In this section, we shall prove our main results by using Linking Theorem.

Assumptions (*T*_{1}) and (*T*_{2}) imply that *F*(*k*, 0, ⋯, 0) = 0 and *f*(*k*, 0, ⋯, 0) = 0 for *k* ∈ **Z**. Then *u* = 0 is a trivial *mT*-periodic solution of Eq. (1.1). It suffices to prove that *J* has at least two nontrivial critical points on *E*_{mT}.

Firstly, we show the existence of one nontrivial critical point. By Lemma 3.4, *J* is bounded from above on *E*_{mT}. The proof of it implies
$\begin{array}{}\underset{\phantom{\rule{thinmathspace}{0ex}}|u\phantom{\rule{thinmathspace}{0ex}}|\to +\mathrm{\infty}}{lim}J(u)=-\mathrm{\infty}\end{array}$.
This means that –*J*(*u*) is coercive. Let we define
$\begin{array}{}{c}_{0}=\underset{u\in {E}_{mT}}{sup}J(u).\end{array}$

There exists *ū* ∈ *E*_{mT} such that *J*(*ū*) = *c*_{0} by the continuity of *J*(*u*). Clearly, *ū* is a critical point of *J*.

We claim that *c*_{0} > 0, which implies that *ū* is a nontrivial crtical point of *J*. Indeed, by (*T*_{2}), there exist two positive constants *δ* and *α*′ ∈
$\begin{array}{}\left(\alpha ,\frac{\underset{\_}{r}}{(\tau +1{)}^{p/2}p}(mT{)}^{\frac{-|2-p|}{2}}(2\mathrm{sin}\frac{\pi}{mT}{)}^{np}\right)\end{array}$
such that
$\begin{array}{}{\displaystyle F(k,{u}_{0},{u}_{1},\cdots ,{u}_{\tau})\u2a7d{\alpha}^{\prime}{\left(\sqrt{\sum _{j=0}^{\tau}{u}_{j}^{2}}\right)}^{p},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathrm{f}\mathrm{o}\mathrm{r}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}k\in \phantom{\rule{thinmathspace}{0ex}}\mathrm{Z}\phantom{\rule{thinmathspace}{0ex}}\mathrm{a}\mathrm{n}\mathrm{d}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\sum _{j=0}^{\tau}{u}_{j}^{2}\u2a7d{\delta}^{2}.}\end{array}$

For any *u* ∈ *V* with ‖u‖ ⩽ *δ*, we have

$$\begin{array}{}{\displaystyle J(u)\u2a7e\left(\frac{\underset{\_}{r}}{p}{c}_{1}^{p}(p){\underset{\_}{\lambda}}^{\frac{np}{2}}-(\tau +1{)}^{\frac{p}{2}}{\alpha}^{\prime}{c}_{2}^{p}(p)\right)\parallel u{\parallel}^{p}.}\end{array}$$(4.1)

Taking
$\begin{array}{}\sigma =\left(\frac{\underset{\_}{r}}{p}{c}_{1}^{p}(p){\underset{\_}{\lambda}}^{\frac{np}{2}}-(\tau +1{)}^{\frac{p}{2}}{\alpha}^{\prime}{c}_{2}^{p}(p)\right){\delta}^{p},\end{array}$
then by Eq. (4.1),

$$\begin{array}{}{\displaystyle J(u)\u2a7e\sigma ,\phantom{\rule{thinmathspace}{0ex}}\mathrm{\forall}\phantom{\rule{thinmathspace}{0ex}}u\in V\cap \mathrm{\partial}{B}_{\delta}.}\end{array}$$

This implies that
$\begin{array}{}{c}_{0}=\underset{u\in {E}_{mT}}{sup}J(u)\u2a7e\sigma >0.\end{array}$
Hence the claim is proved. At the same time, we have proved that there exist constants *σ* > 0 and *δ* > 0 such that *J*|_{∂Bδ} ∩ *V* ⩾ *σ*. In the other word, *J* satisfies the condition (*J*_{1}) of Linking Theorem.

