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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

$x\left(t\right)={f}_{1}\left(t,{\int }_{0}^{t}k\left(t,x\right){f}_{2}\left(s,x\left(s\right)\right)\phantom{\rule{0.166667em}{0ex}}ds\right),\phantom{\rule{1.em}{0ex}}t\in {ℝ}_{+}·\phantom{\rule{2.em}{0ex}}\left(*\right)$

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 $\left(*\right)$ in ${L}^{1}\left({ℝ}_{+}\right)$. 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 $\left(*\right)$ is relatively compact in ${L}_{1}\left({ℝ}_{+}\right)$ 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