A two dimensional Hammerstein problem: The linear case

Abstract

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 $gcirc u$ 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.

Keywords

• Hammerstein problem
• nonlinear differential equation.