Multiple solutions to a boundary value problem for an n-th order nonlinear difference equation

Abstract

We seek multiple solutions to the n-th order nonlinear difference equation $$Delta^n x(t)= (-1)^{n-k} f(t,x(t)),quad t in [0,T]$$ satisfying the boundary conditions $$x(0) = x(1) = cdots = x(k - 1) = x(T + k + 1) = cdots = x(T+ n) = 0,.$$ Guo's fixed point theorem is applied multiple times to an operator defined on annular regions in a cone. In addition, the hypotheses invoked to obtain multiple solutions to this problem involves the condition (A) $f:[0,T] imes {mathbb R}^+ o {mathbb R}^+$ is continuous in $x$, as well as one of the following: (B) $f$ is sublinear at $0$ and superlinear at $infty$, or (C) $f$ is superlinear at $0$ and sublinear at $infty$.

Keywords

• n-th order difference equation
• boundary value problem
• superlinear
• sublinear
• fixed point theorem
• Green's function
• discrete
• nonlinear.