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Zbl 1168.45005
Banaś, Józef; Chlebowicz, Agnieszka
On existence of integrable solutions of a functional integral equation under Carathéodory conditions. (English)
[J] Nonlinear Anal., Theory Methods Appl. 70, No. 9, A, 3172-3179 (2009). ISSN 0362-546X

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.
[Marianne Reichert (Frankfurt am Main)]

MSC 2000:: *45G10 Nonsingular nonlinear integral equations
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces

Keywords: Lebesgue integrable function; Carathéodory conditions; functional integral equation; superposition operator; Schauder fixed point principle

Citations: Zbl 0699.45002

Cited in: Zbl 1203.45004

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