PeriodicSolutionswithPrescribedMinimalPeriod for2n th-Order Nonlinear Discrete Systems

In this paper, by using the critical point theory, some new results of the existence of at least two nontrivial periodic solutions with prescribed minimal period to a class of 2nth-order nonlinear discrete system are obtained.)emain approach used in our paper is variational technique and the linking theorem.)e problem is to solve the existence of periodic solutions with prescribed minimal period of 2nth-order discrete systems.


Introduction
In the following and in the sequel, we denote by N, Z, and R the sets of all natural numbers, integers, and real numbers, respectively. e symbol * is defined by the transpose of a vector. Let [·] be the greatest integer function. For any integers a and b with a ≤ b, we let Z(a, b) denote the discrete interval a, a + 1, . . . , b { } and Z(a) � a, a + 1, . . . , { }. Now, we concern with the following 2nth-order nonlinear discrete system: where Δ is the forward difference operator defined by Δx k � x k+1 − x k , Δ 0 x k � x k , and Δ j x k � Δ j− 1 (Δx k ) for j ≥ 2, g(t, x) ∈ C 1 (R 2 , R), and g(t + M, x) � g(t, x) for a given integer M ≥ 3. We may think of (1) as a discrete analogue of the following 2nth-order differential equation: dt 2n � (− 1) n g(t, x(t)), t ∈ R. (2) Equations similar in structure to (2) have been studied by many authors [1][2][3][4][5][6]. e difficulty of this paper comparing with system (2) is that there are few known techniques for studying the existence of periodic solutions with minimal period of (1). Denote roughout this paper, we suppose that there is a function G(t, x) ∈ C 2 (R 2 , R) with G(t + M, x) � G(t, x), G(− t, − x) � G(t, x), G(t, x) ≥ 0, and Bin [7] in 2013 considered the second-order discrete Hamiltonian systems: By using Morse theory, some new results concerning the existence of nontrivial periodic solution are obtained.
Using the critical point method, Liu et al. [8] studied the following forward and backward difference equation: Some new criteria for the existence and multiplicity of periodic and subharmonic solutions are established.
In 2016, Leng [9] established some new criteria for the existence and multiplicity of periodic and subharmonic solutions to the 2nth-order difference equation with ϕ c -Laplacian using the linking theorem in combination with variational technique.
By establishing a new proper variational framework and using the critical point theory, He [10] established some new existence criteria to guarantee that the 2nth-order nonlinear difference equation containing both advance and retardation with p-Laplacian has infinitely many homoclinic orbits. Lin and Zhou [11] concerned with the existence and multiplicity of periodic and subharmonic solutions of the following 2nth-order difference equation By using the critical point theory, some new sufficient conditions are obtained and some previous results are generalized.
Xia [12] in 2018 considered the existence of periodic solutions for a higher order nonlinear difference equation by making use of the critical point theory. Some existence criteria are established. Some results are generalized.
In 1978, Rabinowitz [13] proposed a conjecture that the Hamiltonian system has a nonconstant periodic solution with prescribed minimal period under some given conditions. From then on, there has been much progress [14][15][16] on Rabinowitz's conjecture under various conditions. Ambrosetti and Mancini [14] assumed that the dual functional is bounded from below and the Hamiltonian system has a minimum to which correspond a solution with given minimal period. Ekeland and Hofer [16] proved that if the Hamiltonian system is flat near an equilibrium and superquadratic near infinity, it has a periodic solution with minimal period. e estimate of number of periodic solutions was established in [15]. In contrast to differential equations, the research on periodic solutions with prescribed minimal period of higher order discrete systems is fresh and there are very few literature (see [7][8][9][10][11][12][17][18][19][20][21][22][23][24][25][26][27][28][29]) on it. Comparing this paper with references [8][9][10][11], the advantages and differences of this paper are that the existence of periodic solutions with prescribed minimal period of (1) is obtained in this paper; however, only periodic solutions are obtained in the references [8][9][10][11]. Given integer T ≥ 2, Long [23] considered the following T-cycle discrete Hamiltonian systems: By making use of minimax theory and geometrical index theory, some results on the existence and multiplicity of subharmonic solutions with prescribed minimal period to the abovementioned discrete Hamiltonian systems are obtained. Yu et al. [27] in 2004 obtained some sufficient conditions on the existence of subharmonic solutions with prescribed minimal period for the second-order difference equation by using variational methods. erefore, there is still spacious room to explore the periodic solutions with prescribed minimal period of higher order discrete systems. Motivated by the papers [9,12], for a given integer s with s > 1, the aim of this paper is to obtain some new results for the existence of at least two nontrivial periodic solutions with minimal period sM to a 2nth-order discrete system by using critical point method.
Here, we give the existence results of at least two nontrivial periodic solutions with minimal period sM as follows.
(G 5 ) Denote by m s the least prime factor of s: en, (1) possesses at least two nontrivial periodic solutions with minimal period sM.
then there is S > 0 such that for any prime integer s > S, (1) possesses at least two nontrivial periodic solutions with minimal period sM.
) and the following assumptions: en, (1) has at least two nontrivial periodic solutions with minimal period sM.
then (1) has at least two nontrivial periodic solutions with minimal period sM.
then there is S > 0 such that for any prime integer s > S, (1) possesses at least two nontrivial periodic solutions with minimal period sM. e remainder of this paper is organized as follows. In Section 2, we build the variational functional and gather some basic notations that are necessary in the proofs of our main theorems. In Section 3, we state some useful lemmas. In Section 4, the main results will be proved.
Regarding the basis for variational methods, we refer the reader to [30]. Regarding the basic knowledge of integral inequalities and extended hypergeometric functions, the reader is referred to [31].

