 Research
 Open access
 Published:
On mountain pass theorem and its application to periodic solutions of some nonlinear discrete systems
Advances in Difference Equations volume 2019, Article number: 150 (2019)
Abstract
In this paper, we prove a new quantitative deformation lemma, and then gain a new mountain pass theorem in Hilbert spaces. By using the new mountain pass theorem, we obtain the new existence of two nontrivial periodic solutions for a class of nonlinear secondorder discrete systems, which greatly improves the result in (Zhou et al. in Proc. R. Soc. Edinb. A 134:1013–1022, 2004).
1 Introduction and main results
It is well known that the classical mountain pass theorem of Ambrosetti–Rabinowitz [2] has proved to be a powerful tool in applications of many areas to obtain the existence of periodic solutions. We first recall the famous theorem.
Theorem 1.1
([2])
Let X be a Banach space, \(\varphi \in C^{1}(X, R)\). Suppose there exist \(e\in X\) and two real numbers \(\alpha >0\) and \(r>0\) such that \(\e\>r\) and

(i)
\(\varphi (u) \geq \alpha >0\) on \(\{u\in X\mid \u\ \leq r\} \setminus \{0\}\);

(ii)
\(\varphi (0)=\varphi (e)=0\);

(iii)
if \((u_{n}) \subset X\) with \(0<\varphi (u_{n}), \varphi (u_{n})\) bounded above, and \(\varphi ^{\prime }(u_{n})\rightarrow 0\), then \((u_{n})\) possesses a convergent subsequence.
Then \(c:=\inf_{\gamma \in \varGamma }\max_{t\in [0, 1]}\varphi ( \gamma (t) )\), where
is a critical value of φ.
Since then, there have been many variant generalizations for the above mountain pass theorem [3,4,5]. Elegant work was done by Willem [3]. When proving the mountain pass theorem, we should introduce a quantitative deformation lemma first (also see [6, 7]).
Lemma 1.1
(Quantitative deformation lemma [3])
Let X be a Hilbert space, \(\varphi \in C^{2}(X, R)\), \(c\in R\), \(\varepsilon >0\). Assume that
Then there exists η∈ \(C(X, X)\), such that

(i)
\(\eta (u)=u\), \(\forall u\notin \varphi ^{1} ([c2 \varepsilon , c+2\varepsilon ] )\);

(ii)
\(\eta (\varphi ^{c+\varepsilon })\subset \varphi ^{c\varepsilon }\), where \(\varphi ^{c\varepsilon }:=\varphi ^{1} ((\infty , c\varepsilon ] )\).
Using Lemma 1.1, Willem established the following mountain pass type theorem.
Theorem 1.2
([3])
Let X be a Hilbert space, \(\varphi \in C^{2}(X, R)\), \(c_{0}>c_{1}\), \(c:= \inf_{\gamma \in \varGamma }\max_{t\in [0, 1]}\varphi (\gamma (t) )\), where \(c_{1}:=\max \{\varphi (0), \varphi (e)\}\), \(c_{0}:=\inf_{\u\=r}\varphi (u)\) and
Suppose

(i)
φ satisfies the \((P.S.)\) condition (φ is said to satisfy the \((P.S.)\) condition, if any sequence \(\{u^{(k)}\} \subset X\) satisfying \(\varphi (u^{(k)})\) is bounded and \(\varphi ^{\prime }(u^{(k)})\rightarrow 0\) as \(k\rightarrow +\infty \), possesses a convergent subsequence);

(ii)
there exist \(e\in X\) and \(r>0\) be such that \(\e\>r\) and \(c_{0}>\varphi (0)\geq \varphi (e)\).
Then c is a critical value of φ.
Let \(c_{0}\) and \(c_{1}\) be stated in Theorem 1.2. We see that Theorems 1.1 and 1.2 hold for \(c_{0}> c_{1}\). A few years later, this condition is relaxed to \(c_{0}\geq c_{1}\) in [8] (also see [9]):
Theorem 1.3
([8])
Let X be a Banach space and X has finite dimension, \(\varphi \in C^{1}(X, R)\). Suppose there exist \(e\in X\) and two real numbers a and \(r>0\) such that \(\e\>r\) and

