On integrable solutions of a nonlinear Volterra integral equation under Carathéodory conditions

In this paper we study the existence of solutions of a nonlinear Volterra integral equation in the space of Lebesgue integrable functions on an unbounded interval. Our existence result is obtained under the assumption that functions involved in the considered integral equation satisfy conditions of Carathéodory type. The main tool used in the investigations of the paper is the combination of the technique of measures of weak noncompactness with the classical Schauder fixed point principle. The obtained result generalizes several ones obtained earlier in many papers and monographs.