## Existence of solutions of quasilinear integrodifferential equations with nonlocal condition.(English)Zbl 0976.45011

The authors prove the existence and uniqueness of mild and classical solutions of the quasilinear abstract differential equation $du(t)/dt + Au(t)=f (t, u(t))\quad (t\in (0, a])$ with the nonlocal condition $u (0) + g(t_1, t_2,\dots , t_p, u(t_1),\dots , u(t_p))=u_0$ where $$-A$$ is the infinitesimal generator of a $$C_0$$-semigroup in a Banach space $$X$$, $$u_0\in X$$ and $$f : [0, a] \times X\rightarrow X, f : [0, a]^p \times X^p\rightarrow X$$ are given functions, and $0\leq t_1\leq t_2\leq \dots \leq t_p\leq a.$ The main results of the paper are obtained by using $$C_0$$-semigroups and the Banach fixed point theorem.

### MSC:

 45N05 Abstract integral equations, integral equations in abstract spaces 45K05 Integro-partial differential equations 34G20 Nonlinear differential equations in abstract spaces 45G10 Other nonlinear integral equations
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