}>-\mathrm{\infty }$;

• (ii) $\mathrm{\Psi }((y,v))\to -\mathrm{\infty }$ uniformly for $(y,v)\in Y\times T^{r_1+r_2}$ as $|y|\to \mathrm{\infty }$.

For $(z,v)\in Z\times T^{r_1+r_2}$, set $u=u(t)=z(t)+v={(z_1(t)+v_1,z_2(t)+v_2)}^{\tau }$. Then $z_m(t)={\tilde{u}}_m(t)$, $v_m=Q_m{\overline{u}}_m$, and $u_m(t)=z_m(t)+v_m$, $m=1,2$. By ($\mathcal{F}3$) and Lemma 2.1, we have

$$\begin{array}{r}\left|\sum _{t=1}^T[F(t,u_1(t),u_2(t))-F(t,0,0)]\right|\\ \le \sum _{t=1}^T|F(t,u_1(t),u_2(t))-F(t,0,u_2(t))|+\sum _{t=1}^T|F(t,0,u_2(t))-F(t,0,0)|\\ \le \sum _{t=1}^T\left|{\int }_0^1({\mathrm{\nabla }}_{x_1}F(t,s(Q_1{\overline{u}}_1+{\tilde{u}}_1(t)),u_2(t)),Q_1{\overline{u}}_1+{\tilde{u}}_1(t))ds\right|\\ +\sum _{t=1}^T\left|{\int }_0^1({\mathrm{\nabla }}_{x_2}F(t,0,s(Q_2{\overline{u}}_2+{\tilde{u}}_2(t))),Q_2{\overline{u}}_2+{\tilde{u}}_2(t))ds\right|\\ \le (|Q_1{\overline{u}}_1|+{\parallel {\tilde{u}}_1\parallel }_{\mathrm{\infty }})\sum _{t=1}^T{\int }_0^1\left|{\mathrm{\nabla }}_{x_1}F(t,s(Q_1{\overline{u}}_1+{\tilde{u}}_1(t)),u_2(t))\right|ds\\ +(|Q_2{\overline{u}}_2|+{\parallel {\tilde{u}}_2\parallel }_{\mathrm{\infty }})\sum _{t=1}^T{\int }_0^1\left|{\mathrm{\nabla }}_{x_2}F(t,0,s(Q_2{\overline{u}}_2+{\tilde{u}}_2(t)))\right|ds\\ \le (M_{13}+{\parallel {\tilde{u}}_1\parallel }_{\mathrm{\infty }})\sum _{t=1}^T{\int }_0^1[f_1(t)w_1\left(\left|s(Q_1{\overline{u}}_1+{\tilde{u}}_1(t))\right|\right)+g_1(t)]ds\\ +(M_{23}+{\parallel {\tilde{u}}_2\parallel }_{\mathrm{\infty }})\sum _{t=1}^T{\int }_0^1[f_2(t)w_2\left(\left|s(Q_2{\overline{u}}_2+{\tilde{u}}_2(t))\right|\right)+g_2(t)]ds\\ \le (M_{13}+{\parallel {\tilde{u}}_1\parallel }_{\mathrm{\infty }})\sum _{t=1}^T{\int }_0^1[k_{11}{f}_1(t){\left|s(Q_1{\overline{u}}_1+{\tilde{u}}_1(t))\right|}^{{\alpha }_1}+k_{12}{f}_1(t)+g_1(t)]ds\\ +(M_{23}+{\parallel {\tilde{u}}_2\parallel }_{\mathrm{\infty }})\sum _{t=1}^T{\int }_0^1[k_{21}{f}_2(t){\left|s(Q_2{\overline{u}}_2+{\tilde{u}}_2(t))\right|}^{{\alpha }_2}+k_{22}{f}_2(t)+g_2(t)]ds\\ \le (M_{13}+{\parallel {\tilde{u}}_1\parallel }_{\mathrm{\infty }})\sum _{t=1}^T[k_{11}{f}_1(t){|Q_1{\overline{u}}_1+{\tilde{u}}_1(t)|}^{{\alpha }_1}+k_{12}{f}_1(t)+g_1(t)]\\ +(M_{23}+{\parallel {\tilde{u}}_2\parallel }_{\mathrm{\infty }})\sum _{t=1}^T[k_{21}{f}_2(t){|Q_2{\overline{u}}_2+{\tilde{u}}_2(t)|}^{{\alpha }_2}+k_{22}{f}_2(t)+g_2(t)]\\ \le D_{11}{\parallel {\tilde{u}}_1\parallel }_{\mathrm{\infty }}^{{\alpha }_1+1}+D_{21}{\parallel {\tilde{u}}_2\parallel }_{\mathrm{\infty }}^{{\alpha }_2+1}+D_{12}{\parallel {\tilde{u}}_1\parallel }_{\mathrm{\infty }}^{{\alpha }_1}+D_{22}{\parallel {\tilde{u}}_2\parallel }_{\mathrm{\infty }}^{{\alpha }_2}\\ +D_{13}{\parallel {\tilde{u}}_1\parallel }_{\mathrm{\infty }}+D_{23}{\parallel {\tilde{u}}_2\parallel }_{\mathrm{\infty }}+D_4\\ \le D_{11}C\left(p^{\prime }\right){\left(\sum _{s=1}^T{|\mathrm{\Delta }{u}_1(s)|}^p\right)}^{\frac{{\alpha }_1+1}{p}}+D_{21}C\left(q^{\prime }\right){\left(\sum _{s=1}^T{|\mathrm{\Delta }{u}_2(s)|}^q\right)}^{\frac{{\alpha }_2+1}{q}}\\ +D_{12}C\left(p^{\prime }\right){\left(\sum _{s=1}^T{|\mathrm{\Delta }{u}_1(s)|}^p\right)}^{\frac{{\alpha }_1}{p}}+D_{22}C\left(q^{\prime }\right){\left(\sum _{s=1}^T{|\mathrm{\Delta }{u}_2(s)|}^q\right)}^{\frac{{\alpha }_2}{q}}\\ +D_{13}C\left(p^{\prime }\right){\left(\sum _{s=1}^T{|\mathrm{\Delta }{u}_1(s)|}^p\right)}^{\frac{1}{p}}+D_{23}C\left(q^{\prime }\right){\left(\sum _{s=1}^T{|\mathrm{\Delta }{u}_2(s)|}^q\right)}^{\frac{1}{q}}+D_4.\end{array}$$

