Existence and boundedness of solutions of abstract nonlinear integrodifferential equations of nonconvolution type

Abstract

Existence and boundedness theorems are given for solutions of nonlinear integrodifferential equations of type d dt u(t) + Bu(t) + ∝ 0 t a(t, s) Au(s) ds ϵ f(t) (t > 0) , (1.1) u(0) = u 0, Here A and B are nonlinear, possibly multivalued, operators on a Banach space W and a Hilbert space H, where W ⊆ H. The function f (0, ∞) → H and the kernel a( t, s): R × R → R are known functions. The results of this paper extend the results of Crandall, Londen, and Nohel [4] for equation (1.1). They assumed the kernel to be of the type a( t, s) = a( t − s). We relax this assumption and obtain similar results. Examples of kernels satisfying the conditions we require are given in section 4.