Existence and multiple solutions to a discrete fourth order boundary value problem

In this article, we study a discrete fourth order boundary value problem. By making use of variational methods and critical point theory, we obtain some criteria for the existence and multiple solutions. Moreover, two examples are included to illustrate the applicability of the main results.


Introduction and statement of the main results
In this article, we are interested in the existence and multiple solutions to the discrete fourth order nonlinear equation 4 u n-2 -(r n-1 u n-1 ) = f (n, u n ), n ∈ Z [1, k], (1.1) with boundary value conditions i u -1 = i u k-1 , i = 0, 1, 2, 3, (1.2) where j u n = ( j-1 u n ) (j = 2, 3, 4), 0 u n = u n , u n = u n+1u n , f (s, u) ∈ C(R 2 , R), r n > 0 is real-valued for each n ∈ Z[0, k], r 0 = r k , k ≥ 1 is an integer. Here, Z denotes the sets of integers, R denotes the sets of real numbers, N denotes the sets of natural numbers. Given Let u* denote the transpose of a vector u. Boundary value problem (1.1) with (1.2) can be regarded as being a discrete analogue of the fourth order differential equation u (4) (s) = f s, u(s) , s ∈ R, (1.5) which is used to describe the stationary states of the deflection of an elastic beam [29]. Differential equations similar to (1.3) and special cases of it have been studied using a number of different methods in the literature, we refer the reader to papers [1, 2, 11-14, 24, 25] and the references contained therein. Difference equations [1-10, 15-20, 22, 26-28, 30, 31] appear in numerous settings and forms, both as a fundamental tool in the discrete analogue of a differential equation and as a useful model for several economical and population problems.
If f (n, u n ) = q n u n , Peterson and Ridenhour [22] considered the fourth order difference equation 4 u n-2 + q n u n = 0, n ∈ [a + 2, a + 3, . . . , b + 2], (1.6) and gave some conditions on q n that ensure (1.6) is (2,2)-disconjugate on [a, b + 4] utilizing an appropriately defined quadratic form. Making use of the symmetric mountain pass lemma, Chen and Tang [5] established some existence criteria to guarantee the fourth order difference system 4 u n-2 + q n u n = f (n, u n+1 , u n , u n-1 ), n ∈ Z (1.7) has infinitely many homoclinic orbits.
In [16], the existence, multiplicity, and nonexistence results of nontrivial solutions for discrete nonlinear fourth order boundary value problems 4 u n-2 + η 2 u n-1ξ u n = λf (n, u n ), n ∈ Z[a + 1, b + 1], with u a = 2 u a-1 = 0, u b+2 = 2 u b+1 = 0, are obtained. The methods used here are based on the critical point theory and monotone operator theory. Positive solutions of the following fourth order nonlinear difference equations with a deviating argument a n b n c n ( u n ) γ β α + d n u λ n+τ = 0 (1.8) are investigated. Došlá, Krejčová, and Marini [8] introduced for (1.8) the notions of a minimal solution and a maximal solution, and gave necessary and sufficient conditions for their existence. Some relationships with nonoscillatory solutions, which have a different growth at infinity, were presented as well.
Graef, Kong, and Wang [10] studied the discrete fourth order periodic boundary value problem with a parameter ⎧ ⎨ ⎩ 4 u n-2 -(p n-1 u n-1 ) + q n u n = λf (n, u n ), n ∈ Z(1, N), By using variational methods and the mountain pass lemma, sufficient conditions are found under which the above problem has at least two nontrivial solutions.
In 2015, Liu, Zhang, and Shi [19] considered the following fourth order nonlinear difference equation: with boundary value conditions Using the critical point theory, the authors established various sets of sufficient conditions for the existence and nonexistence of solutions for the Dirichlet boundary value problem and gave some new results. By using the invariant set of descending flow and variational method, Long and Chen [20] in 2018 established the existence of multiple solutions to a class of second order discrete Neumann boundary value problem ⎧ ⎨ ⎩ -(p n-1 u n-1 ) + q n u n = kf (n, u n ), n ∈ Z(1, N), The solutions included sign-changing solutions, positive solutions, and negative solutions. Moreover, an example was given to illustrate our results.
In the last few years, variational methods and critical point theory have been used to study the existence and multiple solutions of discrete boundary value problems. In this article, we utilize this approach to obtain some sufficient conditions for the existence and multiple solutions to the boundary value problem (BVP for short) (1.1) with (1.2). What is more, two examples are included to illustrate the applicability of the main results.
Throughout this article, assume that there is a function F(s, u) such that Our main results are the following theorems.
(F 2 ) There exist two constants a 2 ∈ ( λ max 2 , +∞) and a 3 > 0 such that where λ min and λ max are constants which can be referred to (2.4) and (2.5). Then BVP (1.1) with (1.2) admits at least three solutions which are a trivial solution and two nontrivial solutions. Remark 1.1 In [10], the authors considered the discrete fourth order periodic boundary value problem with a parameter (1.9) The following hypotheses are satisfied in [10]: (H1) p n > 0 for n ∈ Z(0, N + 1) and q n > 0 for n ∈ Z(0, N); Note that (F 2 ) of Theorem 1.1 does not satisfy (H2). At least two nontrivial solutions of (1.9) are obtained by the mountain pass lemma in [10]. However, in our paper, we employ a linking theorem to obtain at least two nontrivial solutions. Furthermore, our conditions on the nonlinear term are weaker than [10]. Theorem 1.2 Assume that the function F(s, u) ≥ 0 satisfies the following assumptions: There exist three constants a 4 > 0, γ > 2, and a 5 > 0 such that Then BVP (1.1) with (1.2) admits at least three solutions which are a trivial solution and two nontrivial solutions. Theorem 1.3 Assume that the function F(s, u) ≥ 0, (F 1 ) and (F 2 ) and the following assumptions are satisfied: Then BVP (1.1) with (1.2) admits at least q distinct pairs of nontrivial solutions, where q is the dimension of Q which can be referred to (2.6).
Remark 1.2 In [9], the authors considered the fourth order nonlinear difference equation The following hypotheses are satisfied in [9]: (H 1 ) p n ≥ 0 and q n ≥ 0 for n ∈ Z(1, N) and there exists η with η < q such that Note that (F 1 ) of Theorem 1.3 does not satisfy (H 3 ). Furthermore, our conditions on the nonlinear term are weaker than [9].
If f (n, u n ) = τ n ψ(u n ), (1.1) reduces to the following fourth order nonlinear equation: where ψ ∈ C(R, R), τ n > 0 is real-valued for each n ∈ Z [1, k]. Therefore, we can easily obtain the following results.
Then BVP (1.10) with (1.2) admits at least q distinct pairs of nontrivial solutions, where q is the dimension of Q which can be referred to (2.6).

