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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)

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