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.
a r t i c l e i n f o a b s t r a c t In this paper we are interested in monotone versions of partitionability of topological spaces and weak versions thereof. We identify several classes of spaces with these properties by constructing trees of open sets with various properties.