*(English)*Zbl 1168.45005

The authors consider the functional integral equation

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