(i)
\(c_{0}\geq a\), \(\varphi (0)\leq a\), \(\varphi (e)\leq a\);

(ii)
any sequence \((u_{n})\) in X such that \(\varphi (u_{n}) \rightarrow \mathit{limit} \geq a\), and \(\varphi ^{\prime }(u_{n})\rightarrow 0\) possesses a convergent subsequence.
Then c is a critical value of φ.
Clearly, all above results are based upon the relationship between \(c_{0}\) and \(c_{1}\). Then an interesting question is raised: can we obtain a mountain pass type theorem, which is independent of \(c_{0}\)? In this paper, we give a positive answer and the mountain pass type theorem is given by:
Theorem 1.4
Let X be a Hilbert space, \(\varphi \in C^{2}(X, R)\), \(e, e_{1} \in X\), \(r>0\) be such that \(0<\e_{1}\<r\) and \(\e\>r\), and \(\varphi (0)<\varphi (e)=\varphi (e_{1})\). Then, for each small enough \(\varepsilon >0\), there exists \(\hat{u}\in X\) such that

(i)
\(\hat{c}2\varepsilon \leq \varphi (\hat{u})\leq \hat{c}+2 \varepsilon \);

(ii)
\(\\varphi ^{\prime }(\hat{u})\<2\varepsilon \),
where \(\hat{c}:=\inf_{\gamma \in \hat{\varGamma }}\max_{t\in [0, 1]} \varphi (\gamma (t) )\) and
Remark 1.1
The new mountain pass theorem is independent of \(c_{0}\), and if φ satisfies the \((P.S.)\) condition, there exists \(\hat{u}\in X\) such that \(\varphi (\hat{u})= \hat{c}\).
Now, we turn to an application of this mountain pass type theorem to the existence of periodic solutions of discrete systems, which have been appeared in computer science, economic, neural networks, ecology, cybernetics, etc. and which were extensively investigated in [1, 10,11,12,13,14,15,16,17,18].
Let Z, N, R be the set of all integers, natural numbers and real numbers, respectively. In [13], by critical point theory, Guo and Yu established the existence of two nontrivial Mperiodic solutions to the following discrete difference equations:
where \(u_{n}=u(n)\in R\) and
and \(F: Z\times R\rightarrow R\), \(F(n,x)\) is continuously differentiable in x for every \(n\in Z\) and Tperiodic (\(0< T\in N\)) in n for all \(x\in R\).
If \(u_{n}\not \equiv 0\), \(\forall n\in Z\), is a Mperiodic solution of system (1.1), then we call \(u_{n}\) a nontrivial Mperiodic solution of system (1.1). It is well known that [13] is one of the original papers to study the existence of nontrivial periodic solutions of system (1.1) for superlinear \(f(n,u_{n})\) at \(u_{n}\) (also see [12, 17]). When \(f(n, u_{n})\) is sublinear in the second variable \(u_{n}\), we refer the reader to [14, 16], and for the case of \(f(n, u_{n})\) is neither superlinear nor sublinear, we refer to [1]. For more details in this direction, one consults to [10, 11, 15, 18]. It is remarked that, in [1], under the assumptions described below:
 (A_{1}):