In the remaining of the proof, we shall use Lemma 3.2 to obtain another nontrivial critical point. We have known that *J* satisfies the P.S. condition on *E*_{mT}. In the following, we shall verify the condition (*J*_{2}).

Take *e* ∈*∂* *B*_{1} ∩ *V*. For any *z* ∈ *W* and *s* ∈**R**, we denote *u* = *se* + *z*. Then we have

$$\begin{array}{}{\displaystyle J(u)=\frac{1}{p}\sum _{k=1}^{mT}{r}_{k-1}{\left|{\mathit{\Delta}}^{n}{u}_{k-1}\right|}^{p}-\sum _{k=1}^{mT}F(k,{u}_{k},{u}_{k-1},\cdots ,{u}_{k-\tau})}\\ {\displaystyle \phantom{\rule{1em}{0ex}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\u2a7d\frac{\overline{r}}{p}{s}^{p}\sum _{k=1}^{mT}{\left|{\mathit{\Delta}}^{n}{e}_{k}\right|}^{p}-\sum _{k=1}^{mT}F(k,s{e}_{k}+{z}_{k},s{e}_{k-1}+{z}_{k-1},\cdots ,s{e}_{k-\tau}+{z}_{k-\tau})}\\ {\displaystyle \phantom{\rule{1em}{0ex}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\u2a7d\frac{\overline{r}}{p}{s}^{p}{c}_{2}^{p}(p){\left[\sum _{k=1}^{mT}{\left({\mathit{\Delta}}^{n}{e}_{k}\right)}^{2}\right]}^{\frac{p}{2}}-{\beta}^{\prime}{c}_{1}^{p}(p){\left\{\sum _{k=1}^{mT}\left[\sum _{i=-\tau}^{0}(s{e}_{k+i}+{z}_{k+i}{)}^{2}\right]\right\}}^{\frac{p}{2}}+mT\zeta}\\ {\displaystyle \phantom{\rule{1em}{0ex}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\u2a7d\frac{\overline{r}}{p}{s}^{p}{c}_{2}^{p}(p){\overline{\lambda}}^{\frac{np}{2}}-{\beta}^{\prime}{c}_{1}^{p}(p){\left[(\tau +1)\sum _{k=1}^{mT}{\left(s{e}_{k}+{z}_{k}\right)}^{2}\right]}^{\frac{p}{2}}+mT\zeta}\\ {\displaystyle \phantom{\rule{1em}{0ex}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\u2a7d\frac{\overline{r}}{p}{c}_{2}^{p}(p){\overline{\lambda}}^{\frac{np}{2}}{s}^{p}-(\tau +1{)}^{\frac{p}{2}}{\beta}^{\prime}{c}_{1}^{p}(p){\left({s}^{2}+\parallel z{\parallel}^{2}\right)}^{\frac{p}{2}}+mT\zeta .}\end{array}$$

Noting that for all
$\begin{array}{}u\in W,\sum _{k=1}^{mT}{r}_{k-1}{\left|{\mathit{\Delta}}^{n}{u}_{k-1}\right|}^{p}=0,\end{array}$
we have

$$\begin{array}{}{\displaystyle J(u)=\frac{1}{p}\sum _{k=1}^{mT}{r}_{k-1}{\left|{\mathit{\Delta}}^{n}{u}_{k-1}\right|}^{p}-\sum _{k=1}^{mT}F(k,{u}_{k},{u}_{k-1},\cdots ,{u}_{k-\tau})=-\sum _{k=1}^{mT}F(k,{u}_{k},{u}_{k-1},\cdots ,{u}_{k-\tau})\u2a7d0.}\end{array}$$

