Abstract
By using topological degree theory and fixed point index theory, we consider a discrete fourth-order Neumann boundary value problem. We provide sufficient conditions for the existence of sign-changing solutions, positive solutions, and negative solutions.
Highlights
In this paper, we consider the existence of sign-changing solutions to the following discrete nonlinear fourth-order boundary value problem (BVP): u(t – ) – α u(t – ) + βu(t) = f t, u(t), t ∈ [, T]Z, ( . )u( ) = u(T) = u( ) = u(T – ) =, where denotes the forward difference operator defined by u(t) = u(t + ) – u(t), T > is an integer, f : [, T]Z × R → R is continuous, and α, β are real parameters and satisfy α ≥ β, and α – α – β > – sin π . (T – )Let a, b be two integers with a < b
1 Introduction In this paper, we consider the existence of sign-changing solutions to the following discrete nonlinear fourth-order boundary value problem (BVP):
We may think of BVP ( . ), ( . ) as a discrete analogue of the fourth-order boundary value problem u( )(t) – αu( )(t) + βu(t) = λf t, u(t), t ∈ (, ), ( . )
Summary
We consider the existence of sign-changing solutions to the following discrete nonlinear fourth-order boundary value problem (BVP): u(t – ) – α u(t – ) + βu(t) = f t, u(t) , t ∈ [ , T]Z,. ) has been studied by many authors using various approaches; for example, see [ , ] It seems that there is no similar result in the literature on the existence of sign-changing solutions, positive solutions, and negative solutions for BVP ). Motivated by [ ], our purpose is to apply some basic theorems in topological degree theory and fixed point index theory to establish some conditions for the nonlinear function f , which are able to guarantee the existence of sign-changing solutions, positive solutions, and negative solutions for the above discrete boundary value problem.
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