An existence theorem for solutions of an integro-differential equation in Banach spaces

Abstract

Abstract. The paper contains an existence theorem for local solutions of an initial value problem for a nonlinear integro-differential equation in Banach spaces. The assumptions and proofs are expressed in terms of measures of noncompactness.Consider the following Cauchy problemin a Banach space E, where m ≥ 1 is a natural number. Throughout this paper we shall assume that D = [0, a] is a compact interval in R, B = {x ∈ E : x ≤ b}, f : D × B × E → E is a continuous function, and g : D 2 × B → E is a bounded continuous function. Moreover, we suppose that f (t, x, z) ≤ M for t ∈ D, x ∈ B, z ∈ W , whereDenote by α the Kuratowski measure of noncompactness in E (cf.[1]). Main resultIn this section we shall prove an existence theorem for local solutions of the above initial value problem for the nonlinear integro-differential equation in Banach spaces.2000 Mathematics Subject Classification: 47G20, 45J05.