We consider the integral equation h(u(t)) = f Ê I g(t, x) u(x) dx ¡ , with t ∈ [0, 1], and prove an existence theorem for bounded solutions where f is not assumed to be continuous.
We deal with the integral equation u(t) = f (Ê I g(t, z) u(z) dz), with t ∈ I = [0, 1], f : R n → R n and g : I ×I → [0, +∞[. We prove an existence theorem for solutions u ∈ L ∞ (I, R n) where the function f is not assumed to be continuous, extending a result previously obtained for the case n = 1.
We deal with the integral equation u(t) = f (t, Ê f is not assumed to be continuous in the second variable. Our result extends a result recently obtained for the special case where f does not depend explicitly on the first variable t ∈ I.
We deal with the implicit integral equation h(u(t)) = f (t , I g(t, z) u(z) dz) for a.a. t ∈ I, We prove an existence theorem for solutions u ∈ L s (I) where the contituity of f with respect to the second variable is not assumed.
We establish a fixed point theorem for a continuous function f : X → E, where E is a Banach space and X ⊆ E. Our result, which involves multivalued contractions, contains the classical Schauder fixed point theorem as a special case. An application is presented.