\(F\geq 0\), \(F\in C^{1}(R\times R, R)\), and for every \((n, x)\in Z\times R\), there is a positive integer \(M\geq 3\) such that \(F(n+M,x)=F(n,x)\);
 (A_{2}):

there exist constants \(\delta >0\), \(\alpha \in (0,1 \cos \frac{2\pi }{M})\) such that
$$ F(n,x)\leq \alpha x^{2}\quad \mbox{for } n\in N, x\in R \mbox{ and } \vert x \vert \leq \delta ; $$  (A_{3}):

there exist constants \(w_{1}>0\), \(w_{2}>0\) and \(w_{3}\in (2, +\infty )\) when M is even or \(w_{3}\in (1+\cos \frac{ \pi }{M}, +\infty )\) when M is odd, such that
$$ F(n,x) \geq w_{3}x^{2}w_{2}\quad \mbox{for } n \in N, \vert x \vert \geq w_{1}, $$
by using the linking theorem in [15], Zhou, Yu and Guo derived the existence of two nontrivial Mperiodic solutions for system (1.1), and they gave an example:
Example 1.1
Take \(F(t, x)=a(x^{2}/2+\cos x1)(\phi (t)+K)\) with \(x\in R\), constant \(K>0\), where constant a and positive integer \(M\geq 3\) satisfy
\(\phi (t)\in C^{1}({R,R})\), and \(\phi (t)\) is a Mperiodic function satisfying \(\phi (t)< K\).
Obviously, by conditions (A_{1}) and (A_{2}), we deduce that \(F(n,0)=0\), \(\forall n\in Z\). However, the opposite conclusion does not hold. So, it is also natural to ask: if \(f(n, u_{n})\) is neither superlinear nor sublinear such that \(F(n,0)=0\), \(\forall n\in Z\), can we still obtain the existence of two nontrivial periodic solutions?
Moreover, in the original paper [13], we note that \(F\in R\) but not \(F\geq 0\). So, another interesting question is raised: for the case of \(f(n, u_{n})\) is neither superlinear nor sublinear at \(u_{n}\), if we omit this nonnegative restriction on F, can we still obtain two nontrivial Mperiodic solutions for system (1.1)?
In this paper, by employing the mountain pass type theorem 1.4, for the case of \(f(n, u_{n})\) is neither superlinear nor sublinear, we will derive the new existence of two nontrivial periodic solutions for secondorder discrete system (1.1), and our result is the following.
Theorem 1.5
Let \(F\in C^{1}(R\times R,R)\) and for every \(n\in Z\), \(F(n,x)\) is twice continuously differentiable in x. Suppose that there is a positive integer \(M\geq 3\) satisfying condition (A_{3}) and the following conditions:
 (W_{1}):

for every \((n,x)\in Z\times R\), \(F(n+M,x)=F(n,x)\);
 (W_{2}):

\(F(n,0)=0\), \(\forall n\in Z\),
hold. Then system (1.1) has at least two nontrivial Mperiodic solutions.
Remark 1.2
For the case of \(f(n, u_{n})\) is neither superlinear nor sublinear at \(u_{n}\), by using linking theorem in [15], Zhou–Yu–Guo [1] obtained two nontrivial Mperiodic solutions for system (1.1). Here, by using a different philosophy, i.e. an extension of the mountain pass type theorem, we also obtain the existence of two nontrivial Mperiodic solutions.
Remark 1.3
Theorem 1.5 is concerned with \(f(n, u_{n})\) which is neither superlinear nor sublinear, and under condition (W_{1}) of Theorem 1.5, we do not need \(F\geq 0\).
Remark 1.4
We only need \(F(n,0)=0\), \(\forall n\in Z\) in Theorem 1.5 (W_{2}), which is also weaker than condition (A_{2}).
Example 1.2
Let \(F(t,x)=a( x^{2}/2+x+\cos x1)(\phi (t)+K)\) with \(x\in R\), \(K>0\) such that the constant a and the positive integer \(M\geq 3\) satisfy
Let \(\phi (t)\in C^{1}({R,R})\) so that \(\phi (t)\) is a Mperiodic function satisfying \(\phi (t)< K\). Then all assumptions in Theorem 1.5 are satisfied. Thus (1.1) has at least two nontrivial Mperiodic solutions.
Example 1.3
Let \(F(t,x)=a( \mu x^{2}+x+\cos x1)(\phi (t)+K)\) with \(x\in R\), \(\mu \geq 1/2\), \(K>0\), such that the constant a and the positive integer \(M\geq 3\) satisfy
\(\phi (t)\in C^{1}({R,R})\), and \(\phi (t)\) is a Mperiodic function satisfying \(\phi (t)< K\). Then F satisfies all assumptions in Theorem 1.5. Therefore, (1.1) has at least two nontrivial Mperiodic solutions.
Remark 1.5
Obviously, F in Examples 1.1, 1.2 and 1.3 satisfy all the assumptions in Theorem 1.5, but in Examples 1.2 and 1.3, F satisfies neither condition (A_{1}) nor condition (A_{2}).
The paper is organized as follows: Sect. 2 is devoted to establishing a new quantitative deformation lemma. In Sect. 3, by using the new quantitative deformation lemma, we derive our new mountain pass theorem (Theorem 1.4). In Sect. 4, as an application of our new mountain pass theorem, we prove Theorem 1.5.
2 New quantitative deformation lemma
Lemma 2.1
Let X be a Hilbert space and ε be a small enough positive number. Let \(\varphi \in C^{2}(X, R)\), \(h\in R\). Assume that
Then there exists η∈ \(C(X, X)\), such that

