We consider the integral equation $h(u(t))=f\big(\int_I g(t,x)\,u(x)\,dx\big)$, with $t\in[0,1]$, and prove an existence theorem for bounded solutions where $f$ is not assumed to be continuous.
@article{118921, author = {Filippo Cammaroto and Paolo Cubiotti}, title = {Implicit integral equations with discontinuous right-hand side}, journal = {Commentationes Mathematicae Universitatis Carolinae}, volume = {38}, year = {1997}, pages = {241-246}, zbl = {0886.47031}, mrnumber = {1455490}, language = {en}, url = {http://dml.mathdoc.fr/item/118921} }
Cammaroto, Filippo; Cubiotti, Paolo. Implicit integral equations with discontinuous right-hand side. Commentationes Mathematicae Universitatis Carolinae, Tome 38 (1997) pp. 241-246. http://gdmltest.u-ga.fr/item/118921/
Differential Inclusions, Springer-Verlag, Berlin, 1984. | MR 0755330 | Zbl 0538.34007
Set-Valued Analysis, Birkhäuser, Boston, 1990. | MR 1048347 | Zbl 1168.49014
Integrable solutions of a functional-integral equation, Rev. Mat. Univ. Complut. Madrid 2 (1989), 31-38. (1989) | MR 1012104 | Zbl 0679.45003
About the existence of integrable solutions of a functional-integral equation, Rev. Mat. Univ. Complut. Madrid 4 (1991), 65-69. (1991) | MR 1142550 | Zbl 0746.45004
Integrable solutions of a functional-integral equation, J. Integral Equations Appl. 4 (1992), 89-94. (1992) | MR 1160090 | Zbl 0755.45005
Nonnegative solutions of nonlinear integral equations, Comment. Math. Univ. Carolinae 36 (1995), 615-627. (1995) | MR 1378685
Real and Abstract Analysis, Springer-Verlag, Berlin, 1965. | MR 0367121 | Zbl 0307.28001
Functional Analysis in Normed Spaces, Pergamon Press, Oxford, 1964. | MR 0213845 | Zbl 0127.06104
Theory of Correspondences, John Wiley and Sons, New York, 1984. | MR 0752692 | Zbl 0556.28012
Real and Functional Analysis, Springer-Verlag, New York, 1993. | MR 1216137 | Zbl 0831.46001
An existence theorem for inclusions of the type $\Psi(u)(t)\in F(t,\Phi(u)(t))$ and application to a multivalued boundary value problem, Appl. Anal. 38 (1990), 259-270. (1990) | MR 1116184
Sur la semi-continuité inférieure de certaines multifonctions, C.R. Acad. Sci. Paris, Série I 294 (1982), 265-267. (1982) | MR 0653748 | Zbl 0483.54010
Riemann-measurable selections, Set-Valued Anal. 2 (1994), 481-485. (1994) | MR 1304050 | Zbl 0851.54021
On Lusin's condition for the inverse function, Rend. Circ. Mat. Palermo 33 (1984), 331-335. (1984) | MR 0779937 | Zbl 0562.26002