A NOTE ON NON-AUTONOMOUS IMPLICIT INTEGRAL EQUATIONS WITH DISCONTINUOUS RIGHT-HAND SIDE

• Published in: The Journal of integral equations and applications Vol. 19; no. 4; pp. 391 - 403
• Main Authors: ANELLO, GIOVANNI, CUBIOTTI, PAOLO
• Format: Journal Article
• Language: English
• Published: The Rocky Mountain Mathematics Consortium 01-12-2007; Rocky Mountain Mathematics Consortium
• Subjects: 45G10; 45P05; 47G10; Differential equations; discontinuity; Function discontinuity; Implicit integral equations; Local maximum; Local minimum; lower semicontinuous multi-functions; Mathematical discontinuity; Mathematical functions; Mathematical theorems; operator inclusions; selections; Separable spaces; Topological spaces
• ISSN: 0897-3962; 1938-2626
• DOI: 10.1216/jiea/1192628615

Let I := [0, 1], f : I × [0, σ] → R, g : I × I → [0, +∞[ and h : I× ] 0, +∞ [ → R. In this note we prove an existence result for solutions u ∈ Ls(I) of the integral equation $\mathrm{h}(\mathrm{t},\mathrm{u}(\mathrm{t}\left)\right)=\mathrm{f}(\mathrm{t},{\int }_{\mathrm{I}}\mathrm{g}(\mathrm{t},\mathrm{z}\left)\mathrm{u}\right(\mathrm{z} \left) \ \mathrm{d}\mathrm{z}\right) \ \mathrm{f}\mathrm{o}\mathrm{r} \ \mathrm{a}\mathrm{.}\mathrm{a}\mathrm{.} \ \mathrm{t}\in \mathrm{I}$ for a.a. t ∈ I where, in particular, the continuity of f with respect to the second variable is not assumed. Our result is a partial extension of a previous result of the same authors [1], where the function h was not allowed to depend explicitly on t.