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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\mathbb{R}_+.\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 \(L^1 (\mathbb{R}_+)\). 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 \(L_1(\mathbb{R}_+)\) several considerations are necessary. Finally, two examples are given.

MSC:

45G10 Other nonlinear integral equations
47H10 Fixed-point theorems

Citations:

Zbl 0699.45002
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References:

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