EXISTENCE AND MULTIPLICITY OF SIGN-CHANGING SOLUTIONS FOR THE DISCRETE PERIODIC PROBLEMS WITH MINKOWSKI-CURVATURE OPERATOR

We are concerned with the discrete periodic problems with Minkowski-curvature operator

The existence and multiplicity of solutions for the prescribed mean curvature equations with Dirichlet boundary conditions and the discrete analogue have been widely discussed by various methods, see [9,10,12,15,25] and the references therein.
In [14] some general solvability results for (1.2) with Dirichlet boundary conditions were proved under the assumption that the function g is globally bounded. Yet, as all spacelike solutions of the Dirichlet problem are uniformly bounded by the quantity 1 2 d(Ω), with d(Ω) the diameter of Ω, one can always reduce to that situation by truncation, see Coelho et al. [9,10].
Nevertheless it should be observed that the situation differs substantially for periodic solutions of (1.2). In the continuous case, existence and multiplicity of periodic solutions for the equations with mean-curvature operator have been studied extensively, see Benevieri, Marcos do Ó and de Medeiros [3], Bereanu, Jebelean and Mawhin [4], Bereanu and Mawhin [5], Bereanu and Zamora [7], Bosecaggin and Feltrin [8].
However, there are many differences, even essential differences, between the difference equation and the corresponding differential equation. From the definition of Strogatz [28], chaos sensitivity depends on initial conditions. That is shown that nearby trajectories diverge exponentially. Continuous systems in a 2-dimensional phase space cannot experience such a divergence, hence chaotic behaviors can only be observed in deterministic continuous systems with a phase space of dimension 3, at least. On the other hand, in a discrete map it is well known that chaos occurs also in one-dimension. Therefore, discrete chaotic systems exhibit chaos whatever their dimension.
In addition, unlike the continuous case, the discrete linear eigenvalue problem has only a finite number of eigenvalues. Let as T is even, From [19] and [20], the eigenvalues of (1.3) are the following The eigenspace corresponding to λ 0 is span{1}, the eigenspace corresponding to λ j , and the eigenspace corresponding to λ N is Semilinear discrete boundary value problems was studied by Atici and Cabada [1], Atici, Cabada and Otero-Espinar [2], Henderson and Thompson [16], Ma and Ma [19,20]. For example, Atici and Cabada [1] concerned with proving the existence of positive solutions of a periodic boundary value problem for a discrete nonlinear equation where f is a continuous function on R. Under certain conditions of f , they obtained that there exists at least one positive solution of (1.4) for λ belonging to a given interval.
Existence and multiplicity of solutions for discrete problems with periodic boundary condition involving mean curvature operator have been studied by some authors, see Bereanu and Thompson [6], Mawhin [22,23] and the references therein. In particular, Bereanu and Thompson [6] considered the forced equation involving the discrete φ-Laplacian where φ can be the mean curvature operator and h = (h 2 , · · · , h n−1 ) satisfies n−1 ∑ k=2 h k = 0. They obtained the result as following.
Theorem A. Assume that there exist numbers α, β such that Then problem (1.5) has at least one solution with In fact, most people only obtain the existence of positive solutions, or even the existence of solutions. Very little is known about existence and multiplicity of signchanging periodic solutions for the problem (1.1). The likely reason is that the multiplicity of higher eigenvalues, λ k (k ∈ {1, 2, · · · , N − 1}), is even. However, the key of Rabinowitz global bifurcation theorem is dim M k = 1, see Rabinowitz [26,27].
In order to apply the bifurcation techniques, we have to work in and which are two subspaces of The above method of constructing invariant subspaces is motivated by Coron [11] and Marlin [21].
Therefore, the purpose of this paper is to use the bifurcation techniques to investigate the existence and multiplicity of sign-changing solutions for (1.1), which is a discrete periodic problem with mean curvature operators in the Minkowski space.
Due to g 0 = +∞, the global bifurcation techniques cannot be used directly in the case. Therefore, we referred to [18] and applied some properties of the superior limit of a certain infinity collection of connected sets. In addition, we also made some estimates on the properties of solutions of (1.1), and proved the existence of a priori bound for the solutions of (1.1).
Assume that g(t, ·) is odd for all t ∈T, and satisfies either (H1) there exists 0 < s 0 < ∞ such that Theorem 1.1. Assume that uniformly for t ∈T. Then By a component of solution set S we mean a continuum which is maximal with respect to inclusion ordering.

Preliminary results
is called a nodal point of y(t). The simple zero and the nodal point are called the simple generalized zero of y(t).

Definition 2.2 ( [29]
). Let X be a Banach space and {C n | n = 1, 2, · · · } be a family of subsets of X. Then the the superior limit D of C n is defined by . Let X be a Banach space, and let {C n } be a family of connected subsets of X. Assume that (a) there exist z n ∈ C n , n = 1, 2, · · · , and z * ∈ X, such that z n → z * ; Then there exists an unbounded component C in D and z * ∈ C. Proof. If u(0) = u(T ) = u , then u = 0. The conclusion is clearly correct. If Due to u is a sign-changing solution of (1.1), then |∆u(t)| < 1 for any t ∈T and Therefore, (ii) Assume that u(t 0 ) < 0. Similarly, Hence, the proof is completed.

From Lemma 2.5 and the fact
This is a contradiction. Obviously, (2.2) can be proved by the similar method. Therefore, the claim is proved to be true.
Next, we will show that max t∈T u(t) ≤ s 0 .
Assume that u(0) > s 0 . Then from (2.2), Due to u is even, then This is a contradiction with max t∈T u(t) = u(0).
This is a contradiction with (2.2). Similarly, min t∈T u(t) ≥ −s 0 . The proof is completed.
Then for any nonconstant solution u of (1.1), Proof. Claim. u is a solution with the change of sign inT. Suppose on the contrary that u(t) > 0, t ∈T. Then g(t, u) ≥ 0 and ∇(∆u(t)) ≤ 0 for all t ∈T. This suggests that Similarly, u(t) < 0, t ∈T can derive a contradiction. Therefore, the claim is proved to be true.
Assume that max . Then u(t 2 ) < 0 < u(t 1 ) and Obviously, The proof is completed.
Define the operator L : D(L) → H, where We introduce two subspaces of H which will play important roles in the proofs.

(3.2)
From Lemma 2.5, it can be verified that (1.1) is equivalent to Now let us consider the auxiliary family of the equations Let ζ ∈ C(T × R) be such that uniformly for t ∈T.

In the subspace E 1 of H
Let us consider (3.7) as a bifurcation problem from the trivial solution u ≡ 0.
Eq. (3.7) can be converted to the equivalent equation where G(t, i) is the Green function of −∇(∆u(t)) + λσu(t) = 0 with the periodic boundary condition. Define the operator H 1 : Obviously, H 1 is completely continuous. From (3.5) and (3.6), for any t ∈T, uniformly in λ of any bounded set.

In the subspace E 2 of H
Let us consider ) (3.8) as a bifurcation problem from the trivial solution v ≡ 0.