1 Introduction and statement of main results

The interest in the second-order difference systems

$$ \textstyle\begin{cases} \triangle^{2}u(t-1)+\nabla F(t,u(t))=0,\quad t\in\mathbb{Z}[1,T],\\ u(0)=u(T), \end{cases} $$
(1)

has been aroused, where \(\bigtriangleup u(t)=u(t+1)-u(t)\), \(\bigtriangleup^{2}u(t)=\bigtriangleup(\bigtriangleup u(t))\), \(\nabla F(t, x)= \frac{\partial F(t,x)}{\partial x}\), T is a positive integer, and \(\mathbb{Z}\) and R denote the set of integers and the set of real numbers, respectively, and \(\mathbb{Z}[a,b]=\{a,a+1,\ldots, b\}\), for \(a,b\in \mathbb{Z}\) and \(a\leq b\). Assume that \(F(t,x)\) is T-periodic in t for all \(x\in R^{N}\) and \(F(t,x)\in C^{1}(\mathbb{Z}\times R^{N},R)\).

In 2003, Yu and Guo [1–3] established a variational structure and variational methods to study discrete Hamiltonian systems and obtain the solvability condition of a periodic solution for discrete systems, based on operator theory. Since then more and more authors have contributed to study second-order discrete Hamiltonian systems, with an effective tool named the critical point theory, and one obtained many interesting results [4–11]. In [8], with operator theory, Xue and Tang constructed a variational setting unlike the one in [1] to study the second-order superquadratic discrete Hamiltonian systems (1) and obtained the existence of periodic solutions. This result generalized the one in [4]. In [7], Xue and Tang obtained the existence of one periodic solution of systems (1) under the hypothesis there exist \(M_{1}>0\), \(M_{2}>0\) and \(0\leq\alpha<1\) such that

$$\begin{aligned} \bigl\vert \nabla F(t,x)\bigr\vert \leq M_{1}\vert x\vert ^{\alpha}+M_{2}, \end{aligned}$$

for all \((t,x)\in\mathbb{Z}[1,T]\times R^{N}\), and the condition

$${ \vert x\vert ^{-2\alpha}\sum_{t=1}^{T}F(t,x) \rightarrow +\infty}\quad \mbox{as } \vert x \vert \rightarrow\infty. $$

Subsequently, in [9], Yan, Wu and Tang extended these results in [7] to the case that \(F(t,x)\) is \(T_{i}\)-periodic in \(x_{i}\) and obtained the existence of multiple periodic solutions, where \(x_{i}\) is the ith component of \(x=(x_{1},x_{2},\ldots,x_{N})\), \(i\in[1,N]\). Especially in [12], Che and Xue obtained the existence of infinitely many periodic solutions for systems in the case that:

(F1) there exist \(f, g: \mathbb{Z}[0,T]\rightarrow R^{+}\) and \(\alpha\in[0,1)\) such that

$$\bigl\vert \nabla F(t,x)\bigr\vert \leq f(t)\vert x\vert ^{\alpha}+g(t) \quad\mbox{for all } (t,x)\in \mathbb{Z}[0,T]\times R^{N}, $$

and a suitable oscillating behaviour at infinity,

(F2) \(\liminf_{r\rightarrow\infty}\sup_{x\in R^{N},\vert x\vert =r}{ \vert x\vert ^{-2\alpha}\sum_{t=0}^{T}F(t,x) =-\infty}\),

(H3) \(\limsup_{r\rightarrow\infty}\inf_{x\in R^{N},\vert x\vert =r}\sum_{t=0}^{T}F(t,x) =+\infty\).

Consequently, it is natural to ask if infinitely many solutions still exist when \(\alpha=1\). With the fact that \(\alpha=1\), (F1) and (F2), respectively, change to the linearly bounded gradient condition:

(F1′) there exist \(f, g: \mathbb{Z}[0,T]\rightarrow R^{+}\) such that

$$\bigl\vert \nabla F(t,x)\bigr\vert \leq f(t)\vert x\vert +g(t) \quad\mbox{for all } (t,x)\in \mathbb{Z}[0,T]\times R^{N}, $$

and the condition

(F2′) \(\liminf_{r\rightarrow\infty}\sup_{x\in R^{N},\vert x\vert =r}{ \vert x\vert ^{-2}\sum_{t=0}^{T}F(t,x) =-\infty}\).

