Generalized Hammerstein Equations and Applications

Abstract

In this paper the authors study the Hammerstein generalized integral equation

u(t)=∫₀¹k(t,s) g(s) f(s,u(s),u′(s),…,u^{(m)}(s))ds,

where k:[0,1]²→R are kernel functions, m≥1, g:[0,1]→[0,∞), and f:[0,1]×R^{m+1}→[0,∞) is a L^{∞}-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 the first one to consider discontinuous nonlinearities with dependence on derivatives. Moreover, the kernels functions, and their partial derivatives with respect to the first variable, may be discontinuous and may change sign since they are only required to be positive on some subsets of [0,1] of positive measure. 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