16th Conference on Applied Mathematics, Univ. of Central Oklahoma,
Electron. J. Diff. Eqns., Conf. 07, 2001, pp. 71-88.

A two dimensional Hammerstein problem: The linear case

Jun Hua & James L. Moseley

Nonlinear equations of the form $L[u]=\lambda g(u)$ where $L$ is a linear operator on a function space and $g$ maps $u$ to the composition function $g\circ u$ arise in the theory of spontaneous combustion. We assume $L$ is invertible so that such an equation can be written as a Hammerstein equation, $u=B[u]$ where $B[u]=\lambda L^{-1}[g(u)]$. To investigate the importance of the growth rate of $g$ and the sign and magnitude of $\lambda $ on the number of solutions of such problems, in a previous paper we considered the one-dimensional problem $L(x)=\lambda g(x)$ where $L(x)=ax$. This paper extends these results to two dimensions for the linear case.

Published July 20, 2001.
Subject lassfications: 47H30.
Key words: Hammerstein problem, nonlinear differential equation.

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Jun Hua
West Virginia University
Morgantown, West Virginia 26506-6310 USA
James L. Moseley
West Virginia University
Morgantown, West Virginia 26506-6310 USA
e-mail: moseley@math.wvu.edu
Telephone: 304-293-2011

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