However, similarly to what was pointed in [13], (F2′) does not hold if \(\sum_{t=0}^{T} f(t)\) is bounded. Therefore, it is necessary to improve condition (F2′). Inspired by [7, 12, 13], in this paper, we will use minmax methods to further study systems (1) under the following assumptions:

  1. (H1)

    there exist \(f, g: \mathbb{Z}[0,T]\rightarrow R^{+}\) with \(\sum_{t=0}^{T} f(t)<\frac{\lambda_{1}}{2}\) such that

    $$\bigl\vert \nabla F(t,x)\bigr\vert \leq f(t)\vert x\vert +g(t) \quad\mbox{for all } (t,x)\in \mathbb{Z}[0,T]\times R^{N}, $$
  2. (H2)

    \(\liminf_{r\rightarrow+\infty}\sup_{x\in R^{N},\vert x\vert =r}{ \vert x\vert ^{-2}\sum_{t=0}^{T}F(t,x) <- \frac{ 4\sum_{t=0}^{T} f^{2}(t)}{\lambda_{1}}}\),

where \(\lambda_{k}=2-2\cos\frac{2k\pi}{T} \) satisfy the eigenvalue problem

$$-\triangle^{2}u(t-1)=\lambda_{k}u(t),\quad k\in \mathbb{Z} \biggl[0,\biggl[\frac{T}{2}\biggr]\biggr], $$

and we note

$$0=\lambda_{0}< \lambda_{1}< \cdots< \lambda_{[\frac{T}{2}]} \leq4. $$

The main result on the existence of infinitely many periodic solutions of systems (1) is obtained. The details are described.

Theorem 1.1

Under the hypotheses of (H1), (H2), and (H3):

  1. (a)

    discrete systems (1) have a sequence \(\{u_{n}\}\) of solutions such that \(\{u_{n}\}\) is a critical point of the functional φ and \(\varphi(u_{n})\rightarrow+\infty\) as \(n\rightarrow\infty\);

  2. (b)

    discrete systems (1) have a sequence \(\{u_{n}^{*}\}\) of solutions such that \(\{u_{n}^{*}\}\) is a local minimizer of φ and \(\varphi(u_{n}^{*})\rightarrow-\infty\) as \(n\rightarrow \infty\),

where the variational functional φ is

$$\varphi(u)=\frac{1}{2}\sum_{t=0}^{T} \bigl\vert \triangle u(t)\bigr\vert ^{2}-\sum _{t=0}^{T}F\bigl(t,u(t)\bigr), $$

on Hilbert space \(H_{T}\) defined by

$$H_{T}=\bigl\{ u:\mathbb{Z}\rightarrow R^{N} \vert u(t)=u(t+T), t\in\mathbb {Z}\bigr\} , $$

with the inner product and the norm, respectively, written as

$$\langle u,v\rangle =\sum_{t=0}^{T} \bigl(u(t),v(t)\bigr), \quad\forall u,v \in H_{T} $$

and

$$\Vert u\Vert = \Biggl(\sum^{T}_{t=0} \bigl\vert u(t)\bigr\vert ^{2} \Biggr)^{\frac{1}{2}}. $$

Remark 1.2

As pointed out in [12], the nonlinearity potential F does not need the symmetry condition in the paper. Moreover, Theorem 1.1 is a complement to and development of Theorem 1.1 in [12] corresponding to \(\alpha=1\).

2 Proof of main result

For all \(u\in H_{T}\), \({\Vert u \Vert _{\infty}=\sup_{t\in\mathbb{Z}[0,T]}\vert u(t)\vert }\) is defined. It is obvious that

$$\begin{aligned} \Vert u \Vert _{\infty}\leq \Vert u \Vert = \Biggl(\sum ^{T}_{t=0}\bigl\vert u(t)\bigr\vert ^{2} \Biggr)^{\frac{1}{2}}. \end{aligned}$$
(2)

By the definition of \(H_{T}\), \(H_{T}\) is finite dimensional. By (H1), one gets \(\varphi\in C^{1}(H_{T},R)\) and

$$\bigl\langle \varphi'(u),v\bigr\rangle =\sum _{t=0}^{T}\bigl( \triangle u(t),\triangle v(t)\bigr)- \sum_{t=0}^{T} \bigl(\nabla F\bigl(t,u(t) \bigr),v(t)\bigr), $$

for all \(u,v\in H_{T}\).