(3.14)

Then

$$\begin{array}{rl}\mathrm{\Psi }((z,v))=& \phi (u_1,u_2)\\ =& \sum _{t=1}^T[{\mathrm{\Phi }}_1(\mathrm{\Delta }{u}_1(t))+{\mathrm{\Phi }}_2(\mathrm{\Delta }{u}_2(t))+F(t,u_1(t),u_2(t))\\ +(h_1(t),u_1(t))+(h_2(t),u_2(t))]\\ =& \sum _{t=1}^T[{\mathrm{\Phi }}_1(\mathrm{\Delta }{u}_1(t))+{\mathrm{\Phi }}_2(\mathrm{\Delta }{u}_2(t))+F(t,u_1(t),u_2(t))-F(t,0,0)\\ +F(t,0,0)+(h_1(t),u_1(t))+(h_2(t),u_2(t))]\\ \ge & \sum _{t=1}^T({\gamma }_1{|\mathrm{\Delta }{u}_1(t)|}^p+{\gamma }_3{|\mathrm{\Delta }{u}_2(t)|}^q-{\gamma }_2{|\mathrm{\Delta }{u}_1(t)|}^{{\beta }_1}-{\gamma }_4{|\mathrm{\Delta }{u}_2(t)|}^{{\beta }_2})\\ -D_{11}C\left(p^{\prime }\right){\left(\sum _{s=1}^T{|\mathrm{\Delta }{u}_1(s)|}^p\right)}^{\frac{{\alpha }_1+1}{p}}-D_{21}C\left(q^{\prime }\right){\left(\sum _{s=1}^T{|\mathrm{\Delta }{u}_2(s)|}^q\right)}^{\frac{{\alpha }_2+1}{q}}\\ -D_{12}C\left(p^{\prime }\right){\left(\sum _{s=1}^T{|\mathrm{\Delta }{u}_1(s)|}^p\right)}^{\frac{{\alpha }_1}{p}}-D_{22}C\left(q^{\prime }\right){\left(\sum _{s=1}^T{|\mathrm{\Delta }{u}_2(s)|}^q\right)}^{\frac{{\alpha }_2}{q}}\\ -D_{13}C\left(p^{\prime }\right){\left(\sum _{s=1}^T{|\mathrm{\Delta }{u}_1(s)|}^p\right)}^{\frac{1}{p}}-D_{23}C\left(q^{\prime }\right){\left(\sum _{s=1}^T{|\mathrm{\Delta }{u}_2(s)|}^q\right)}^{\frac{1}{q}}-D_4\\ -M_{14}C\left(p^{\prime }\right){\left(\sum _{s=1}^T{|\mathrm{\Delta }{u}_1(s)|}^p\right)}^{\frac{1}{p}}-M_{24}C\left(q^{\prime }\right){\left(\sum _{s=1}^T{|\mathrm{\Delta }{u}_2(s)|}^q\right)}^{\frac{1}{q}}\\ +\sum _{t=1}^{T}F(t,0,0).\end{array}$$

(3.15)

It is easy to see that conclusion (i) holds from (3.15).

