Differential Equations and Computational Simulations III

Electron. J. Diff. Eqns., Conf. 01, 1997,
pp. 129-136.

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

Susan D. Lauer

**Abstract:**

We seek multiple solutions to the n-th order nonlinear difference equation

*x(t)= (-1)*^{n-k} f(t,x(t)), t in *[0,T]*

satisfying the boundary conditions

*x(0) = x(1) = ... = x(k - 1) = x(T + k + 1) = ... = 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)
is continuous in *x*, as well as one of the following:

(B) *f* is sublinear at 0 and superlinear at infinity, or

(C) *f* is superlinear at 0 and sublinear at infinity.
Published November 12, 1998.

Mathematics Subject Classifications: 39A10, 34B15.

Key words and phrases: n-th order difference equation,
boundary value problem, superlinear, sublinear, fixed point theorem,
Green's function, discrete, nonlinear.

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Susan D. Lauer

Department of Mathematics,
Tuskegee University

Tuskegee, Alabama 36088 USA

E-mail address: lauersd@home.com

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