Subsequently, an important lemma in [8] is shown for the reader’s convenience. The lemma is constructed in a variational setting, with the operator theory, unlike the one in [1]. Details can be found in [8].

Lemma 1

[8]

Let \(N_{k}\) be a subspace of \(H_{T}\) written as

$$N_{k}:=\bigl\{ u\in H_{T}\vert {-}\triangle^{2}u(t-1)= \lambda_{k}u(t)\bigr\} , $$

where \(\lambda_{k}=2-2\cos\frac{2k\pi}{T} \), \(k\in \mathbb{Z}[0,[\frac{T}{2}]]\). Then one has the results as follows:

  1. (i)

    \(N_{k}\mathbin{\bot} N_{j}\), \(k\neq j\), \(k,j\in\mathbb{Z}[0,[\frac{T}{2}]]\).

  2. (ii)

    \({H_{T}=\bigoplus_{k=0}^{[T/2]}N_{k}}\).

Let \(V= N_{0}\) and \(W= {N_{0}^{\bot}=\bigoplus_{k=1}^{[T/2]}N_{k}}\). Via Lemma 2 in [8], we have

$$H_{T}=N_{0}\oplus N_{0}^{\bot}=V \oplus W $$

and

$$\begin{aligned} \sum_{t=0}^{T}\bigl\vert \triangle u(t) \bigr\vert ^{2}\geq\lambda_{1}\Vert u\Vert ^{2}, \quad \forall u\in W=N_{0}^{\bot}. \end{aligned}$$
(3)

From the definition of \(N_{0}\), one gets \(u(t)=u(0)=C\in R^{N}\), for all \(u\in N_{0}\) and \(t\in\mathbb{Z}[0,T]\). By Lemma 1, one rewrites u as \(u=\bar{u}+\tilde{u}\in V\oplus W=N_{0}\oplus N_{0}^{\bot}\), where

$$\bar{u}=\frac{1}{T}\sum_{t=0}^{T} u(t). $$

From the fact that

$$\begin{aligned} \Vert u \Vert =& \Biggl(\sum^{T}_{t=0} \bigl\vert u(t)\bigr\vert ^{2} \Biggr)^{\frac{1}{2}}= \Biggl(\sum ^{T}_{t=0}\bigl\vert \bar{u}+\tilde{u}(t) \bigr\vert ^{2} \Biggr)^{\frac {1}{2}}\\ =& \Biggl(\sum ^{T}_{t=0}\langle\bar{u}+\tilde{u},\bar {u}+\tilde{u} \rangle \Biggr)^{\frac{1}{2}} \\ =& \Biggl(\sum^{T}_{t=0}\bigl(\vert \bar{u}\vert ^{2}+ \bigl\vert \tilde{u}(t)\bigr\vert ^{2}\bigr) \Biggr)^{\frac{1}{2}} = \bigl( T\vert \bar{u}\vert ^{2}+ \Vert \tilde{u}\Vert ^{2} \bigr)^{\frac{1}{2}}, \end{aligned}$$

one has

$$\Vert u \Vert \leq \sqrt{T+1} \bigl(\vert \bar{u}\vert ^{2}+ \Vert \tilde{u}\Vert ^{2} \bigr)^{\frac{1}{2}} \quad\textit{and} \quad \Vert u \Vert \geq \bigl(\vert \bar{u}\vert ^{2}+ \Vert \tilde{u}\Vert ^{2} \bigr)^{\frac{1}{2}}. $$

Thus one obtains that \(\Vert u\Vert \rightarrow\infty\) if and only if \((\vert \bar{u}\vert ^{2}+\Vert \tilde{u}\Vert ^{2} )^{\frac{1}{2}}\rightarrow\infty\).

Proof of Theorem 1.1

The proof of Theorem 1.1 relies on a minimax theorem (Corollary 4.3) in [14]. We complete the proof with a series of statements below.

Step 1, we claim that φ is coercive in the subspace W.

