EXISTENCE OF ABSTRACT SOLUTIONS OF INTEGRO-DIFFERENTIAL OPERATOR EQUATIONS

A. E. Hamza and G. Muraz

Abstract

           We consider the integro-differential operator equations having the form

n ∑ i = 1

α i u( i )(t)-αAu(t)+βμ*u(t) = f(t),      t∈ R

where the free term  f  belongs to a closed subspace M of  L^∞(R,X), A is the generator of a C₀-semigroup of operators defined on a Banach space X, μ is a bounded Borel measure on R and α, β, α _i ∈ C, i =1,2,.., n. Certain conditions will be imposed to guarantee the existence of solutions in the class M.