Thus, there exists a positive constant *R*_{1} > *δ* such that *J*(*u*) ⩽ 0 for any *u* ∈*∂* *Q*, where *Q* = (*B̄*_{R1} ∩ *W*)⊕{*se*|0 < *s* < *R*_{1}}. This verifies the condition (*J*_{2}) of Lemma 3.2. Therefore, *J* possesses a critical value *c* ⩾ *σ* > 0 with

$$\begin{array}{}{\displaystyle c=\underset{h\in \mathit{\Gamma}}{inf}\underset{u\in Q}{sup}J(h(u))\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathrm{a}\mathrm{n}\mathrm{d}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathit{\Gamma}\mathit{=}\{\mathit{h}\in \mathit{C}\mathit{(}\overline{\mathit{Q}}\mathit{,}{\mathit{E}}_{\mathit{m}\mathit{T}}\mathit{)}\mid \mathit{h}{\mathit{|}}_{\mathrm{\partial}\mathit{Q}}\mathit{=}\mathit{i}\mathit{d}\}\mathit{.}}\end{array}$$

Let *ũ* ∈ *E*_{mT} be a critical point associated to the critical value *c* of *J*, i.e., *J*(*ũ*) = *c*. If *ũ* ≠ *ū*, then we are done. Otherwise, if *ũ* = *ū*, it follows that

$$\begin{array}{}{\displaystyle {c}_{0}=\underset{u\in {E}_{mT}}{sup}J(u)=\underset{h\in \mathit{\Gamma}}{inf}\underset{u\in Q}{sup}J(h(u)).}\end{array}$$

In particular, choosing *h* = *id*, we have
$\begin{array}{}\underset{u\in Q}{sup}J(u)={c}_{0}.\end{array}$
Since the choice of *e* ∈ *∂* *B*_{1} ∩ *V* is arbitrary, we can take –*e* ∈*∂* *B*_{1} ∩ *V*. Similarly, there exists a positive number *R*_{2} > *δ*, so that *J*(*u*) ⩽ 0 for any *u* ∈*∂* *Q*_{1}, where *Q*_{1} = (*B̄*_{R2} ∩ *W*) ⊕ {–*se*|0 < *s* < *R*_{2}}.

Using Linking Theorem again, *J* possesses a critical value *c*′⩾ *σ* > 0, where

$$\begin{array}{}{\displaystyle {c}^{\prime}=\underset{h\in {\mathit{\Gamma}}_{1}}{inf}\underset{u\in {Q}_{1}}{sup}J(h(u))\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\mathit{\Gamma}}_{1}=\{h\in C({\overline{Q}}_{1},{E}_{mT})\mid h{|}_{\mathrm{\partial}{Q}_{1}}=id\}.}\end{array}$$

We claim that *c*′ ≠ *c*_{0}. Otherwise, suppose that *c*′ = *c*_{0}, then
$\begin{array}{}\underset{u\in {Q}_{1}}{sup}J(u)={c}_{0}.\end{array}$
Due to the facts *J*|_{∂Q} ⩽ 0 and *J*|_{∂Q1} ⩽ 0, *J* attains its maximum at some points in the interior of sets *Q* and *Q*_{1}. However, *Q* ∩ *Q*_{1}⊂ *W* and *J*(*u*) ⩽0 for any *u* ∈ *W*. This implies that *c*_{0} ⩽0, which contradicts *c*_{0} > 0. This proves the claim and hence the proof is complete.

According to Theorem 1.1, it is easy to obtain the following theorems and corollaries.

#### Theorem 4.1

*Assume that* (*T*_{1}) *and the following conditions hold*:

(T4)

*there exist constants*
$\begin{array}{}{\rho}_{1}>0,\phantom{\rule{thinmathspace}{0ex}}\alpha \in \left(0,\frac{\underset{\_}{r}}{(\tau +1{)}^{p/2}p}(mT{)}^{\frac{-|2-p|}{2}}(2\mathrm{sin}\frac{\pi}{mT}{)}^{np}\right)\end{array}$*such that*

$$\begin{array}{}{\displaystyle F(k,{v}_{0},{v}_{1},\cdots ,{v}_{\tau})\u2a7d\alpha {\left(\sqrt{\sum _{j=0}^{\tau}{v}_{j}^{2}}\right)}^{p},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}for\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}k\in Z\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}and\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\sum _{j=0}^{\tau}{v}_{j}^{2}\u2a7d{\rho}_{1}^{2};}\end{array}$$