Due to (H1), there exists a constant \(C_{1}\) satisfying the following inequality:

$$\begin{aligned} \bigl\vert F(t,x)\bigr\vert \leq& \biggl\vert \int_{0}^{1} \bigl(\nabla F(t,sx),x\bigr)\,ds\biggr\vert +\bigl\vert F(t,0)\bigr\vert \\ \leq& \int_{0}^{1} \bigl\vert \nabla F(t,sx)\bigr\vert \vert x \vert \,ds+ C_{1}\leq\frac{f(t)}{2}\vert x\vert ^{2}+g(t)\vert x\vert +C_{1} \end{aligned}$$
(4)

for all \(t\in\mathbb{Z}[0,T]\) and \(x\in R^{N}\). Hence, using the Hölder inequality, (2), (3), and (4), for all \(u\in W\), one derives

$$\begin{aligned} \varphi(u) =&\frac{1}{2}\sum_{t=0}^{T} \bigl\vert \triangle u(t)\bigr\vert ^{2}-\sum _{t=0}^{T} F(t,u) \\ \geq& \frac{1}{2}\lambda_{1}\Vert u\Vert ^{2} - \sum_{t=0}^{T} \biggl(\frac {1}{2}f(t) \bigl\vert u(t)\bigr\vert ^{2}+g(t)\bigl\vert u(t)\bigr\vert +C_{1} \biggr) \\ \geq& \frac{1}{2}\lambda_{1}\Vert u\Vert ^{2} - \Vert u\Vert ^{2}_{\infty}\sum_{t=0}^{T} \frac{1}{2}f(t) -\Vert u\Vert _{\infty}\sum _{t=0}^{T} g(t)-C_{1}T \\ \geq& \frac{1}{2}\lambda_{1}\Vert u\Vert ^{2} - \Vert u\Vert ^{2}\frac{1}{2} \sum _{t=0}^{T} f(t) -\Vert u\Vert \sum _{t=0}^{T} g(t)-C_{1}T \\ =& \Biggl(\frac{1}{2}\lambda_{1} -\frac{1}{2}\sum _{t=0}^{T} f(t) \Biggr) \Vert u\Vert ^{2} -\Vert u\Vert \sum_{t=0}^{T} g(t)-C_{1}T. \end{aligned}$$

Combining this with the fact \(\sum_{t=0}^{T} f(t)<\frac{1}{2}\lambda _{1}\), one deduces \(\lim_{\Vert u\Vert \rightarrow+\infty}\varphi(u)= +\infty\).

Step 2, we claim that there are positive sequences \(\{a_{n}\}, \{b_{m}\}\) satisfying

  1. (c)

    \(\lim_{n\rightarrow\infty}a_{n}=+\infty \mbox{ and } \lim_{n\rightarrow\infty}\sup_{u\in V, \Vert u\Vert =a_{n}}\varphi(u)=-\infty\),

  2. (d)

    \(\lim_{m\rightarrow\infty}b_{m}=+\infty \mbox{ and } \lim_{m\rightarrow\infty}\inf_{u\in H_{b_{m}}}\varphi(u)=+\infty\),

where \(H_{b_{m}}=\{u\in V, \Vert u\Vert =b_{m}\}\oplus W\).

The detailed proof of (c) can be founded in [12]. On the other hand, by (H2), one can take a constant

$$a>\frac{ 8}{\lambda_{1}}. $$

Thus one gets

$$\begin{aligned} \liminf_{r\rightarrow\infty}\sup_{x\in R^{N},\vert x\vert =r}{ \vert x\vert ^{-2}\sum_{t=0}^{T}F(t,x) < - \frac{a}{2}\sum_{t=0}^{T} f^{2}(t)}. \end{aligned}$$
(5)

For any \(u\in H_{b_{m}} \), rewritten \(u=\tilde{u}+\bar{u}\), where \(\tilde{u}\in W\) and \(\bar{u}\in V\), by the Hölder inequality, (H1), and (H2), one has

