On mountain pass theorem and its application to periodic solutions of some nonlinear discrete systems

We obtain a new quantitative deformation lemma, and then gain a new mountain pass theorem. More precisely, the new mountain pass theorem is independent of the functional value on the boundary of the mountain, which improves the well known results (\cite{AR,PS1,PS2,Qi,Wil}). Moreover, by our new mountain pass theorem, new existence of nontrivial periodic solutions for some nonlinear second-order discrete systems is obtained, which greatly improves the result in \cite{Z04}.


Introduction and main results
It is well known that the classical mountain pass theorem of Ambrosetti-Rabinowitz [1] has been proved to be a powerful tool in applications to many areas of analysis, and quantitative deformation lemma is used to be a very excellent method to derive different kinds of minimax theorems including the mountain pass theorem, we refer the authors to see [3,7,19,21]. Firstly, we recall the famous mountain pass theorem given by Ambrosetti and Rabinowitz [1]: Theorem 1.1 ( [1]) Let X be a Banach space, ϕ ∈ C 1 (X, R), suppose there exist e ∈ X and two real numbers α > 0 and r > 0 such that e > r and  (iii) If (u n ) ⊂ X with 0 < ϕ(u n ), ϕ(u n ) bounded above, and ϕ ′ (u n ) → 0, then (u n ) possesses a convergent subsequence.
Set c 1 := max{ϕ(0), ϕ(e)}, we find that c 0 > c 1 in Theorem 1.1 and since then, there are many variant generalizations on this case for mountain pass theorem [1,2,3,21]. One of the elegant works is founded by Willem [21]. To be make the result more clear, we outline the quantitative deformation lemma first Then there exists η ∈ C(X, X), such that If one lets c 0 be stated in Theorem 1.1, then Willem's result can be described as the following: 21]) (Mountain pass type theorem) Let X be a Hilbert space, ϕ ∈ C 2 (X, R), suppose (i) ϕ satisfies the (P.S.) condition (ϕ is said to satisfies (P.S.) condition, if any sequence {u (k) } ⊂ X satisfying ϕ(u (k) ) → c and ϕ ′ (u (k) ) → 0 as k → +∞ with any constant c, implies a convergent subsequence); (ii) there exist e ∈ X and r > 0 be such that e > r and c 0 > ϕ(0) ϕ(e).
Then, c is a critical value of ϕ.
For the case of c 0 c 1 in mountain pass type theorems, we refer the authors to see [14,15,17]. Specially, we introduce a mountain pass type theorem in [15] and the extension mountain pass type theorem in [17] as follows: 15]) (Mountain pass type theorem) Let X be a Banach space and X has finite dimension, ϕ ∈ C 1 (X, R). Suppose there exist e ∈ X and two real numbers a and r > 0 such that e > r and (i) c 0 a; (ii) ϕ(0) a, ϕ(e) a; (iii) any sequence (u n ) in X such that ϕ(u n ) → limit a, and ϕ ′ (u n ) → 0 possesses a convergent subsequence.
Then, c is a critical value of ϕ.
Remark 1.1 For infinite-dimensional case, Pucci and Serrin in [15] gained that c is a critical value of ϕ when the assumption (i) in Theorem 1.3 was suitably strengthened, more precisely, their conditions depending on c 0 and the neighbor of {ϕ(u), u = r}.
Theorem 1.4 ( [17]) (Extension mountain pass theorem) Let X be a real Hilbert space, ϕ ∈ C 1 (X, R) satisfying the (P. S.) condition, e ∈ X and r > 0 be such that e > r. If c 0 c 1 , c is a critical value of ϕ.

