A one dimensional Hammerstein problem

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. If $L$ is invertible, 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 this paper we consider the one-dimensional problem $L(x)=lambda g(x)$ where $L(x)=ax$.

Keywords

• Hammerstein problem
• nonlinear eigenvalue problem.