$$\begin{aligned} &\Biggl\vert \sum_{t=0}^{T}\bigl(F \bigl(t,u(t)\bigr)-F(t,\bar{u})\bigr)\Biggr\vert \\ &\quad\leq\sum _{t=0}^{T} \int _{0}^{1}\bigl\vert \bigl(\nabla F\bigl(t, \bar{u}+s\tilde{u}(t)\bigr),\tilde{u}(t)\bigr)\bigr\vert \,ds \\ &\quad\leq\sum_{t=0}^{T} \int_{0}^{1} \bigl(f(t)\bigl\vert \bar{u}+s\tilde {u}(t)\bigr\vert +g(t) \bigr)\cdot\bigl\vert \tilde{u}(t)\bigr\vert \,ds \\ &\quad\leq\sum_{t=0}^{T} f(t) \biggl(\vert \bar{u}\vert +\frac{1}{2}\bigl\vert \tilde{u}(t)\bigr\vert \biggr) \bigl\vert \tilde{u}(t)\bigr\vert +\sum_{t=0}^{T}g(t) \bigl\vert \tilde{u}(t)\bigr\vert \\ &\quad\leq \vert \bar{u}\vert \Biggl(\sum_{t=0}^{T} f^{2}(t) \Biggr)^{\frac{1}{2}} \Biggl(\sum _{t=0}^{T}\bigl\vert \tilde{u}(t)\bigr\vert ^{2} \Biggr)^{\frac{1}{2}} +\frac {1}{2}\Vert \tilde{u}\Vert ^{2}_{\infty}\sum_{t=0}^{T} f(t)+\Vert \tilde{u}\Vert _{\infty}\sum_{t=0}^{T}g(t) \\ &\quad\leq\frac{1}{2a} \Vert \tilde{u}\Vert ^{2}+ \frac{a}{2}\sum_{t=0}^{T} f^{2}(t) \vert \bar{u}\vert ^{2}+\frac{1}{2}\sum _{t=0}^{T} f(t)\Vert \tilde{u}\Vert ^{2}+\Vert \tilde{u}\Vert \sum_{t=0}^{T}g(t) \\ &\quad\leq \Biggl(\frac{1}{2a}+\frac{1}{2}\sum _{t=0}^{T} f(t) \Biggr) \Vert \tilde{u}\Vert ^{2}+\frac{a}{2}\sum_{t=0}^{T} f^{2}(t) \vert \bar{u}\vert ^{2}+\Vert \tilde {u} \Vert \sum_{t=0}^{T}g(t). \end{aligned}$$
(6)

Hence, for all \(u\in H_{b_{m}} \), it follows from inequalities (3) and (6) that

$$\begin{aligned} \varphi(u) ={}&\frac{1}{2}\sum_{t=0}^{T} \bigl\vert \triangle u(t)\bigr\vert ^{2}-\sum _{t=0}^{T} F\bigl(t,u(t)\bigr) \\ ={}& \frac{1}{2}\sum_{t=0}^{T}\bigl\vert \triangle\tilde{u}(t)\bigr\vert ^{2}-\sum _{t=0}^{T}\bigl(F\bigl(t,u(t)\bigr)-F(t,\bar{u})\bigr) -\sum_{t=0}^{T} F(t,\bar{u}) \\ \geq{}& \frac{1}{2}\lambda_{1}\Vert \tilde{u }\Vert ^{2}- \Biggl(\frac {1}{2a}+\frac{1}{2}\sum _{t=0}^{T} f(t) \Biggr) \Vert \tilde{u}\Vert ^{2}-\frac {a}{2} \sum_{t=0}^{T} f^{2}(t)\vert \bar{u}\vert ^{2} \\ &{}-\Vert \tilde{u}\Vert \sum_{t=0}^{T}g(t)-\sum _{t=0}^{T} F(t,\bar{u}) \\ ={}& \Biggl(\frac{\lambda_{1}}{2}-\frac{1}{2a}-\frac{1}{2}\sum _{t=0}^{T} f(t) \Biggr)\Vert \tilde{u} \Vert ^{2}-\Vert \tilde{u}\Vert \sum_{t=0}^{T}g(t) \\ &{}- \vert \bar{u}\vert ^{2} \Biggl(\frac{\sum_{t=0}^{T} F(t,\bar{u})}{\vert \bar {u}\vert ^{2}}+ \frac{a}{2}\sum_{t=0}^{T} f^{2}(t) \Biggr). \end{aligned}$$
(7)

By \(\sum_{t=0}^{T} f(t)<\frac{\lambda_{1}}{2}\) and \(a>\frac {8}{\lambda_{1}}\),

$$\frac{\lambda_{1}}{2}-\frac{1}{2a}-\frac{1}{2}\sum _{t=0}^{T} f(t)>0 $$

is verified. By (5), (7), and the fact \(\Vert u\Vert \rightarrow\infty\) if and only if \((\vert \bar{u}\vert ^{2}+\Vert \tilde{u}\Vert ^{2} )^{\frac{1}{2}}\rightarrow \infty\), the conclusion (d) is achieved.

