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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 $$x(t)= f_1\left(t, \int^t_0k(t,x)f_2(s,x(s))\,ds\right),\quad t\in\bbfR_+.\tag{*}$$ 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 $L^1 (\bbfR_+)$. The main tool of the proof is the measure of weak noncompactnes developed by {\it J. Banas} and {\it 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 $L_1(\bbfR_+)$ several considerations are necessary. Finally, two examples are given.

##### MSC:
 45G10 Nonsingular nonlinear integral equations 47H10 Fixed-point theorems for nonlinear operators on topological linear spaces
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##### References:
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