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 and 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 . 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 several considerations are necessary. Finally, two examples are given.