For any $(y,v)\in Y\times T^{r_1+r_2}$, it follows from (1.2) and (2.11) that

$$\begin{array}{r}\mathrm{\Psi }((y,v))\\ =\phi (y_1+v_1,y_2+v_2)\\ =\sum _{t=1}^{T}F(t,y_1+v_1,y_2+v_2)\\ =\sum _{t=1}^{T}F(t,y_1+v_1,y_2+v_2)-\sum _{t=1}^{T}F(t,y_1,y_2+v_2)\\ +\sum _{t=1}^{T}F(t,y_1,y_2+v_2)-\sum _{t=1}^{T}F(t,y_1,y_2)+\sum _{t=1}^{T}F(t,y_1,y_2)\\ =\sum _{t=1}^{T}F(t,y_1,y_2)+\sum _{t=1}^T{\int }_0^1(\mathrm{\nabla }F(t,y_1+s{v}_1,y_2+v_2),v_1)ds\\ +\sum _{t=1}^T{\int }_0^1(\mathrm{\nabla }F(t,y_1,y_2+s{v}_2),v_2)ds\\ \le \sum _{t=1}^{T}F(t,y_1,y_2)+|v_1|\sum _{t=1}^T{\int }_0^1{f}_1(t)w_1(|y_1+s{v}_1|)ds+|v_1|\sum _{t=1}^T{g}_1(t)\\ +|v_2|\sum _{n=1}^T{\int }_0^1{f}_2(t)w_2(|y_2+s{v}_2|)ds+|v_2|\sum _{t=1}^T{g}_2(t)\\ \le \sum _{t=1}^{T}F(t,y_1,y_2)+|v_1|\sum _{t=1}^T{f}_1(t)w_1(|y_1|)+|v_1|\sum _{t=1}^T{\int }_0^1{w}_1(|s{v}_1|)ds\\ +|v_1|\sum _{t=1}^T{g}_1(t)+|v_2|\sum _{t=1}^T{f}_2(t)w_2(|y_2|)\\ +|v_2|\sum _{t=1}^T{\int }_0^1{w}_2(|s{v}_2|)ds+|v_2|\sum _{t=1}^T{g}_2(t)\\ \le \sum _{t=1}^{T}F(t,y_1,y_2)+D_5{w}_1(|y_1|)+D_6{w}_2(|y_2|)+D_7\\ =[w_1^{p^{\prime }{\alpha }_1}(|y_1|)+w_2^{q^{\prime }{\alpha }_2}(|y_2|)]({[w_1^{p^{\prime }{\alpha }_1}(|y_1|)+w_2^{q^{\prime }{\alpha }_2}(|y_2|)]}^{-1}\sum _{t=1}^{T}F(t,y_1,y_2))\\ +D_5{w}_1(|y_1|)+D_6{w}_2(|y_2|)+D_7\end{array}$$

for positive constants $D_5$, $D_6$, and $D_7$. Hence, the above inequality, (3.5) and ($\mathcal{A}3$) imply that conclusion (ii) holds. It follows from Lemma 2.6 that Ψ has at least $r_1+r_2+1$ critical points. Hence φ has at least $r_1+r_2+1$ geometrically distinct critical points. Therefore, system (1.1) has at least $r_1+r_2+1$ geometrically distinct solutions in ℋ. The proof is complete. □

Proof of Theorem 1.3 Note that ${\mathrm{\Phi }}_m$ are coercive, $m=1,2$. Then by Remark 1.1, we know that (1.2) holds. Hence, it follows from (1.2), ($\mathcal{F}5$), ($\mathcal{F}6$), and (ℰ) that

$$\begin{array}{r}\left|\sum _{t=1}^T[F(t,u_1(t),u_2(t))-F(t,P_1{\overline{u}}_1,P_2{\overline{u}}_2)]\right|\\ \le \sum _{t=1}^T|F(t,u_1(t),u_2(t))-F(t,P_1{\overline{u}}_1,u_2(t))|\\ +\sum _{t=1}^T|F(t,P_1{\overline{u}}_1,u_2(t)-F(t,P_1{\overline{u}}_1,P_2{\overline{u}}_2)|\\ \le \sum _{t=1}^T\left|{\int }_0^1({\mathrm{\nabla }}_{x_1}F(t,P_1{\overline{u}}_1+s(Q_1{\overline{u}}_1+{\tilde{u}}_1(t)),u_2(t)),Q_1{\overline{u}}_1+{\tilde{u}}_1(t))ds\right|\\ +\sum _{t=1}^T\left|{\int }_0^1({\mathrm{\nabla }}_{x_2}F(t,P_1{\overline{u}}_1,P_2{\overline{u}}_2+s(Q_2{\overline{u}}_2+{\tilde{u}}_2(t))),Q_2{\overline{u}}_2+{\tilde{u}}_2(t))ds\right|\\ \le \sum _{t=1}^T{b}_1(t)|Q_1{\overline{u}}_1+{\tilde{u}}_1(t)|+\sum _{t=1}^T{b}_2(t)|Q_2{\overline{u}}_2+{\tilde{u}}_2(t)|\\ \le (|Q_1{\overline{u}}_1|+{\parallel {\tilde{u}}_1\parallel }_{\mathrm{\infty }})\sum _{t=1}^T{b}_1(t)+(|Q_2{\overline{u}}_2|+{\parallel {\tilde{u}}_2\parallel }_{\mathrm{\infty }})\sum _{t=1}^T{b}_2(t)\\ \le M_{13}{M}_{15}+C\left(p^{\prime }\right)M_{15}{\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_1(t)|}^p\right)}^{1/p}+M_{23}{M}_{25}\\ +C\left(q^{\prime }\right)M_{25}{\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_2(t)|}^q\right)}^{1/q}.\end{array}$$