(i)
\(\eta (u)=u\), \(\forall u\notin \varphi ^{1} ([h2 \varepsilon , h+2\varepsilon ] )\setminus D\), where D is any subset of X satisfying \(D\subset \varphi ^{1} ([h\frac{1}{3} \varepsilon ^{2}, h+\frac{1}{3}\varepsilon ^{2}] )\);

(ii)
\(\eta (\varphi ^{1}[h+\frac{1}{2}\varepsilon ^{2}, h+ \varepsilon ^{2}] )\subset \varphi ^{1} ([h\frac{3}{2} \varepsilon ^{2}, h\frac{1}{2}\varepsilon ^{2}] )\).
Proof
Let us define
Then ψ is a locally Lipschitz continuous function such that
Let us also define a locally Lipschitz continuous vector field
It is clear that \(\f(u)\\leq (2\varepsilon )^{1}\) on X. For each u∈ X, the Cauchy problem
has a unique solution \(\sigma (\cdot , u)\) defined on R. Moreover, σ is continuous on \(R\times X\) (see e.g. [7]). By (2.2) and (2.3), then
On the other hand, with the help of (2.1), \(\psi =0\) on X∖ A, so the map η defined on X by \(\eta (u):=\sigma (2\varepsilon ^{2}, u)\) satisfies (i).
By (2.3), we have
Combining \(\f(u)\\leq (2\varepsilon )^{1}\), we know
Let \(u\in \varphi ^{1} ([h+\frac{1}{2}\varepsilon ^{2}, h+ \varepsilon ^{2}] )=C\), then \(\psi (\sigma (0, u) )=\psi (u)=1\). Clearly, \(\sigma (\cdot , u)\) is continuous on R, ψ is locally Lipschitz continuous on σ, and \(\psi (\sigma (0, u) )=1\) for \(u\in C\). Thus, by (2.5), for ε is small enough, we have
Therefore, it follows from (2.5) that
and
So, (ii) is proved. □
Remark 2.1
The conclusion (ii) in Lemma 2.1 is different from conclusion (ii) in in Lemma 1.1.
Now, by Lemma 2.1, we can prove our mountain pass type theorem which is independent of \(c_{0}\).
3 Proof of Theorem 1.4
Conclusion (i) is obvious. Suppose that conclusion (ii) does not hold. Consider \(\beta =\eta \circ \gamma \), where η is given by Lemma 2.1, and then we need to check two cases.
Case 1. \(\varphi (0)<\varphi (e)=\varphi (e_{1})<\hat{c}\).
By an analogous argument of conclusion (i) of Lemma 2.1, we have
So, \(\beta \in \hat{\varGamma }\). By the definition of ĉ, there exists \(\gamma \in \hat{\varGamma }\) such that
Then, from conclusion (ii) of Lemma 2.1 and (3.1), we have \(\hat{c}\leq \max_{t\in [0, 1]}\varphi (\beta (t) ) \leq \hat{c}\varepsilon \). This is a contradiction.
Case 2. \(\varphi (0)<\varphi (e)=\varphi (e_{1})=\hat{c}\).
Let us assume that \(\max_{t\in [0, 1]}\varphi (\gamma (t)) \equiv \hat{c}\) for every \(\gamma \in \hat{\varGamma }\). Since \(\varphi (e_{1})=\hat{c}\) for every \(0<\e_{1}\<r\) and \(\varphi \in C^{2}(X, R)\), we see that conclusion (ii) holds. Thus, in the following discussion, we only need to argue the case of \(\max_{t\in [0, 1]}\varphi (\gamma (t)) \not \equiv \hat{c}\). Observing that \(\max_{t\in [0, 1]}\varphi (\gamma (t)) \geq \varphi (e_{1})=\hat{c}\), there exists \(\gamma _{1}\in \hat{\varGamma }\) such that \(\max_{t\in [0, 1]}\varphi (\gamma _{1}(t)) >\varphi (e_{1})\). Note that \(\gamma _{1}(\frac{1}{2})=e_{1}\), \(\varphi (e_{1})=\hat{c}\) (\(0<\e_{1}\<r\)), and \(\gamma _{1}(t)\in C ([0, 1], X )\) and \(\varphi \in C^{2}(X, R)\), there exists \(t_{1}\in [0, 1]\) such that