Remark 1.2
The new mountain pass theorem is independent of c 0 , and if ϕ satisfies the (P.S.) condition, there existsû ∈ X such that ϕ(û) =ĉ . Now, we turn to an application of our new mountain pass theorem to the existence of periodic solutions of discrete systems, which has been appeared in computer science, economic, neural networks, ecology, cybernetics, etc and extensively investigated in [4,5,6,8,9,18,22,23,24,25].
Let Z, N, R be the set of all integers, natural numbers and real numbers, respectively. In [8], by critical point theory, Guo and Yu established the existence of periodic solutions to the below discrete difference equations So far as we know, [8] is the first paper to study the existence of periodic solutions of system (1.1) for superlinear f (n, u n ). For more results when f (n, u n ) is superlinear in the second variable u n , one consults to [6,23]. When f (n, u n ) is sublinear in the second variable u n , we refer the authors to see [9] and [22] and for the case of f (n, u n ) is neither suplinear nor sublinear, we refer to see [25]. For more details in this direction, one consults to [4,5,18,24]. It is remarked that, in [25], under the assumptions described below: x ∈ R and |x| δ; by using linking theorem [18], Zhou, Yu and Guo derived the existence of nontrivial M -periodic solutions for system (1.1), and they give an example: where constant a and positive integer M 3 satisfy a > 2, when M is even, Obviously, F (n, 0) = 0, ∀n ∈ Z, is weaker than condition (A 2 ), and the following example is failure to satisfy condition (A 2 ), but satisfy F (n, 0) = 0, ∀n ∈ Z: where constant a and positive integer M 6 satisfy a > 2, when M is even, So, the second interesting question is raised: for f (n, u n ) is neither superlinear nor sublinear, when condition (A 2 ) is replaced by F (n, 0) = 0, ∀n ∈ Z, can we still obtain the existence of nontrivial periodic solutions ?
In this paper, employing our new mountain pass theorem, we obtain new existence of nontrivial periodic solutions for discrete second-order discrete system (1.1), and our result is that: Theorem 1.6 Assume that F ∈ C 2 (R × R, R) and there is a positive integer M 6 satisfying condition (A 3 ) and the following conditions: (W 1 ) F 0, and for every (n, x) ∈ Z × R, F (n + M, x) = F (n, x); (W 2 ) F (n, 0) = 0, ∀ n ∈ Z.
Then, system (1.1) has at least one nontrivial M -periodic solutions. Remark 1.3 Obviously, condition (W 2 ) is weaker than condition (A 2 ), and it is easy to verify that F (t, x) in Example 1.2 satisfies all the conditions of Theorem 1.6, but does not satisfy condition (A 2 ). So, by a different method, i.e., Theorem 1.5, we greatly improve the result in [25].
The paper is organized as follows: Section 2 is devoted to establish a new quantitative deformation lemma. In Section 3, by using the new quantitative deformation lemma, we derive our new mountain pass theorem (Theorem 1.5). In Section 4, as an application of our new mountain pass theorem, we prove Theorem 1.6.

New quantitative deformation lemma
Then there exists η ∈ C(X, X), such that

Proof. Let us define
so that ψ is locally Lipschitz continuous, ψ = 1 on B, ψ = −1 on C and ψ = 0 on X\ A.
Step 1: We make some notations.
• For x, y ∈ S, a, b ∈ 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 inner product x s y s , ∀ x, y ∈ E M , then E M with the inner product given above is a finite dimensional Hilbert space and linearly homeomorphic to R M . And the norms · and · β induced by are equivalent, i.e., there exist constants 0 < C 1 C 2 such that • For a given matrix then by the results in [25], we have all the eigenvalues of B are 0, λ 1 , λ 2 , . . . , λ M −1 and λ j > 0 for all j ∈ Z[1, M − 1]. Moreover, , when M is even, 2(1 + cos π M ), when M is odd.
Step 2: Let the functional Then, by condition (A 3 ), we say ϕ(u) is bounded from above on E M . In fact, according to condition Then, by condition (A 3 ), we have ϕ(u) w ′ . So, ϕ(u) is bounded from above on E M .
Step 3: We claim that ϕ(u) satisfies the (P.S.) condition. In fact, let u (k) ∈ E M , for all k ∈ N, be such that {ϕ(u (k) )} is bounded. Then, by Step 2, there exists M 1 > 0, such that which implies that , which is convergent in E M , so the (P.S.) condition is verified.