(3.16)

Then

$$\begin{array}{rcl}\phi (u)& =& \phi (u_1,u_2)\\ =& \sum _{t=1}^T[{\mathrm{\Phi }}_1(\mathrm{\Delta }{u}_1(t))+{\mathrm{\Phi }}_2(\mathrm{\Delta }{u}_2(t))+F(t,u_1(t),u_2(t))-F(t,P_1{\overline{u}}_1,P_2{\overline{u}}_2)\\ +F(t,P_1{\overline{u}}_1,P_2{\overline{u}}_2)+(h_1(t),u_1(t))+(h_2(t),u_2(t))]\\ \ge & {\delta }_1\sum _{t=1}^T|\mathrm{\Delta }{u}_1(t)|+{\delta }_2\sum _{t=1}^T|\mathrm{\Delta }{u}_2(t)|-C\left(p^{\prime }\right)M_{15}{\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_1(t)|}^p\right)}^{1/p}\\ -C\left(q^{\prime }\right)M_{25}{\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_2(t)|}^q\right)}^{1/q}+\sum _{t=1}^{T}F(t,P_1{\overline{u}}_1,P_2{\overline{u}}_2)\\ -C\left(p^{\prime }\right)M_{14}{\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_1(t)|}^p\right)}^{1/p}-C\left(q^{\prime }\right)M_{24}{\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_2(t)|}^q\right)}^{1/q}\\ -M_{13}{M}_{15}-M_{23}{M}_{25}-({\delta }_1+{\delta }_2)T\\ \ge & {\delta }_1\sum _{t=1}^T|\mathrm{\Delta }{u}_1(t)|+{\delta }_2\sum _{t=1}^T|\mathrm{\Delta }{u}_2(t)|-M_{13}{M}_{15}-C\left(p^{\prime }\right)M_{15}\sum _{t=1}^T|\mathrm{\Delta }{u}_1(t)|\\ -M_{23}{M}_{25}-C\left(q^{\prime }\right)M_{25}\sum _{t=1}^T|\mathrm{\Delta }{u}_2(t)|+\sum _{t=1}^{T}F(t,P_1{\overline{u}}_1,P_2{\overline{u}}_2)-({\delta }_1+{\delta }_2)T\\ -C\left(p^{\prime }\right)M_{14}\sum _{t=1}^T|\mathrm{\Delta }{u}_1(t)|-C\left(q^{\prime }\right)M_{24}\sum _{t=1}^T|\mathrm{\Delta }{u}_2(t)|.\end{array}$$

(3.17)

The features ($\mathcal{F}6$) and ($\mathcal{F}7$) imply that φ is bounded from below. Similar to the proof of Theorem 1.1, we can prove that φ is G-invariant and satisfies the (PS) _G condition. Then by Lemma 2.5, we obtain the conclusion. □

Proof of Theorem 1.4 First, we prove that Ψ defined by (2.11) satisfies the (PS) condition. Assume that ${\{(y^{[n]}+z^{[n]},v^{[n]})\}}_{n=1}^{\mathrm{\infty }}\subset X\times T^{r_1+r_2}$ is a (PS) sequence for Ψ, that is, $\{\mathrm{\Psi }((y^{[n]}+z^{[n]},v^{[n]}))\}$ is bounded and ${\mathrm{\Psi }}^{\prime }((y^{[n]}+z^{[n]},v^{[n]}))\to 0$, where $y^{[n]}={(y_1^{[n]},y_2^{[n]})}^{\tau }\in Y$, $z^{[n]}=z^{[n]}(t)={(z_1^{[n]}(t),z_2^{[n]}(t))}^{\tau }\in Z$, $v^{[n]}={(v_1^{[n]},v_2^{[n]})}^{\tau }\in T^{r_1+r_2}$ for $n=1,2,\dots$ . Let

$$u^{[n]}=y^{[n]}+v^{[n]}+z^{[n]}={(y_1^{[n]}+v_1^{[n]}+z_1^{[n]},y_2^{[n]}+v_2^{[n]}+z_2^{[n]})}^{\tau },n=1,2,\dots .$$