Now we have a family of maps \(\Gamma_{n}\) expressed as

$$\Gamma_{n} = \bigl\{ \gamma\in C(B_{a_{n}} ,H_{T}) \vert \gamma \vert _{\partial B_{a_{n}}} = \operatorname{Id}\vert _{\partial B_{a_{n}}} \bigr\} $$

and minimax values \(c_{n}\) formulated as

$$c_{n} = \inf_{\gamma\in\Gamma_{n}}\max_{u\in B_{a_{n}}} \varphi\bigl(\gamma(u)\bigr) $$

for each n, where \(B_{a_{n}}\) is a ball in V and \(a_{n}\) is the radius of \(B_{a_{n}}\). One gets

$$\gamma( B_{a_{n}})\cap W \neq\emptyset $$

for any \(\gamma\in\Gamma_{n}\) from Theorem 4.6 in [14].

Step 3, we claim that, for sufficiently large n, there exist sequences \(\{\gamma_{k}\} \subset\Gamma_{n}\) and \(\{\nu_{k}\}\) in \(H_{T}\), respectively, satisfying

$$\begin{aligned} &\max_{u\in B_{a_{n}}} \varphi\bigl(\gamma_{k}(u)\bigr) \rightarrow c_{n}, \\ & \varphi(\nu_{k})\rightarrow c_{n},\qquad \varphi'( \nu_{k})\rightarrow0,\qquad \operatorname{dist}\bigl(\nu_{k}, \gamma_{k}( B_{a_{n}})\bigr)\rightarrow0 \quad\mbox{as } k\rightarrow \infty. \end{aligned}$$
(8)

By Step 1, we know \(\varphi(u)\rightarrow+\infty\) as \(\Vert u\Vert \rightarrow+\infty\), \(u\in W\). Therefore there exists a constant \(C_{2}\) satisfying

$$\max_{u\in B_{a_{n}}}\varphi\bigl(\gamma(u)\bigr)\geq\inf _{u\in W} \varphi(u) \geq C_{2}. $$

Furthermore, one has

$$c_{n}\geq\inf_{u\in W} \varphi(u) \geq C_{2}, $$

for sufficiently large n. By the fact \(\gamma( B_{a_{n}})\cap W \neq \emptyset\) and the conclusion of Step 2, one obtains

$$c_{n}>\max_{u\in\partial B_{a_{n}}} \varphi(u) $$

for sufficiently large n. Therefore, for a fixed n, this claim is proved from Theorem 4.3 and Corollary 4.3 in [14].

Step 4, we draw the conclusion that the sequence \(\{\nu_{k}\}\) is bounded in \(H_{T}\).

For sufficiently large k, by (8), one has

$$c_{n} \leq\max_{u\in B_{a_{n}}}\varphi\bigl( \gamma_{k}(u)\bigr) \leq c_{n} + 1. $$

We choose \(w_{k}\in\gamma_{k}( B_{a_{n}})\) satisfying

$$\begin{aligned} \Vert \nu_{k} - w_{k}\Vert \leq1. \end{aligned}$$
(9)

From the conclusion (d) of Step 2, for a fixed n, a sufficiently large m exists, rendering the formula

$$b_{m} >a_{n} \quad\mbox{and}\quad \inf_{u\in H_{b_{m}}} \varphi(u) > c_{n} + 1. $$