Take \(D = \{u\in X\mid h=\varphi (u)=\hat{c}\}\) in Lemma 2.1. Then, by \(\varphi (0)<\varphi (e)=\varphi (e_{1})=\hat{c}\) and conclusion (i) of Lemma 2.1, we have
So, \(\beta \in \hat{\varGamma }\). It follows from (3.2) and the conclusion (ii) of Lemma 2.1 that \(\max_{t\in [0, 1]} \varphi (\beta (t))\leq \hat{c}\frac{1}{2}\varepsilon ^{2}\). Combining with \(\beta (1) = e\), we must have \(\max_{t\in [0, 1]}\varphi (\beta (t))\geq \varphi (e)=\hat{c}\), which implies a contradiction.
Combining Case 1 and Case 2, the proof for our new mountain pass theorem is complete.
4 Proof of Theorem 1.5
We divide the proof into eight steps.
Step 1: We introduce some notations.
• For \(a, b \in Z\), define \(Z [a] = \{a, a+1, \ldots \}\), \(Z [a, b] = \{a, a+1, \ldots , b\}\) for \(a\leq b\).
• Let the set of sequences \(S= \{u=\{u_{n}\}= (\ldots , u_{n}, \ldots , u_{0}, \ldots , u_{n}, \ldots), u_{n} \in R, n \in Z \}\). For any given positive integer M, \(E_{M}\) is defined as a subspace of S by
• For \(x,y \in S\), \(a, b \in R\), \(ax+by\) is defined by
then S is a vector space. Clearly, \(E_{M}\) is isomorphic to \(R^{M}\), \(E_{M}\) can be equipped with the inner product
then \(E_{M}\) with the inner product given above is a finite dimensional Hilbert space and linearly homeomorphic to \(R^{M}\). And we denote the norm \(\x\ = (\sum_{j=1}^{M}x_{j}^{2} )^{\frac{1}{2}}\).
• For a given matrix
in view of the results established in [1], all the eigenvalues of B are 0, \(\lambda _{1}, \lambda _{2}, \ldots , \lambda _{M1}\) and \(\lambda _{j}>0\) for all \(j\in Z[1, M1]\). Moreover,
Step 2: Let the functional
where
According to condition (A_{3}), if we let
and \(\tilde{w}=w+w_{2}\), then
Combining with the fact
for all \(u\in E_{M}\), there exists a constant \(w^{\prime }> \tilde{w}\) such that
Notice that if M is even then \(w_{3}\in (2, +\infty )\), and if M is odd, then \(w_{3}\in (1+\cos \frac{\pi }{M}, +\infty )\), and
we have \(\lambda _{\max }/2w_{3}<0\), which implies that \(\varphi (u) \leq w^{\prime }\). Therefore, \(\varphi (u)\) is bounded from above on \(E_{M}\).
Step 3: Set \(\tilde{c}=\sup_{u\in E_{M}}\varphi (u)\). From \(\lambda _{\max }/2w_{3}<0\) and