Then it is easy to see that

$$y_m^{[n]}=P_m{\overline{u}}_m^{[n]},v_m^{[n]}=Q_m{\overline{u}}_m^{[n]},z_m^{[n]}(t)={\tilde{u}}_m^{[n]}(t),m=1,2,n=1,2,\dots .$$

By (2.12) and (2.13), we find that $\{\phi (u_1^{[n]},u_2^{[n]})\}$ is bounded and ${\phi }^{\prime }(u_1^{[n]},u_2^{[n]})\to 0$. Then there exists a positive constant $G_0$ such that

$$\left|\phi (u_1^{[n]},u_2^{[n]})\right|\le G_0,\parallel {\phi }^{\prime }(u_1^{[n]},u_2^{[n]})\parallel \le G_0,\mathrm{\forall }n\in \mathbb{N}.$$

(3.18)

It follows from ($\mathcal{F}3$), Lemma 2.1, and Young’s inequality that, for all $(u_1,u_2)\in \mathcal{H}$,

$$\begin{array}{r}|\sum _{t=1}^T({\mathrm{\nabla }}_{x_1}F(t,{\hat{u}}_1(t),{\hat{u}}_2(t)),{\tilde{u}}_1(t))+\sum _{t=1}^T({\mathrm{\nabla }}_{x_2}F(t,{\hat{u}}_1(t),{\hat{u}}_2(t)),{\tilde{u}}_2(t))|\\ \le \left|\sum _{t=1}^T({\mathrm{\nabla }}_{x_1}F(t,P_1{\overline{u}}_1+Q_1{\overline{u}}_1+{\tilde{u}}_1(t),{\hat{u}}_2(t)),{\tilde{u}}_1(t))\right|\\ +\left|\sum _{t=1}^T({\mathrm{\nabla }}_{x_2}F(t,{\hat{u}}_1(t),P_2{\overline{u}}_2+Q_2{\overline{u}}_2+{\tilde{u}}_2(t)),{\tilde{u}}_2(t))\right|\\ \le \sum _{t=1}^T{b}_1(t)|{\tilde{u}}_1(t)|+\sum _{t=1}^T{b}_2(t)|{\tilde{u}}_2(t)|\\ \le {\parallel {\tilde{u}}_1\parallel }_{\mathrm{\infty }}\sum _{t=1}^T{b}_1(t)+{\parallel {\tilde{u}}_2\parallel }_{\mathrm{\infty }}\sum _{t=1}^T{b}_2(t)\\ \le C\left(p^{\prime }\right){\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_1(t)|}^p\right)}^{1/p}\sum _{t=1}^T{b}_1(t)\\ +C\left(q^{\prime }\right){\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_2(t)|}^q\right)}^{1/q}\sum _{t=1}^T{b}_2(t).\end{array}$$

(3.19)

Hence we have

$$\begin{array}{r}{\parallel {\tilde{u}}_1^{[n]}\parallel }_p+{\parallel {\tilde{u}}_2^{[n]}\parallel }_q\\ \ge \left|〈{\phi }^{\prime }(u_1^{[n]},u_2^{[n]}),({\tilde{u}}_1^{[n]},{\tilde{u}}_2^{[n]})〉\right|\\ =\left|\sum _{t=1}^T\right[({\varphi }_1(\mathrm{\Delta }{u}_1^{[n]}(t)),\mathrm{\Delta }{u}_1^{[n]}(t))+({\varphi }_2(\mathrm{\Delta }{u}_2^{[n]}(t)),\mathrm{\Delta }{u}_2^{[n]}(t))\\ +({\mathrm{\nabla }}_{x_1}F(t,u_1^{[n]}(t),u_2^{[n]}(t)),{\tilde{u}}_1^{[n]}(t))+({\mathrm{\nabla }}_{x_2}F(t,u_1^{[n]}(t),u_2^{[n]}(t)),{\tilde{u}}_2^{[n]}(t))\\ +(h_1(t),{\tilde{u}}_1^{[n]}(t))+(h_2(t),{\tilde{u}}_2^{[n]}(t))\left]\right|\\ \ge \sum _{t=1}^T{\delta }_1(|\mathrm{\Delta }{u}_1^{[n]}(t)|-1)+\sum _{t=1}^T{\delta }_2(|\mathrm{\Delta }{u}_2^{[n]}(t)|-1)\\ -C\left(p^{\prime }\right){\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_1^{[n]}(t)|}^p\right)}^{1/p}\sum _{t=1}^T{b}_1(t)-C\left(q^{\prime }\right){\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_2^{[n]}(t)|}^q\right)}^{1/q}\sum _{t=1}^T{b}_2(t)\\ -M_{14}C\left(p^{\prime }\right){\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_1^{[n]}(t)|}^p\right)}^{1/p}-M_{15}C\left(q^{\prime }\right){\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_2^{[n]}(t)|}^q\right)}^{1/q}\\ \ge {\delta }_1{\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_1^{[n]}(t)|}^p\right)}^{1/p}+{\delta }_2{\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_2^{[n]}(t)|}^q\right)}^{1/q}-({\delta }_1+{\delta }_2)T\\ -C\left(p^{\prime }\right){\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_1^{[n]}(t)|}^p\right)}^{1/p}\sum _{t=1}^T{b}_1(t)-C\left(q^{\prime }\right){\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_2^{[n]}(t)|}^q\right)}^{1/q}\sum _{t=1}^T{b}_2(t)\\ -M_{14}C\left(p^{\prime }\right){\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_1^{[n]}(t)|}^p\right)}^{1/p}-M_{15}C\left(q^{\prime }\right){\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_2^{[n]}(t)|}^q\right)}^{1/q}\end{array}$$