T5)

*there exist constants*
$\begin{array}{}{\rho}_{2}>0,\phantom{\rule{thinmathspace}{0ex}}{\zeta}_{1}>0,\phantom{\rule{thinmathspace}{0ex}}\beta \in \left(\frac{\overline{r}}{(\tau +1{)}^{p/2}p}(mT{)}^{\frac{|2-p|}{2}}(2\mathrm{cos}\frac{1-(-1{)}^{mT}}{2mT}\pi ){)}^{np},+\mathrm{\infty}\right)\end{array}$*such that*

$$\begin{array}{}{\displaystyle F(k,{v}_{0},{v}_{1},\cdots ,{v}_{\tau})\u2a7e\beta {\left(\sqrt{\sum _{j=0}^{\tau}{v}_{j}^{2}}\right)}^{p}-{\zeta}_{1},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}for\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}k\in Z\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}and\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\sum _{j=0}^{\tau}{v}_{j}^{2}\u2a7e{\rho}_{2}^{2}.}\end{array}$$

*Then for any given positive integer m*, *Eq. (1.1) has at least two mT*-*periodic nontrivial solutions*.

#### Theorem 4.2

*Assume that* (*T*_{1}) *iand the following conditions hold*:

(T6)

$$\begin{array}{}{\displaystyle \underset{{\delta}_{1}\to 0}{lim}\frac{F(k,{v}_{0},{v}_{1},\cdots ,{v}_{\tau})}{{\delta}_{1}^{p}}=0,\phantom{\rule{thinmathspace}{0ex}}{\delta}_{1}=\sqrt{\sum _{j=0}^{\tau}{v}_{j}^{2}},\phantom{\rule{thinmathspace}{0ex}}\mathrm{\forall}\phantom{\rule{thinmathspace}{0ex}}(k,{v}_{0},{v}_{1},\cdots ,{v}_{\tau})\in Z\times {R}^{\tau +1};}\end{array}$$

(T7)

*there exist constants ρ*_{3} > 0 *and p*′ > *p such that for k* ∈ **Z** and
$\begin{array}{}\sum _{j=0}^{\tau}{v}_{j}^{2}\u2a7e{\rho}_{3}^{2},\end{array}$

$$\begin{array}{}{\displaystyle 0<{p}^{\prime}F(k,{v}_{0},{v}_{1},\cdots ,{v}_{\tau})\u2a7d\sum _{j=0}^{\tau}\frac{\mathrm{\partial}F(k,{v}_{0},{v}_{1},\cdots ,{v}_{\tau})}{\mathrm{\partial}{v}_{j}}{v}_{j}.}\end{array}$$

*Then for any given positive integer m*, *Eq. (1.1) has at least two mT*-*periodic nontrivial solutions*.

If *τ* = 0 and *f*(*k*, *u*_{k}) = *q*_{kg} (*u*_{k}), Eq. (1.1) reduces to the following 2*n*th-order nonlinear equation with *p*-Laplacian,

$$\begin{array}{}{\displaystyle {\mathit{\Delta}}^{n}\left({r}_{k-n}{\phi}_{p}\left({\mathit{\Delta}}^{n}{u}_{k-n}\right)\right)=(-1{)}^{n}{q}_{k}g\left({u}_{k}\right),\phantom{\rule{thinmathspace}{0ex}}k\in Z,}\end{array}$$(4.2)

where *g* ∈ *C*( **R**, **R**), *q*_{k}+T} = *q*_{k} > 0, for all *k* ∈ **Z**. Then, we have the following results.