we have \(\varphi (u)\rightarrow \infty \) as \(\ u \\rightarrow +\infty \), which implies \(\varphi (u)\rightarrow +\infty \) as \(\ u \\rightarrow +\infty \). Hence, for every \(l>\tilde{c}\), there is a constant \(P>0\) such that, for every \(\u\>P\), \(\varphi (u)>l>\tilde{c}\). With the help of the continuity of \(\varphi (u)\), there must be a point \(\bar{u}\in E_{M}\) such that \(\varphi (\bar{u})=\tilde{c}= \sup_{u\in E_{M}}\varphi (u)\) and \(\u\ \leq P\). Therefore, ū is a critical point of the functional \(\varphi (u)\) on \(E_{M}\) with the critical value c̃.
Next, let us search for the second critical point of the functional \(\varphi (u)\) on \(E_{M}\).
Step 4: Let \(u^{(k)} \in E_{M}\), for all \(k\in N\), be such that \(\{\varphi (u^{(k)})\}\) is bounded. From Step 2, there exists \(M_{1} > 0\), such that
which implies that
That is, \(\{u^{(k)}\}\) is bounded in \(E_{M}\). Since \(E_{M}\) is finite dimensional, there exists a subsequence of \(\{ u^{(k)} \}\) (not labeled), which is convergent in \(E_{M}\), so the \((P.S.)\) condition is verified.
Step 5: Let \(f(t)=t^{2}w_{3}t^{3}\) for \(t\in [0, +\infty ]\). Then \(f^{\prime }(t)=2t3w_{3}t^{2}=t(23w_{3}t)\). So f is increase on \([0, 2/(3w_{3})]\) and decrease on \((2/(3w_{3}), 1/w_{3})\). Combining \(f(0)=0\) and \(f(1/w_{3})=0\), there exist \(\xi \in (0, 2/(3w_{3}))\) and \(\zeta \in ( 2/(3w_{3}), 1/w_{3})\), such that \(f(\xi )=f(\zeta )>0\).
Step 6: By (4.1) and condition (W_{2}), we have \(\varphi (0)=0\). Take
and
Then it is easy to verify that
and
In view of the fact that \(f(\xi )=f(\zeta )>0\), \(\e\=\xi \) and \(\e_{1}\=\zeta \), we have \(\varphi (e)=\varphi (e_{1})>0=\varphi (0)\) and \(\e\\neq \e_{1}\\). Moreover, all the assumptions in Theorem 1.4 are satisfied. Noticing that \(\varphi (u)\) satisfies the \((P.S.)\) condition, then, by Remark 1.1, there exists a critical point û such that \(\varphi (\hat{u})=\hat{c}\) (ĉ is given in Theorem 1.4).
Step 7: In order to obtain two critical points, we also need to prove that \(\hat{u}\neq \bar{u}\). Since \(\varphi (\hat{u})=\hat{c}\) and \(\varphi (\bar{u})=\tilde{c}\), if we can prove \(\hat{c}\neq \tilde{c}\), that also implies \(\hat{u}\neq \bar{u}\). So in the following, we are ready to prove that \(\hat{c}\neq \tilde{c}\). Since \(u\in E_{M}\) and \(E_{M}\) is linearly homeomorphic to \(R^{M}\), in Theorem 1.4, we can take \(X=E_{M}\) and construct
where \(u_{n1}=(2\zeta 4\xi )t^{2}+(\zeta +4\xi )t\), \(t\in [0, 1]\). Obviously, \(\gamma _{1}\in C ([0, 1] )\).
One computes that \(\gamma _{1}(0)=(0,\ldots ,0,0,0,0,\ldots ,0)\),
and
Hence, \(\gamma _{1}(t)\in \hat{\varGamma }\) where
Let \(u=\gamma _{1}(t)\), \(t\in [0, 1]\) be in (4.1). Since \(F(n,0)=0\), we have
Let \(u_{n1}=y\in [0, +\infty ]\), then, by (4.2), we see that when \(y=2/3w_{3}\), \(\varphi (\gamma _{1}(t))=y^{2}w_{3}y^{3}\) takes the maximum value \(4/(27w^{2}_{3})\). We must point that when \(t\in [0, 1]\), \(u_{n1}\) can take the value of \(2/3w_{3}\). In fact, when \(t=0\),
