UNIFORM EXPONENTIAL STABILITY AND APPLICATIONS TO BOUNDED SOLUTIONS OF INTEGRO-DIFFERENTIAL EQUATIONS IN BANACH SPACES

Let X be a Banach space. Let A be the generator of an immediately norm continuous C-0-semigroup defined on X. We study the uniform exponential stability of solutions of the Volterra equation u'(t) = Au(t) + integral(t)(0) a(t - s)Au(s)ds, t >= 0, u(0) = x, where a is a suitable kernel and x is an element of X. Using a matrix operator, we obtain some spectral conditions on A that ensure the existence of constants C, omega > 0 such that parallel to u( t)parallel to <= Ce (-omega t) parallel to x parallel to, for each x is an element of D ( A) and all t >= 0. With these results, we prove the existence of a uniformly exponential stable resolvent family to an integro-differential equation with in finite delay. Finally, sufficient conditions are established for the existence and uniqueness of bounded mild solutions to this equation.