#### Corollary 4.3

*Assume that the following hypotheses hold*:

(G1)

*there exists a functional G*(*v*) ∈ *C*^{1}(*R*, *R*) *with G*(*v*)⩾0 *and it satisfies*

$$\begin{array}{}{\displaystyle {G}^{\prime}(v)=g(v);}\end{array}$$

(G2)

*there exists a constant*
$\begin{array}{}\alpha \in \left(0,\frac{\underset{\_}{r}}{(\tau +1{)}^{p/2}p}(mT{)}^{\frac{-|2-p|}{2}}(2\mathrm{sin}\frac{\pi}{mT}{)}^{np}\right)\end{array}$ *such that*

$$\begin{array}{}{\displaystyle \underset{|v|\to 0}{lim\u2006sup}\frac{G(v)}{|v{|}^{p}}\u2a7d\alpha ;}\end{array}$$

(G3)

*there exists a constant*
$\begin{array}{}\beta \in \left(\frac{\overline{r}}{(\tau +1{)}^{p/2}p}(mT{)}^{\frac{|2-p|}{2}}(2\mathrm{cos}\frac{1-(-1{)}^{mT}}{2mT}\pi ){)}^{np},+\mathrm{\infty}\right)\end{array}$
*such that*

$$\begin{array}{}{\displaystyle \underset{|v|\to \mathrm{\infty}}{lim\u2006inf}\frac{G(v)}{|v{|}^{p}}\u2a7e\beta .}\end{array}$$

*Then for any given positive integer m*, *Eq. (4.2) has at least two mT*-*periodic nontrivial solutions*.

#### Corollary 4.4

*Assume that* (*G*_{1}) *and the following conditions hold*:

(G4)

*there exist constants*
$\begin{array}{}{\rho}_{1}^{\prime}>0,\alpha \in \left(0,\frac{\underset{\_}{r}}{p}(mT{)}^{\frac{-|2-p|}{2}}(2\mathrm{sin}\frac{\pi}{mT}{)}^{np}\right)\end{array}$ *such that*

$$\begin{array}{}{\displaystyle G(v)\u2a7d\alpha |v{|}^{p},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}for\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}|v|\u2a7d{\rho}_{1}^{\prime};}\end{array}$$

(G5)

*there exist constants*
$\begin{array}{}{\rho}_{2}^{\prime}>0,\phantom{\rule{thinmathspace}{0ex}}{\zeta}_{1}^{\prime}>0,\phantom{\rule{thinmathspace}{0ex}}\beta \in \left(\frac{\overline{r}}{p}(mT{)}^{\frac{|2-p|}{2}}(2\mathrm{cos}\frac{1-(-1{)}^{mT}}{2mT}\pi ){)}^{np},+\mathrm{\infty}\right)\end{array}$ *such that*

$$\begin{array}{}{\displaystyle G(v)\u2a7e\beta |v{|}^{p}-{\zeta}_{1}^{\prime},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}for\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}|v|\u2a7e{\rho}_{2}^{\prime}.}\end{array}$$

*Then for any given positive integer m*, *Eq. (4.2) has at least two mT*-*periodic nontrivial solutions*.

#### Corollary 4.5

*Assume that* (*G*_{1}) *and the following conditions hold*:

(G6)

$$\begin{array}{}{\displaystyle \underset{|v|\to 0}{lim}\frac{G(v)}{|v{|}^{p}}=0,\phantom{\rule{thinmathspace}{0ex}}\mathrm{\forall}\phantom{\rule{thinmathspace}{0ex}}v\in R;}\end{array}$$

(G7)

*there exist constants*
$\begin{array}{}{\rho}_{3}^{\prime}\end{array}$
> 0 and *p͠*′ > *p such that for k* ∈ **Z** and |*v*|⩾
$\begin{array}{}{\rho}_{3}^{\prime}\end{array}$,

$$\begin{array}{}{\displaystyle 0<\stackrel{~}{{p}^{\prime}}G(v)\u2a7dvg(v).}\end{array}$$

*Then for any given positive integer m*, *Eq. (4.2) has at least two mT*-*periodic nontrivial solutions*.

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