Preliminaries
In this section, we shall establish the variational framework associated with (1) and gather some basic notations that are necessary in the proofs of our main theorems.
Let the vector space B be defined by and for any x ∈ B, define the inner product Discrete Dynamics in Nature and Society 3 and the norm For any x ∈ B, let I be the functional defined by en, I is continuously differentiable and sM). (25) us, x is a critical point of I(x) on B if and only if erefore, we reduce the problem of finding sM-periodic solutions of (1) to that of seeking critical points of the functional I on B. Denote · · · · · · · · · · · · · · · · · · 0 0 0 · · · 2 − 1 It is clear that the eigenvalues of L are Furthermore, L is positively semidefinite and all of eigenvalues of L are positive except for 0, and Obviously, 0 is an eigenvalue of L and (1, 1, . . . , 1) * is an eigenvector associated to 0. Let A � (c, c, . . . , c) If sM is odd, then B � A⊕Y⊕Z. For any x ∈ B and i ∈ Z, where p, q, p k , and q k are constants. If sM is even, then 4 is the eigenvalue of L. Let η denote the eigenvector corresponding to 4, and W �span η . We have B � A⊕Y⊕Z⊕W. For any x ∈ B and i ∈ Z, where p, p k , and q k are constants. Suppose that B is a real Banach space and I ∈ C 1 (B, R). As usual, I is said to satisfy the Palais-Smale condition if every sequence x (j) ⊂ B, such that I(x (j) ) is bounded and I ′ (x (j) ) ⟶ 0, (j ⟶ ∞), has a convergent subsequence.
e sequence x (j) is called a Palais-Smale sequence.

Some Lemmas
To apply critical point theory to study the existence of periodic solutions with minimal period sM of (1), some lemmas should be stated in this section which will be used in proofs of our main results. 4 Discrete Dynamics in Nature and Society Below, we denote by B r (x) the open ball centered at x ∈ B with radius r > 0, B r (x) as its closure, and zB r (x) as its boundary.
Lemma 1 (linking theorem [30]). Let B be a real Banach space, B � B 1 ⊕B 2 , where B 1 is finite dimensional. Suppose that I ∈ C 1 (B, R) satisfies the Palais-Smale condition and the following: ere are positive constants c and r such that We have B � Z, then Lemma 2. Suppose that G(t, x) satisfies (G 1 ) − (G 5 ). en, is bounded from above in B.

Lemma 3. Suppose that
satisfies the Palais-Smale condition.
Proof. Assume that I(x (j) ) is a bounded sequence from the lower bound. en, there is a positive constant c 1 > 0 such that e proof of Lemma 2 implies that erefore, It comes from ϱ > (4 n /2) that we can find a positive constant c 2 such that for any j ∈ N, ‖x (j) ‖ ≤ c 2 . As a consequence of this, we know that the sequence x (j) j∈N is a bounded in the finite-dimensional space B. us, it has a convergent subsequence.
e Palais-Smale condition is verified.

Lemma 4. Suppose that x is a critical point of I(x) on B.
en, x is a critical point of I(x) on B.
e proof of Lemma 4 is similar to the proof of Lemma 2.2 in [12]. For simplicity, we omit its proof.
Let x is a critical point of I(x) on B, then x has a minimal period sM.
Proof. Suppose, for the sake of contradiction, that x exists a minimal period (sM/ϖ). In view of the condition (G 4 ), we have ϖ ≥ m s .
Similarly, x i can be written in the form of us, Discrete Dynamics in Nature and Society where y � (Δx 1 , Δx 2 , . . . , Δx sM ) * . It is obvious that erefore, by (G 5 ), at is, I(x) ≥ Θ m s which is a contradiction with the condition I(x) < Θ m s , and the proof of Lemma 5 is now complete.

Proofs of the Main Theorems
In this section, the proofs of eorems 1-3 and Corollaries 1 and 2 are given by using the critical point theory.
Proof. of eorem 1. In view of Lemma 2, I(x) is bounded from above in B. Set us, there is a sequence x (i) on B such that In addition, by the proof of Lemma 2, for any x ∈ B, erefore, lim ‖x‖⟶+∞ I(x) � − ∞. is means that x (i) is bounded. Consequently, x (i) has a convergent subsequence. We define it as x (i k ) . Denote On account of the continuity of For any x ∈ B, ‖x‖ ≤ δ 1 , from the condition (G 1 ), where y � (Δx 1 , Δx 2 , . . . , Δx sM ) * . It is easy to see that erefore, Take c � [((4 sin 2 (π/sM)) n /2) − ρ]δ 2 1 . us, for any Consequently, Furthermore, there are positive constants c > 0 and δ 1 > 0 such that for any x ∈zB δ 1 (0) ∩ B, For any x ∈ A, note that sM k�1 (Δ n x k− 1 , Δ n x k− 1 ) � 0, we have us, x ∉ A and the critical point x of I(x) corresponding to the critical value m is a nontrivial periodic solution of (1) with period sM.
Proof. of Corollary 1. Since s is a prime integer and s > 0, it is easy to see that m s � s. erefore, In virtue of eorem 1, the conclusion of Corollary 1 is obtained. e proof of Corollary 1 is fulfilled. Discrete Dynamics in Nature and Society