Nonexistence and existence results for a class of fourth-order difference mixed boundary value problems

In this paper, a class of fourth-order nonlinear difference equations are considered. By making use of the critical point method, we establish various sets of sufficient conditions for the nonexistence and existence of solutions for mixed boundary value problems and give some new results. Our results successfully complement the existing results in the literature.

The present paper considers the fourth-order nonlinear difference equation 2 p n−1 2 u n−2 − q n ( u n−1 ) δ + r n u δ n = f (n, u n ), n ∈ Z(1, k), (1.1) with boundary value conditions where p n is nonzero and real valued for each n ∈ Z(0, k + 1), {q n } n∈Z (1,k+1) and {r n } n∈Z (1,k) are real sequences, δ is the ratio of odd positive integers, f ∈ C(R 2 , R). We may think of (1.1) with (1.2) as being a discrete analogue of the following fourth-order nonlinear differential equation   (4) (t) = f t, u(t) , t ∈ R, (1.5) which is used to describe the bending of an elastic beam; see, for example, [5,16,20,22,25,39] and the references therein. Owing to its importance in physics, many methods are applied to study fourth-order boundary value problems by many authors.
In recent years, the study of boundary value problems for differential equations develops at relatively rapid rate. By using various methods and techniques, such as fixed point theory, topological degree theory, coincidence degree theory, a series of existence results of nontrivial solutions for differential equations have been obtained in literatures, we refer to [2-4, 7, 21, 37]. And critical point theory is also an important tool to deal with problems on differential equations [11,14,15,28,33,43]. Only since 2003, critical point theory has been employed to establish sufficient conditions on the existence of periodic solutions of difference equations. By using the critical point theory, Guo and Yu [17][18][19] and Shi et al. [35] have successfully proved the existence of periodic solutions of second-order nonlinear difference equations. We also refer to [40,41] for the discrete boundary value problems. Compared to firstorder or second-order difference equations, the study of higher-order equations, and in particular, fourth-order equations, has received considerably less attention (see, for example, [9,10,13,31,32,36,38] and the references contained therein). Yan, Liu [38] in 1997 and Thandapani, Arockiasamy [36] in 2001 studied the following fourth-order difference equation of form, 2 p n 2 u n + f (n, u n ) = 0, n∈ Z. (1.6) The authors obtain criteria for the oscillation and nonoscillation of solutions for equation (1.6). In 2005, Cai, Yu and Guo [6] have obtained some criteria for the existence of periodic solutions of the fourth-order difference equation In 1995, Peterson and Ridenhour considered the disconjugacy of equation (1.7) when p n ≡ 1 and f (n, u n ) = q n u n (see [31]). The boundary value problem (BVP) for determining the existence of solutions of difference equations has been a very active area of research in the last twenty years, and for surveys of recent results, we refer the reader to the monographs by Agarwal et al. [1,12,24,29,34]. We use the critical point theory to give some sufficient conditions for the nonexistence and existence of solutions for the BVP (1.1) with (1.2). We shall study the suplinear and sublinear cases. The main idea in this paper is to transfer the existence of the BVP (1.1) with (1.2) into the existence of the critical points of some functional. The proof is based on the notable Mountain Pass lemma in combination with variational technique. The purpose of this paper is two-folded. On one hand, we shall further demonstrate the powerfulness of critical point theory in the study of solutions for boundary value problems of difference equations. On the other hand, we shall complement existing ones. The motivation for the present work stems from the recent paper in [11]. Letp = max p n : n ∈ Z(1, k + 1) , p= min p n : n ∈ Z(1, k + 1) , q = max q n : n ∈ Z(2, k + 1) , q= min q n : n ∈ Z(2, k + 1) , r = max r n : n ∈ Z(1, k) , r= min r n : n ∈ Z(1, k) .
Our main results are as follows.
implies that for each n ∈ Z(1, k) there exist constants a 3 > 0 and a 4 > 0 such that Theorem 1.4 Suppose that (p), (q), (r), (F 1 ) and the following assumption are satisfied: The remainder of this paper is organized as follows. First, in Sect. 2, we shall establish the variational framework for the BVP (1.1) with (1.2) and transfer the problem of the existence of the BVP (1.1) with (1.2) into that of the existence of critical points of the corresponding functional. Some related fundamental results will also be recalled. Then, in Sect. 3, we shall complete the proof of the results by using the critical point method. Finally, in Sect. 4, we shall give three examples to illustrate the main results.
About the basic knowledge for variational methods, we refer the reader to [28,30,33,43].

