Abstract

We study the existence, uniqueness, and exponential decay of solutions for a semi-linear integrodifferential equation with a nonlocal initial condition u ′ (t) = Au(t) + Z t 0 F(t − s)Au(s)ds + f(t, u(t)), t ≥ 0, u(0) = Z ∞ 0 g(s)u(s)ds + u0, in a Banach space X, with A the generator of a strongly continuous semigroup. The nonlocal condition can be applied in physics with better effect than the “classical” Cauchy problem u(0) = u0 since more measurements at t ≥ 0 are allowed. The variation of constants formula for solutions via a resolvent operator and the iteration techniques are used in the study.