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

• Main Author: Susan D. Lauer
• Format: Article
• Language: English
• Published: Texas State University 1998-11-01
• Series: Electronic Journal of Differential Equations
• Subjects: n-th order difference equation; boundary value problem; superlinear; sublinear; fixed point theorem; Green's function; discrete; nonlinear.
• Online Access: http://ejde.math.txstate.edu/conf-proc/01/l1/abstr.html

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 $infty$.