Resolvent Operators and Weak Solutions of Integrodifferential Equations

Equations from heat conduction and viscoelasticity are written as x′(t) = A [ x(t) + ∫ t 0 F (t− s)x(s)ds ] + f(t), t ≥ 0, x(0) = x0, (1) in a Banach space X, where A is a closed and densely defined operator and F (t) is a bounded operator for t ≥ 0. We obtain the equivalence between the resolvent operator of Eq.(1) and solutions of d dt 〈x(t), v〉 = 〈x(t) + ∫ t 0 F (t− s)x(s)ds,A∗v〉+ 〈f(t), v〉, t ≥ 0, x(0) = x0. (2) where A∗ is the adjoint of A, v ∈ D(A∗), and 〈, 〉 denotes the pairing between X and its dual X∗. The result also enables us to unify many concepts about solutions of Eq.(1). AMS Subject Classification : 45D, 45J, 45N.