Electronic Journal of Differential Equations,
Conference 01 (1998), pp. 129-136. 

Title: Multiple Solutions to a Boundary Value Problem for an n-th Order Nonlinear Difference Equation

Authors: Susan D. Lauer (Tuskegee Univ., Tuskegee, Alabama, USA)
 
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) $f:[0,T] \times {\mathbb R}^+ \to
{\mathbb R}^+$ 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 $\infty$.



Published November 12, 1998.
Math Subject Classifications: 39A10, 34B15.
Key Words: n-th order difference equation;
boundary value problem; superlinear; sublinear; fixed point theorem;
Green's function; discrete; nonlinear.