Existence of integrable solutions of an integral equation of Hammerstein type on an unbounded interval (Q1762381)

The authors investigate the existence of solutions of the following perturbed Hammerstein integral equation \[ x(t)= g(t,x(t))+ \int^\infty_0 k(t,s) f(s,x(s))\,ds\tag{1} \] for \(t\in\mathbb{R}_+= [0,\infty)\). Solutions of (1) are sought in the Lebesgue space \(L^1(\mathbb{R}_+)\) consisting of all real functions which are measurable and integrable over \(\mathbb{R}_+\). The assumptions imposed on the functions involved in (1) are expressed in terms of Carathéodory conditions and the requirement that \(g\) is a so-called separate contraction. Those assumptions enable the authors to apply the technique of measures of weak noncompactness and a fixed point theorem for the sum of two operators which was recently proved by the second author [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71, No. 9, A, 4131--4136 (2009; Zbl 1203.45004)].