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On existence of integrable solutions of a functional integral equation under Carathéodory conditions. (English) Zbl 1168.45005

The authors consider the functional integral equation

x(t)=f 1 t, 0 t k(t,x)f 2 (s,x(s))ds,t + ·(*)

which generalizes several equations arising in mechanics, physics, engineering etc. and have been studied in the literature. The functions f 1 ,f 2 and k are supposed to satisfy Carathéodory conditions and some other technical assumptions. The authors prove by using the Schauder fixed point principle the existence of at least one solution of (*) in L 1 ( + ). The main tool of the proof is the measure of weak noncompactnes developed by J. Banas and Z. Knap [J. Math. Anal. Appl. 146, No. 2, 353–362 (1990; Zbl 0699.45002)]. To prove that the image of the operator associated to (*) is relatively compact in L 1 ( + ) several considerations are necessary. Finally, two examples are given.

45G10Nonsingular nonlinear integral equations
47H10Fixed point theorems for nonlinear operators on topological linear spaces