These inequalities imply that \(\gamma_{k}(B_{a_{n}}) \cap H_{b_{m}}=\emptyset\) for each k. We now write \(w_{k} = \bar{w}_{k} + \tilde{w}_{k} \), where \(\bar{w}_{k} \in V\) and \(\tilde{w}_{k}\in W\). Then one has

$$\begin{aligned} \Vert \bar{w}_{k}\Vert < b_{m} \end{aligned}$$
(10)

for each k. Moreover, by (2), (3), (4), and (10), one gets

$$\begin{aligned} 1+c_{n}\geq{}&\varphi(w_{k})=\frac{1}{2}\sum _{t=0}^{T}\bigl\vert \triangle w_{k}(t) \bigr\vert ^{2}-\sum_{t=0}^{T} F \bigl(t,w_{k}(t)\bigr) \\ \geq{}& \frac{1}{2}\lambda_{1}\Vert \tilde{w}_{k} \Vert ^{2} -\sum_{t=0}^{T} \biggl(\frac{1}{2}f(t)\bigl\vert w_{k}(t)\bigr\vert ^{2}+g(t)\bigl\vert w_{k}(t)\bigr\vert +C_{1} \biggr) \\ \geq{}& \frac{1}{2}\lambda_{1}\Vert \tilde{w}_{k} \Vert ^{2} -\sum_{t=0}^{T} f(t) \bigl[\vert \bar{w}_{k}\vert ^{2}+\bigl\vert \tilde{w}_{k}(t)\bigr\vert ^{2}\bigr]-\sum _{t=0}^{T}g(t) \bigl(\vert \bar{w}_{k} \vert +\bigl\vert \tilde{w}_{k}(t)\bigr\vert \bigr)-C_{1}T \\ \geq{}& \frac{1}{2}\lambda_{1}\Vert \tilde{w}_{k} \Vert ^{2} -\Vert \tilde{w}_{k}\Vert ^{2}_{\infty}\sum_{t=0}^{T} f(t) - \Vert \bar{w}_{k}\Vert ^{2}\sum _{t=0}^{T} f(t) \\ &{}-\Vert \tilde{w}_{k} \Vert _{\infty}\sum_{t=0}^{T} g(t)- \Vert \bar{w}_{k}\Vert \sum_{t=0}^{T} g(t) -C_{1}T \\ \geq{}& \frac{1}{2}\lambda_{1}\Vert \tilde{w}_{k} \Vert ^{2} -\Vert \tilde{w}_{k}\Vert ^{2} \sum_{t=0}^{T} f(t) - b_{m}^{2} \sum_{t=0}^{T} f(t) \\ &{} -\Vert \tilde{w}_{k}\Vert \sum_{t=0}^{T} g(t)-b_{m}\sum_{t=0}^{T} g(t)-C_{1}T \\ ={}& \Biggl(\frac{\lambda_{1}}{2} -\sum_{t=0}^{T} f(t) \Biggr) \Vert \tilde {w}_{k}\Vert ^{2} -\Vert \tilde{w}_{k}\Vert \sum_{t=0}^{T} g(t)-b_{m}^{2}\sum_{t=0}^{T} f(t)-b_{m}\sum_{t=0}^{T} g(t)-C_{1}T. \end{aligned}$$
(11)

We can combine equation (11) and the fact that \(\sum_{t=0}^{T} f(t) <\frac{\lambda_{1}}{2}\), \(\Vert \tilde{w}_{k}\Vert \) is bounded. Thus, by combining (10) and the fact that \(\Vert w_{k}\Vert = (T\vert \bar {w}_{k}\vert ^{2}+\Vert \tilde{w}_{k}\Vert ^{2} )^{\frac{1}{2}}\), \(\{w_{k}\}\) is bounded. Then \(\{\nu_{k}\}\) is bounded in \(H_{T}\) via (9). The conclusion is proved.

Step 5, we claim that \(c_{n}\) is a critical value of φ.

Since \(\{\nu_{k}\} \) is bounded and \(H_{T}\) is finite dimensional space, \(\{\nu_{k}\} \) contains a convergent subsequence that is still denoted as \(\{\nu_{k}\} \) for convenience, meeting

$$\lim_{k\rightarrow\infty}{\nu_{k}}=u_{n}. $$

Then, by (8), one has

$$\varphi(u_{n}) = c_{n} \quad\mbox{and}\quad \varphi'(u_{n}) = 0. $$

Thus φ has a critical point \(u_{n}\).

We prove part (a) of Theorem 1.1. One chooses sufficiently large n satisfying \(a_{n} >b_{m}\), then one has \(\gamma(B_{a_{n}}) \cap H_{b_{m}}\neq\emptyset\) for any \(\gamma\in \Gamma_{n}\). It follows that

$$\max_{u\in B_{a_{n}}}\varphi\bigl(\gamma(u)\bigr)\geq\inf _{u\in H_{b_{m}}} \varphi(u). $$

With this and the conclusion (d) of Step 2,

$$\lim_{n\rightarrow\infty}c_{n}=+\infty $$

is implied. Part (a) of Theorem 1.1 is proved.

