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Theorem about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space. (English) Zbl 0694.34001
The aim of the paper is to give a theorem about the existence and uniqueness of a solution of the following nonlocal abstract Cauchy problem in a Banach space: $$ x'=f(t,x),\quad t\in I,\quad x(t\sb 0)+g(t\sb 1,...,t\sb p,x(\cdot))=x\sb 0, $$ where $I=[t\sb 0,T]$, $t\sb 0<t\sb 1<...<t\sb p\le T(p\in {\bbfN})$, $x=(x\sb 1,...,x\sb n)\in \Omega$, $x\sb 0=(x\sb{10},...,x\sb{no})\in \Omega$, $f=(f\sb 1,...,f\sb n)\in C(I\times \Omega,E)$, $g=(g\sb 1,...,g\sb n):$ $I\sp p\times \Omega \to E$, $g(t\sb 1,...,t\sb p,\cdot)\in C(\Omega,E)$, $\Omega$ $\subset E$ and $E=E\sb 1\times...\times E\sb n$, where $E\sb i(i=1,...,n)$ are Banach spaces with norms $\Vert \cdot \Vert$. The Banach theorem about the fixed point is used to prove the existence and uniqueness of a solution of the problem considered. The results obtained can be applied among other things to the description of motion phenomena with better effect than the classical Cauchy problem. They are a continuation of those given by the first author [Z. Angew. Math. Mech. 70, 3, 202-206 (1990); J. Appl. Math. Stochastic Anal. 3, No.3, 65-79 (1990); J. Math. Anal. Appl. (to appear) (1990); J. Appl. Math. Stochastic Anal. (to appear) (1990); Appl. Anal. (to appear) (1990)] and generalize the known theorem about the existence and uniqueness of the solution considered by the second author and {\it S. Leela} [Nonlinear Differential Equations in Abstract Spaces (1981; Zbl 0456.34002)] and by {\it W. Kołodziej} [Mathematical Analysis (1970; Zbl 0209.362)].
Reviewer: L.Byszewski

34G20Nonlinear ODE in abstract spaces
34A12Initial value problems for ODE, existence, uniqueness, etc. of solutions
45N05Abstract integral equations, integral equations in abstract spaces
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
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