A construction method for the set of all solutions and Kneser's theorem for nonlinear operator equations
We consider a Volterra operator S : M->C = C([0,T],E), where E is a Banach space and M is a subset of C and show that the set L of all fixed points of S can be represented by an "Iteration" formula using certain mappings defined by S. (Theorem (a), (b)). Now we introduce an asymptotic compactness hypothesis and prove that, under this hypothesis, L is nonempty, compact and, if M is connected, L is also connected (Theorem (c), (d)). Using the Hausdorff measure of noncompactness on E we give a sufficient condition for this hypothesis (Corollary 1). Finally we apply the results to functional differential equations thus giving a simple example for our abstract theory that extends earlier results of Deimling and Szufla for ODE's in a Banach space. AMS-Classification: 47H17