Bounded solutions to a class of semilinear integro-differential equations in Banach spaces

Abstract

Let A be the generator of an immediately norm continuous C 0 -semigroup defined on a Banach space X . We study the existence and uniqueness of bounded solutions for the semilinear integro-differential equation with infinite delay u ′ ( t ) = A u ( t ) + α ∫ − ∞ t e − β ( t − s ) A u ( s ) d s + f ( t , u ( t ) ) t ∈ R ; α , β ∈ R , for each f : R × X → X satisfying diverse Lipschitz type conditions. Sufficient conditions are established for the existence and uniqueness of an almost periodic, almost automorphic and asymptotically almost periodic solution, among other types of distinguished solutions. These results have significance in viscoelasticity theory. Finally, an example is presented to illustrate the feasibility and effectiveness of the results.