Method of semidiscretization in time for quasilinearintegrodifferential equations

where 0< T <∞, A(u) is a linear operator in X for each u in an open subset W of Y , G is a nonlinear Volterra operator defined from C([0,T ];X) into itself, and the nonlinear map f is defined from [0,T ]×Y ×Y into Y . By a strong solution to (1.1) on [0,T ′], 0< T ′ ≤ T , we mean an absolutely continuous function u from [0,T ′] into X such that u(t)∈W for almost every t ∈ [0,T ′] and satisfies (1.1) a.e. on [0,T ′]. We use the ideas and techniques of Zeidler [10] and the method of semidiscretization in time to establish existence, uniqueness, and continuous dependence on initial data of strong solutions to (1.1) on [0,T ′] for some 0 < T ′ ≤ T . For the study of particular cases of (1.1) in which f(t,u,v) ≡ 0 and f(t,u,v) ≡ f(t,u), we refer to Crandall and Souganidis [2], Kato [6], and references cited therein. The crucial assumption in these works is that there exists an open subset W of Y such that for each w ∈ W , A(w) generates a C0-semigroup in X, A(·) is locally Lipschitz continuous on W from X into itself, f , defined from W into Y , is bounded and globally Lipschitz continuous from Y into itself, and there exists an isometric isomorphism S : Y →X such that