(3.20)

for large $n\in \mathbb{N}$ by the fact ${\phi }^{\prime }(u_1^{[n]},u_2^{[n]})\to 0$ as $n\to \mathrm{\infty }$. Moreover, by Lemma 2.1, we have

$$\begin{array}{r}{\parallel {\tilde{u}}_1^{[n]}\parallel }_p+{\parallel {\tilde{u}}_2^{[n]}\parallel }_q\\ \le {(C(p,p^{\prime })+1)}^{1/p}{\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_1^{[n]}(t)|}^p\right)}^{1/p}\\ +{(C(q,q^{\prime })+1)}^{1/q}{\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_2^{[n]}(t)|}^q\right)}^{1/q}.\end{array}$$

(3.21)

Then ($\mathcal{F}7$)′, (3.20), and (3.21) imply that there exists a positive constant $G_1$ such that

$$\sum _{t=1}^T{|\mathrm{\Delta }{u}_1^{[n]}(t)|}^p\le G_1,\sum _{t=1}^T{|\mathrm{\Delta }{u}_2^{[n]}(t)|}^q\le G_1,\mathrm{\forall }n\in \mathbb{N}.$$

(3.22)

By (3.16) and the above inequality, we know that there exists a positive constant $G_2$ such that

$$\begin{array}{r}\sum _{t=1}^T|F(t,u_1^{[n]}(t),u_2^{[n]}(t))-F(t,P_1{\overline{u}}_1^{[n]},P_2{\overline{u}}_2^{[n]})|\\ \le M_{13}{M}_{15}+C\left(p^{\prime }\right)M_{15}{\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_1^{[n]}(t)|}^p\right)}^{1/p}\\ +M_{23}{M}_{25}+C\left(q^{\prime }\right)M_{25}{\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_2^{[n]}(t)|}^q\right)}^{1/q}\\ \le G_2.\end{array}$$

(3.23)

By ($\mathcal{A}$0) and (3.22), there exists a positive constant $G_3$ such that

$${\mathrm{\Phi }}_1(\mathrm{\Delta }{u}_1^{[n]}(t))\le G_3,{\mathrm{\Phi }}_2(\mathrm{\Delta }{u}_2^{[n]}(t))\le G_3.$$

(3.24)

Then it follows from (3.18), (3.22), (3.23), (3.24), and Lemma 2.1 that

$$\begin{array}{rl}-G_0\le & \phi \left(u^{[n]}\right)\\ =& \phi (u_1^{[n]},u_2^{[n]})\\ =& \sum _{t=1}^T[{\mathrm{\Phi }}_1(\mathrm{\Delta }{u}_1^{[n]}(t))+{\mathrm{\Phi }}_2(\mathrm{\Delta }{u}_2^{[n]}(t))+F(t,u_1^{[n]}(t),u_2^{[n]}(t))-F(t,P_1{\overline{u}}_1^{[n]},P_2{\overline{u}}_2^{[n]})\\ +F(t,P_1{\overline{u}}_1^{[n]},P_2{\overline{u}}_2^{[n]})+(h_1(t),u_1^{[n]}(t))+(h_2(t),u_2^{[n]}(t))]\\ \le & 2G_{3}T+G_2+\sum _{t=1}^{T}F(t,P_1{\overline{u}}_1^{[n]},P_2{\overline{u}}_2^{[n]})+M_{14}{\parallel {\tilde{u}}_1\parallel }_{\mathrm{\infty }}+M_{24}{\parallel {\tilde{u}}_1\parallel }_{\mathrm{\infty }}\\ \le & 2G_{3}T+G_2+\sum _{t=1}^{T}F(t,P_1{\overline{u}}_1^{[n]},P_2{\overline{u}}_2^{[n]})+M_{14}C\left(p^{\prime }\right)G_1^{1/p}+M_{24}C\left(q^{\prime }\right)G_1^{1/q}.\end{array}$$

