## 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

Zbl 0699.45002
Full Text:

### References:

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