Singular perturbation for Volterra equations of convolution type

Abstract

Under the assumption that A is the generator of a twice integrated cosine family and K is a scalar valued kernel, we solve the singular perturbation problem ( E ϵ ) ϵ 2 u ϵ ″ ( t ) + u ϵ ′ ( t ) = Au ϵ ( t ) + ( K ∗ Au ϵ ) ( t ) + f ϵ ( t ) , ( t ⩾ 0 ) ( ϵ > 0 ) , when ϵ → 0 +, for the integrodifferential equation ( E) w ′ ( t ) = Aw ( t ) + ( K ∗ Aw ) ( t ) + f ( t ) , ( t ⩾ 0 ) , on a Banach space. If the kernel K verifies some regularity conditions, then we show that problem ( E ϵ ) has a unique solution u ϵ ( t) for each small ϵ > 0. Moreover u ϵ ( t) converges as ϵ → 0 +, to the unique solution u( t) of equation ( E).