A follow-up is to prove part (b) in Theorem 1.1. For a given m, let \(P_{m}\) be a subset of \(H_{T}\), where

$$P_{m} =\bigl\{ u =\bar{u}+ \tilde{u} \in H_{T} \vert \bar{u} \in V, \Vert \bar{u}\Vert \leq b_{m}, \tilde{ u }\in W\bigr\} . $$

For all \(u\in P_{m}\), similar to (11), one obtains

$$\begin{aligned} \varphi(u) =&\frac{1}{2}\sum_{t=0}^{T} \bigl\vert \triangle u(t)\bigr\vert ^{2}-\sum _{t=0}^{T} F\bigl(t,u(t)\bigr) \\ \geq& \frac{1}{2}\lambda_{1}\Vert \tilde{u}\Vert ^{2} -\sum_{t=0}^{T} \biggl( \frac{1}{2}f(t)\bigl\vert u(t)\bigr\vert ^{2}+g(t)\bigl\vert u(t)\bigr\vert +C_{1} \biggr) \\ \geq& \Biggl(\frac{\lambda_{1}}{2} -\sum_{t=0}^{T} f(t) \Biggr) \Vert \tilde{u}\Vert ^{2} -\Vert \tilde{u}\Vert \sum_{t=0}^{T} g(t)-b_{m}^{2} \sum_{t=0}^{T} f(t)-b_{m}\sum _{t=0}^{T} g(t)-C_{1}T. \end{aligned}$$
(12)

Due to (12), φ is bounded below on \(P_{m}\). Take

$$\mu_{m} = \inf_{u\in P_{m}}\varphi(u) $$

and a sequence \(\{u_{k}\}\subset P_{m}\), satisfying

$$\varphi(u_{k})\rightarrow\mu_{m} \quad\mbox{as } k\rightarrow \infty. $$

Similar to the proof of the boundedness of \(\{w_{k}\}\) in Step 4, \(\{u_{k}\} \) is bounded in \(H_{T}\) via (12). Then \(\{u_{k}\}\) contains a convergent subsequence that is still denoted \(\{u_{k}\} \) for convenience, satisfying

$$u_{k} \rightharpoonup u^{*}_{m} \quad\mbox{weakly in } H_{T},\quad \mbox{as } k\rightarrow\infty. $$

Noting that \(P_{m}\) is convex and closed in \(H_{T}\), one has \(u^{*}_{m}\in P_{m}\). Moreover, in view of the weakly lower semi-continuity of φ, one has

$$\mu_{m}=\lim_{k\rightarrow\infty}\varphi(u_{k})\geq \varphi\bigl(u^{*}_{m}\bigr) $$

and

$$\mu_{m}=\varphi\bigl(u^{*}_{m}\bigr). $$

Next, we draw the conclusion that \(u^{*}_{m}\) is an interior point of \(P_{m}\). Thus \(u^{*}_{m}\) is a critical point of φ.

Taking

$$u^{*}_{m}=\bar{u}^{*}_{m}+\tilde{u}^{*}_{m}, $$

where \(\bar{u}^{*}_{m}\in V\), \(\tilde{u}^{*}_{m}\in W\). If \(a_{n} < b_{m}\), one has \(\partial B_{a_{n}}\subset P_{m}\), which implies that

$$\varphi\bigl(u^{*}_{m}\bigr) = \inf_{u\in P_{m}} \varphi(u) \leq\sup_{u\in\partial B_{a_{n}}}\varphi(u). $$

From the inequality above and the result (d) of Step 2, one gets

$$\varphi\bigl(u^{*}_{m}\bigr)\rightarrow-\infty \quad\mbox{as } m \rightarrow \infty. $$

By the conclusion of Step 3, one has \(\bar{u}^{*}_{m}\neq b_{m}\) for large m. From this one deduces that \(u^{*}_{m}\) is an interior point of \(P_{m}\) and \(u^{*}_{m}\) is a critical point of φ. Then, the proof of Theorem 1.1 is completed. □