Generalized Hammerstein Equations and Applications

Abstract

In this paper the authors study the Hammerstein generalized integral equation

$$\begin{aligned} u(t)=\int _{0}^{1}k(t,s)\text { }g(s)\text { }f(s,u(s),u^{\prime }(s),\dots ,u^{(m)}(s))\,ds, \end{aligned}$$

where $k:[0,1]^{2}\rightarrow {\mathbb {R}}$ are kernel functions, $m\ge 1$, $g:[0,1] \rightarrow [0,\infty )$, and $f:[0,1]\times {\mathbb {R}}^{m+1} \rightarrow [0,\infty )$ is a $L^{\infty }-$Carathéodory function. The existence of solutions of integral equations has been studied in concrete and abstract cases, by different methods and techniques. However, in the existing literature, the nonlinearity depends only on the unknown function. This paper is one of a very few to consider equations having discontinuous nonlinearities that depend on the derivatives of the unknown function and having discontinuous kernels functions that have discontinuities in the partial derivatives with respect to their first variable. Our approach is based on the Krasnosel’skiĭ–Guo compression/expansion theorem on cones and it can be applied to boundary value problems of arbitrary order $n>m$. The last two sections of the paper contain an application to a third order nonlinear boundary value problem and a concrete example.