Abstract
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.
Highlights
∆4u(k − 2) − α∆2u(k − 1) + βu(k) = f (k, u(k)), k ∈ [2, T ]Z ∆u(0) = ∆u(T ) = ∆3u(0) = ∆3u(T − 1) = 0, 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
Proposition 2.1. [see 14] Let E be a real reflexive Banach space and E∗ be the dual space of 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 Gateaux differentiable and sequentially weakly lower semicontinuous functional that is bounded on subsets of E and whose Gateaux derivative admits a continuous inverse on E∗. ψ : E → R be a continuously Gateaux differentiable functional whose Gateaux derivative is compact such that φ(0) = ψ(0) = 0
Summary
Let us collect some theorems and lemmas that will be used below. [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 Gateaux differentiable and sequentially weakly lower semicontinuous functional that is bounded on subsets of E and whose Gateaux derivative admits a continuous inverse on E∗. Assume that there exist r > 0 and u ∈ E with r < φ(u) such that sup ψ(u)
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