(3.25)

Then ($\mathcal{F}6$)′ implies that $\{P_1{\overline{u}}_1^{[n]}\}$ and $\{P_2{\overline{u}}_2^{[n]}\}$ are bounded. Then (3.22) implies that $\{u^{[n]}\}$ is bounded in ℋ. Since $dim\mathcal{H}<\mathrm{\infty }$, $\{u^{[n]}\}$ has a convergent subsequence. Hence, Ψ satisfies the (PS) condition.

Next we prove the following conclusions:

• (i) $inf\{\mathrm{\Psi }((z,v))|(z,v)\in Z\times T^{r_1+r_2}\}>-\mathrm{\infty }$;

• (ii) $\mathrm{\Psi }((y,v))\to -\mathrm{\infty }$ uniformly for $(y,v)\in Y\times T^{r_1+r_2}$ as $|y|\to \mathrm{\infty }$.

For $(z,v)\in Z\times T^{r_1+r_2}$, set $u=u(t)=z(t)+v={(z_1(t)+v_1,z_2(t)+v_2)}^{\tau }$. Then $z_m(t)={\tilde{u}}_m(t)$, $v_m=Q_m{\overline{u}}_m$, and $u_m(t)=z_m(t)+v_m$, $m=1,2$. By ($\mathcal{F}3$) and Lemma 2.1, we have

$$\begin{array}{r}\left|\sum _{t=1}^T[F(t,u_1(t),u_2(t))-F(t,0,0)]\right|\\ \le \sum _{t=1}^T|F(t,u_1(t),u_2(t))-F(t,0,u_2(t))|+\sum _{t=1}^T|F(t,0,u_2(t))-F(t,0,0)|\\ \le \sum _{t=1}^T\left|{\int }_0^1({\mathrm{\nabla }}_{x_1}F(t,s(Q_1{\overline{u}}_1+{\tilde{u}}_1(t)),u_2(t)),Q_1{\overline{u}}_1+{\tilde{u}}_1(t))ds\right|\\ +\sum _{t=1}^T\left|{\int }_0^1({\mathrm{\nabla }}_{x_2}F(t,0,s(Q_2{\overline{u}}_2+{\tilde{u}}_2(t))),Q_2{\overline{u}}_2+{\tilde{u}}_2(t))ds\right|\\ \le (|Q_1{\overline{u}}_1|+{\parallel {\tilde{u}}_1\parallel }_{\mathrm{\infty }})\sum _{t=1}^T{\int }_0^1\left|{\mathrm{\nabla }}_{x_1}F(t,s(Q_1{\overline{u}}_1+{\tilde{u}}_1(t)),{\hat{u}}_2(t))\right|ds\\ +(|Q_2{\overline{u}}_2|+{\parallel {\tilde{u}}_2\parallel }_{\mathrm{\infty }})\sum _{t=1}^T{\int }_0^1\left|{\mathrm{\nabla }}_{x_2}F(t,0,s(Q_2{\overline{u}}_2+{\tilde{u}}_2(t)))\right|ds\\ \le (M_{13}+{\parallel {\tilde{u}}_1\parallel }_{\mathrm{\infty }})\sum _{t=1}^T{b}_1(t)+(M_{23}+{\parallel {\tilde{u}}_2\parallel }_{\mathrm{\infty }})\sum _{t=1}^T{b}_2(t)\\ \le C\left(p^{\prime }\right){\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_1(t)|}^p\right)}^{1/p}\sum _{t=1}^T{b}_1(t)+C\left(q^{\prime }\right){\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_2(t)|}^q\right)}^{1/q}\sum _{t=1}^T{b}_2(t)\\ +M_{13}\sum _{t=1}^T{b}_1(t)+M_{23}\sum _{t=1}^T{b}_2(t).\end{array}$$

(3.26)

