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Existence of solutions for an integrodifferential equation with nonlocal condition in Banach spaces. (English) Zbl 0862.45016
The purpose of this paper is to prove the existence and uniqueness of a local or global solution of the integrodifferential equation \[ {du(t)\over dt}+Au(t)= f\Biggl(u,u(t),\int^t_0k(t,s,u(s))ds\Biggr) \] with a nonlocal “initial” condition of the form \[ u(0)+g(t_1,t_2,\dots,t_p,u)= u_0. \] Here \(A\) generates a bounded analytic semigroup \(X\) in a Banach space \(Z\). The nonlinear function \(f\) is Hölder continuous in its first argument and Lipschitz continuous in the other arguments from \(\mathbb{R}\times Z_\alpha\times Z_\alpha\) to \(Z\) where \(Z_\alpha\) is the domain of \(A^\alpha\) and \(0\leq \alpha<1\), and \(k\) and \(g\) have similar continuity properties. The proofs are based on the variation of constants formula and the contraction mapping principle.
Reviewer: O.Staffans (Åbo)

45N05 Abstract integral equations, integral equations in abstract spaces
45J05 Integro-ordinary differential equations
45G10 Other nonlinear integral equations
47H20 Semigroups of nonlinear operators