Variational framework
In this section, we shall establish the corresponding variational framework for BVP (1.1) with (1.2) which will be of fundamental importance in proving our main results.
In order to apply the critical point theory, we define a k-dimensional Hilbert space U by , ∀u ∈ U.
Remark 2.1 It is obvious that As a matter of fact, U is isomorphic to R k . Throughout this article, when we say u = (u 1 , u 2 , . . . , u k ) ∈ R k , we always imply that u can be extended to a vector in U so that (2.1) holds.

Some basic lemmas
Assume that U is a real Banach space and J ∈ C 1 (U, R). As usual, J is said to satisfy the Palais-Smale condition if every sequence {u (j) } ⊂ U such that {J(u (j) )} is bounded and J (u (j) ) → 0 (j → ∞) has a convergent subsequence. The sequence {u (j) } is called a Palais-Smale sequence. Let U be a real Banach space. Define the symbol B ρ (u) as the open ball in U about u of radius ρ, ∂B ρ (u) as its boundary, andB ρ (u) as its closure.   [21]) Let U be a real Banach space, J ∈ C 1 (U, R), with J being even, bounded from below and satisfying the Palais-Smale condition. Assume J(0) = 0, there is a set Γ ⊂ U such that Γ is homeomorphic to S k-1 (k -1 dimension unit sphere) by an odd map, and sup Γ J < 0. Then J has at least k distinct pairs of nonzero critical points. Proof Let {u (j) } j∈N ⊂ U be such that {J(u (j) )} j∈N is bounded and J (u (j) ) → 0 as j → ∞. Then there is a constant A > 0 such that

From (F 2 ) and (2.3), for any {u
Then It comes from a 2 ∈ ( λ max 2 , +∞) that we can find a constant B > 0 such that, for any j ∈ N, u (j) ≤ B. Thus, we know that the sequence {u (j) } j∈N is bounded in the k dimensional space U. Therefore, the Palais-Smale condition holds.
It comes from Lemma 3.3 that J(u) is bounded from above in U. Let Therefore, there exists a sequence {u (j) } on U such that What is more, from the proof of Lemma 3.3, we have Due to the continuity of J(u) in u, there must be a pointū ∈ U, J(ū) =J. Clearly,ū ∈ U is a critical point of J(u). From (F 1 ), for any u ∈ Q, u ≤ δ 1 , we have We have Thus, there are constants c > 0 and δ 1 > 0 such that J| ∂B δ 1 (0)∩Q ≥ c. Assumption (J 1 ) of the linking theorem is satisfied. In view of Mu = 0, for all u ∈ P, we have Hence,ū / ∈ P and the critical pointū of J(u) corresponding to the critical valueJ is a nontrivial solution of BVP (1.1) with (1.2).
Remark 4.1 Note that (F 3 ) implies (F 1 ). Similar to the above argument, we can also prove Theorem 1.2. For simplicity, we omit its proof.
Proof of Theorem 1.3 Obviously J ∈ C 1 (U, R), J is even, and J(0) = 0. From Lemma 3.3, J satisfies the Palais-Smale condition. By the proof of Theorem 1.1, we have that J is bounded from below. On account of Lemma 3.2, it is sufficient to find a set Γ and an odd map such that Γ is homeomorphic to S q-1 by an odd map. Choose Clearly, Γ is homeomorphic to S q-1 by an odd map. It is comes from (4.2) that sup Γ (-J) < 0.

Examples
Firstly, our example illustrates Theorem 1.1. As an example of Theorem 1.3, we have the following.