Existence of Three Solutions to the Discrete Fourth-order Boundary Value Problem with Four Parameters

In this work, we willproving the existence of three solutionsf or the discrete nonlinear fourth order boundary value problems with four parameters. The methods used here are based on the critical point theory.

The theory of nonlinear difference equations has been widely used to study the discrete models in many fields such as computer science, economics, neural network, ecology, cybernetics, etc.In recent years, a great deal of work has been done in the study of the existence and multiplicity of solutions for discrete boundary value problem.For the background and recent results, we refer the reader to the monographs [1][2][3][4][5][6][7][8][9][10][11][12][13] and the references therein.In this work , we will examine some applications of the variational methods to study the BVP (1).Depending on the values of the parameters α, β, λ and µ, BVP (1.1) covers many problems .If λ = 1 and µ = 0 the BVP (1) becomes has been recently investigated in [17], and existence results of sign-changing solutions are obtained using a topological degree theory and fixed point index theory.Also, If λ > 0 and µ = 0 this problem has been studed by M.Ousbika and Z.El allali in [18], using the critical point theory and the direct method of calculus variational.Here, we will wish the existence of three solutions for BVP (1) by using some basic theorems in critical point theory and variational methods under some conditions imposed on the nonlinear functions f and g .In this paper, we introduce in section 2 some preliminary theorems, the corresponding variational framwork of BVP (1) and we present some lemmas to prove our main results , in section 3 we obtain the existence of three solutions for BVP (1).

Preliminaries
Let us collect some theorems and lemmas that will be used below.One can refer to [14,19,20] for more details.
Proposition 2.1.[see 14] Let E be a real reflexive Banach space and E * be the dual space of E. Suppose that T : E → E * is a continuous operator and there exists ω > 0 such that Then T : E → E * is a homeomorphism between E and E * Theorem 2.2.[see ,20,theorem 1] Let E be a real reflexive Banach space, E * be the dual space of E , φ : E → R be a continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional that is bounded on subsets of E and whose Gâteaux derivative admits a continuous inverse on E * .ψ : E → R be a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact such that φ(0) = ψ(0) = 0.
Assume that there exist r > 0 and ū ∈ E with r < φ(ū) such that (i) and and assume that δ < η.Then , for each compact interval [a, b] ⊂ ( 1 η , 1 δ ) , there exists K > 0 with the following property : for each µ ∈ [a, b] and every C 1 functional Γ : E → R with compact derivative, there exists ζ > 0 such that, for each λ ∈ (0, ζ], the functional φ − λψ − µΓ has at least three distinct critical points in E whose norms are less than K.
We define the real vector space which is a (T-1)-dimentional Hilbert space , see [17] with the inner product The associated norm is defined by Lemma 2.5.For any u, v ∈ E, we have

2)
Proof: We first prove (2.1).For any u, v ∈ E, by the summation by parts formula and the fact that ∆v(0) = ∆v(T ) = 0 , it follows that in other hand , by the summation by parts formula and the fact that ∆ 3 u(0) = ∆ 3 u(T − 1) = 0, we have i.e.,(2.1)holds.Next, we show (2.2).Again, by the summation by parts formula and the fact that ∆u(T ) = 0 and v(1) = 0, we have This completes the proof of the lemma.✷ Discrete Fourth-order Boundary Value Problem With Four Parameters 181 We consider the functional as follows: ) Lemma 2.6.For any u ∈ E, we have Proof: in fact that ∆u(0) = 0, then by Hölder's inequality, we have Similarly, for any u ∈ E and k ∈ [2, T ] Z , note that in fact that u(1) = 0, then by Hölder's inequality, we have therefore, from (2.3) and by summation the parts inequalities, we deduce that then by (2.1), we deduce that the proof of lemma is completed.✷ Note that , for u ∈ E, and where The functional corresponding of BVP( 1) is given by With any fixed λ > 0 and µ > 0 , the functionals Φ,Ψ 1 , Ψ 2 and I is of class C 1 (E, R), and for u, v ∈ E, we have Discrete Fourth-order Boundary Value Problem With Four Parameters 183 The search of solutions of BVP (1) reduce to finding critical points u ∈ E of the functional I by the following, Lemma 2.7.If u ∈ E is a critical point of the functional I then u is a solution of BVP (1).
so from (2.7) − (2.10) and lemma 2.5 , we deduce that thus by the arbitrarieness of v ∈ E, we have then u ∈ E is a solution of BVP (1).This completes the proof.✷

Main results
Theorem 3.1.Assume that the following conditions holds : (H1) There exists c > 0 and d > 0 such that c < d √ T − 1.
Then for each compact interval there exist ζ > 0 such that , for each λ ∈ [a, b] there exist η > 0 such that , for each µ ∈ (0, η], the BVP(1) has at least three distinct solutions in E whose norms are less than ζ.
Proof: To prove the theorem 3.1, we will apply theorem 2.2 with ψ = Ψ 1 and Γ = Ψ 2 .Firsty, we show that the functionals Φ , Ψ 1 and Ψ 2 satisfy the regularity assumptions of theorem 2.2.By lemma 2.6, we prove that Φ is coercive, sequentielly weakly lower semicontinuous and is bounded on each bounded subset of E. From (2.3) and (2.8), we have Hence by proposition 2.1, (Φ ′ ) −1 : E * → E exist and is continuous.Secondly, we show that Ψ ′ 1 and Ψ ′ 2 are compacts.Suppose that u n → u ∈ E then since f and g are continuous and from (2.9), (2.10), we deduce that Ψ Taking into the fact that , for any k ∈ therefore, it follows from (H 4 ) that It is easy to verify that this imply that the assumption (i) of Theorem 2.2 is verified.
Theorem 2.3.[see,19,theorem2]Let E be a real reflexive Banach space with the norm .E , E * be the dual space of E .Let φ : E → R be a coercive, continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional that is bounded on subsets of E and whose Gâteaux derivative admits a continuous inverse on E * and ψ : E → R be a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact.Assume that φ has a strict local minimum