and when \(t=1\),
Observing that \(\zeta \in (2/(3w_{3}), 1/w_{3})\), then, by the continuity of \(u_{n1}=(2\zeta 4\xi )t^{2}+(\zeta +4\xi )t\), there exists \(\tilde{t}\in [0, 1]\) such that \(u_{n1}=(2\zeta 4\xi )\tilde{t}^{2}+(\zeta +4\xi )\tilde{t}=2/(3w_{3})\).
Since \(\hat{c}:=\inf_{\gamma \in \hat{\varGamma }}\max_{t\in [0, 1]}\varphi (\gamma (t) )\) and \(\gamma _{1}(t)\in \hat{\varGamma }\), we have \(\hat{c}\leq 4/(27w^{2}_{3})\). In order to obtain two critical point of \(\varphi (u)\) on \(E_{M}\), we will show that \(\tilde{c}>4/(27w^{2}_{3})\).
Since \(E_{M}\) is linearly homeomorphic to \(R^{M}\) and \(M\geq 3\), if we choose
then
By (4.1) and \(F(n, 0)=0\), we have
Employing \(\tilde{c}=\sup_{u\in E_{M}}\), we have \(\tilde{c}\geq 1/ (4w^{2}_{3})>4/(27w^{2}_{3})\geq \hat{c}\). Combining \(\varphi (\hat{u})=\hat{c}\), \(\varphi (\bar{u})=\tilde{c}\), both û and ū are critical point of the functional φ, we obtain two different critical points of φ.
Step 8: (1.1) has at least two nontrivial Mperiodic solutions. Since \(\varphi \in \mathcal{C}^{2} (E_{M}, R)\), for any \(u = \{u_{n} \}_{n \in Z} \in E_{M}\), according to \(u_{0} = u_{M}\), \(u_{1} = u _{M+1}\), one computes that
Therefore, the existence of critical points of φ on \(E_{M}\) implies the existence of periodic solutions of system (1.1). Moreover, we obtained two different critical points of \(\varphi (u)\) on \(E_{M}\) in Step 7, so system (1.1) has two different Mperiodic solutions.
Note that in (4.1), \(\varphi (0)=0\). But \(\varphi (\hat{u})=\hat{c}\geq \varphi (e) >\varphi (0)=0\) and \(\varphi (\bar{u})=\tilde{c}=\sup_{u\in E_{M}}>4/(27w^{2}_{3}) \geq \hat{c}\geq \varphi (e)>0\), so any of the above periodic solutions û and ū is nontrivial. From this, Theorem 1.5 is proved.
Remark 4.1
Let \(F(t,x)\) be stated in Example 1.2, from (4.1), we have
where \(u= (\ldots , u_{n}, \ldots , u_{0}, \ldots )\in E_{M}\). We notice that the value of \(\inf_{\u\=r}\varphi (u)\) is very difficult to compute, but fortunately the condition in our new mountain pass theorem (Theorem 1.4) is independent of \(\inf_{\u\=r} \varphi (u)\), and we need not compute it.
References
Zhou, Z., Yu, J.S., Guo, Z.M.: Periodic solutions of higherdimensional discrete systems. Proc. R. Soc. Edinb. A 134, 1013–1022 (2004)
Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 49, 349–381 (1973)
Willem, M.: Minimax Theorems. Birkhäuser, Boston (1996)
Chang, K.C.: Infinite Dimensional Morse Theory and Multiple Solution Problems, vol. 6. Birkhäuser, Boston (1993)