Then

$$\begin{array}{rl}\mathrm{\Psi }((z,v))=& \phi (u_1,u_2)\\ =& \sum _{t=1}^T[{\mathrm{\Phi }}_1(\mathrm{\Delta }{u}_1(t))+{\mathrm{\Phi }}_2(\mathrm{\Delta }{u}_2(t))+F(t,u_1(t),u_2(t))\\ +(h_1(t),u_1(t))+(h_2(t),u_2(t))]\\ =& \sum _{t=1}^T[{\mathrm{\Phi }}_1(\mathrm{\Delta }{u}_1(t))+{\mathrm{\Phi }}_2(\mathrm{\Delta }{u}_2(t))+F(t,u_1(t),u_2(t))-F(t,0,0)\\ +F(t,0,0)+(h_1(t),u_1(t))+(h_2(t),u_2(t))]\\ \ge & \sum _{t=1}^T[{\delta }_1(|\mathrm{\Delta }{u}_1(t)|-1)+{\delta }_2(|\mathrm{\Delta }{u}_2(t)|-1)]+\sum _{t=1}^{T}F(t,0,0)\\ -C\left(p^{\prime }\right){\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_1(t)|}^p\right)}^{1/p}\sum _{t=1}^T{b}_1(t)-C\left(q^{\prime }\right){\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_2(t)|}^q\right)}^{1/q}\sum _{t=1}^T{b}_2(t)\\ -C\left(p^{\prime }\right){\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_1(t)|}^p\right)}^{1/p}\sum _{t=1}^T|h_1(t)|-C\left(q^{\prime }\right){\left(\sum _{t=1}^T{|\mathrm{\Delta }{u}_2(t)|}^q\right)}^{1/q}\sum _{t=1}^T|h_2(t)|\\ -M_{13}\sum _{t=1}^T{b}_1(t)-M_{23}\sum _{t=1}^T{b}_2(t)\\ \ge & \sum _{t=1}^T[{\delta }_1(|\mathrm{\Delta }{u}_1(t)|-1)+{\delta }_2(|\mathrm{\Delta }{u}_2(t)|-1)]+\sum _{t=1}^{T}F(t,0,0)-M_{13}\sum _{t=1}^T{b}_1(t)\\ -C\left(p^{\prime }\right)\left(\sum _{t=1}^T|\mathrm{\Delta }{u}_1(t)|\right)\sum _{t=1}^T{b}_1(t)-C\left(q^{\prime }\right)\left(\sum _{t=1}^T|\mathrm{\Delta }{u}_2(t)|\right)\sum _{t=1}^T{b}_2(t)\\ -C\left(p^{\prime }\right)\left(\sum _{t=1}^T|\mathrm{\Delta }{u}_1(t)|\right)\sum _{t=1}^T|h_1(t)|-C\left(q^{\prime }\right)\left(\sum _{t=1}^T|\mathrm{\Delta }{u}_2(t)|\right)\sum _{t=1}^T|h_2(t)|\\ -M_{23}\sum _{t=1}^T{b}_2(t).\end{array}$$

(3.27)

It is easy to see that conclusion (i) holds from ($\mathcal{F}7$)′.

For any $(y,v)\in Y\times T^{r_1+r_2}$, it follows from (1.2) and (2.11) that

$$\begin{array}{rcl}\mathrm{\Psi }((y,v))& =& \phi (y_1+v_1,y_2+v_2)\\ =& \sum _{t=1}^{T}F(t,y_1+v_1,y_2+v_2)\\ =& \sum _{t=1}^{T}F(t,y_1+v_1,y_2+v_2)-\sum _{t=1}^{T}F(t,y_1,y_2+v_2)+\sum _{t=1}^{T}F(t,y_1,y_2+v_2)\\ -\sum _{t=1}^{T}F(t,y_1,y_2)+\sum _{t=1}^{T}F(t,y_1,y_2)\\ =& \sum _{t=1}^{T}F(t,y_1,y_2)+\sum _{t=1}^T{\int }_0^1(\mathrm{\nabla }F(t,y_1+s{v}_1,y_2+v_2),v_1)ds\\ +\sum _{t=1}^T{\int }_0^1(\mathrm{\nabla }F(t,y_1,y_2+s{v}_2),v_2)ds\\ \le & \sum _{t=1}^{T}F(t,y_1,y_2)+|v_1|\sum _{t=1}^T{g}_1(t)+|v_2|\sum _{t=1}^T{g}_2(t)\\ \le & \sum _{t=1}^{T}F(t,y_1,y_2)+{\left(\sum _{k=1}^{r_1}{T}_{i_k}^2\right)}^{1/2}\sum _{t=1}^T{g}_1(t)+{\left(\sum _{s=1}^{r_2}{T}_{j_s}^2\right)}^{1/2}\sum _{t=1}^T{g}_2(t).\end{array}$$

Hence, the above inequality and ($\mathcal{F}6$)′ imply that conclusion (ii) holds. It follows from Lemma 2.6 that Ψ has at least $r_1+r_2+1$ critical points. Hence φ has at least $r_1+r_2+1$ geometrically distinct critical points. Therefore, system (1.1) has at least $r_1+r_2+1$ geometrically distinct solutions in ℋ. The proof is complete. □