Variational structure and some lemmas
In order to apply the critical point theory, we shall establish the corresponding variational framework for the BVP (1.1) with (1.2) and give some lemmas which will be of fundamental importance in proving our main results. First, we state some basic notations.
Let R k be the real Euclidean space with dimension k. Define the inner product on R k as follows: by which the norm · can be induced by On the other hand, we define the norm · r on R k as follows: for all u ∈ R k and r > 1.
Thus, u is a critical point of J on R k if and only if 2 p n−1 2 u n−2 − q n ( u n−1 ) δ + r n u δ n = f (n, u n ), ∀n ∈ Z(1, k).
We reduce the existence of the BVP (1.1) with (1.2) to the existence of critical points of J on R k . That is, the functional J is just the variational framework of the BVP (1.1) with (1.2). Let P and Q be the k × k matrices defined by Clearly, P and Q are positive definite. Let λ 1 , λ 2 , . . . , λ k be the eigenvalues of P ,λ 1 ,λ 2 , . . . ,λ k be the eigenvalues of Q. Applying matrix theory, we know λ j > 0,λ j > 0, j = 1, 2, . . . , k. Without loss of generality, we may assume that Let E be a real Banach space, J ∈ C 1 (E, R), i.e., J is a continuously Fréchetdifferentiable functional defined on E. J is said to satisfy the Palais-Smale condition (P.S. condition for short) if any sequence {u (l) } ⊂ E for which {J (u (l) )} is bounded and J (u (l) ) → 0(l → ∞) possesses a convergent subsequence in E.
Let B ρ denote the open ball in E about 0 of radius ρ and let ∂B ρ denote its boundary. +r c δ+1 Since β > max{2, δ + 1}, there exists a constant M 3 > 0 such that Therefore, {u (l) } is bounded on R k . As a consequence, {u (l) } possesses a convergence subsequence in R k . Thus the P.S. condition is verified.

Proof of the main results
In this section, we shall prove our main results by using the critical point theory.
Proof of Theorem 1.1 By (F 2 ), for any u = (u 1 , u 2 , . . . , u k ) * ∈ R k , we have The above inequality means that −J (u) is coercive. By the continuity of J (u), J attains its maximum at some point, and we denote itǔ, that is, Clearly,ǔ is a critical point of the functional J . This completes the proof of Theorem 1.1.
For any u = (u 1 , u 2 , . . . , u k ) * ∈ R k and u ≤ ρ, we have |u n | ≤ ρ, n ∈ Z(1, k). For any n ∈ Z(1, k), At the same time, we have also proved that there exist constants a > 0 and ρ > 0 such that J | ∂B ρ ≥ a. That is to say, J satisfies the condition (J 1 ) of the Mountain Pass lemma. For our setting, clearly J (0) = 0. In order to exploit the Mountain Pass lemma in critical point theory, we need to verify other conditions of the Mountain Pass lemma. By Lemma 2.2, J satisfies the P.S. condition. So it suffices to verify the condition (J 2 ).
From the proof of the P.S. condition, we know Since β > max{2, δ + 1}, we can chooseū large enough to ensure that J (ū) < 0. and Letũ ∈ R k be a critical point associated to the critical value c of J , i.e., J (ũ) = c. Similar to the proof of the P.S. condition, we know that there existsû ∈ R k such that Clearly,û = 0. Ifũ =û, then the conclusion of Theorem 1.2 holds. Otherwise, u =û. Then c = J (ũ) = c max = max s∈ [0,1] J (h(s)). That is, as u → +∞. By the continuity of J , we know from the above inequality that there exist lower bounds of values of the functional. And this means that J attains its minimal value at some point which is just the critical point of J with the finite norm.
Proof of Theorem 1.4 Assume, for the sake of contradiction, that the BVP (1.1) with (1.2) has a nontrivial solution. Then J has a nonzero critical point u . Since This contradicts (3.1) and hence the proof is complete.
It is easy to verify all the assumptions of Theorem 1.2 are satisfied and then the BVP    It is easy to verify all the assumptions of Theorem 1.2 are satisfied and then the BVP (4.4) with (1.2) possesses two nontrivial solutions u n = sin( π 2 n) and u n = cos( π 2 n).
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