Brezis, H., Coron, J.M., Nirenberg, L.: Free vibrations for a nonlinear wave equation and theorem of P. Rabinowitz. Commun. Pure Appl. Math. 33, 667–684 (2010)
Ghoussoub, N.: Duality and Perturbation Methods in Critical Point Theory. Cambridge University Press, Cambridge (1993)
Schwartz, L.: Cours d’analyse. Hermann, Paris (1991–1994)
Pucci, P., Serrin, J.: A mountain pass theorem. J. Differ. Equ. 60, 142–149 (1985)
Qi, G.J.: Extension of mountain pass lemma. Chin. Sci. Bull. 32, 798–801 (1987)
Chen, P., Fang, H.: Existence of periodic and subharmonic solutions for secondorder pLaplacian difference equations. Adv. Differ. Equ. 2007, 042530 (2007)
Deng, X.Q., Shi, H.P., Xie, X.L.: Periodic solutions of second order discrete Hamiltonian systems with potential indefinite in sign. Appl. Math. Comput. 218, 148–156 (2011)
Ding, L., Wei, J.L.: Notes on multiple periodic solutions for secondorder discrete Hamiltonian system. Dyn. Syst. 32, 544–552 (2017)
Guo, Z.M., Yu, J.S.: The existence of periodic and subharmonic solutions for secondorder superlinear difference equations. Sci. China Math. 46, 506–515 (2003)
Guo, Z.M., Yu, J.S.: The existence of periodic and subharmonic solutions to subquadratic secondorder difference equations. J. Lond. Math. Soc. 68, 419–430 (2003)
Rabinowitz, P.H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations, vol. 65. Am. Math. Soc., Providence (1986)
Xue, Y.F., Tang, C.L.: Existence of a periodic solution for subquadratic secondorder discrete Hamiltonian system. Nonlinear Anal., Theory Methods Appl. 67, 2072–2080 (2007)
Xue, Y.F., Tang, C.L.: Multiple periodic solutions for superquadratic secondorder discrete Hamiltonian systems. Appl. Math. Comput. 196, 494–500 (2008)
Yu, J.S., Deng, X.Q., Guo, Z.M.: Periodic solutions of a discrete Hamiltonian system with a change of sign in the potential. J. Math. Anal. Appl. 324, 1140–1151 (2006)
Acknowledgements
The authors express their sincere gratitude to the editors and referees for their helpful comments and suggestions.
Funding
The first author is partially supported by Graduate Science Innovation Foundation of Sichuan University (2018YJSY045). The second author is partially supported by National Science Foundation of China (11501577), and the third author is partially supported by National Science Foundation of China (116712787).
Author information
Authors and Affiliations
Contributions
The authors contributed equally to this paper. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Ding, L., Wei, J. & Zhang, S. On mountain pass theorem and its application to periodic solutions of some nonlinear discrete systems. Adv Differ Equ 2019, 150 (2019). https://doi.org/10.1186/s1366201921028
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s1366201921028