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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:

${x}^{\text{'}}=f\left(t,x\right),\phantom{\rule{1.em}{0ex}}t\in I,\phantom{\rule{1.em}{0ex}}x\left({t}_{0}\right)+g\left({t}_{1},···,{t}_{p},x\left(·\right)\right)={x}_{0},$

where $I=\left[{t}_{0},T\right]$, ${t}_{0}<{t}_{1}<···<{t}_{p}\le T\left(p\in ℕ\right)$, $x=\left({x}_{1},···,{x}_{n}\right)\in {\Omega }$, ${x}_{0}=\left({x}_{10},···,{x}_{no}\right)\in {\Omega }$, $f=\left({f}_{1},···,{f}_{n}\right)\in C\left(I×{\Omega },E\right)$, $g=\left({g}_{1},···,{g}_{n}\right):$ ${I}^{p}×{\Omega }\to E$, $g\left({t}_{1},···,{t}_{p},·\right)\in C\left({\Omega },E\right)$, ${\Omega }$ $\subset E$ and $E={E}_{1}×···×{E}_{n}$, where ${E}_{i}\left(i=1,···,n\right)$ are Banach spaces with norms $\parallel ·\parallel$. 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 S. Leela [Nonlinear Differential Equations in Abstract Spaces (1981; Zbl 0456.34002)] and by W. Kołodziej [Mathematical Analysis (1970; Zbl 0209.362)].

Reviewer: L.Byszewski

##### MSC:
 34G20 Nonlinear ODE in abstract spaces 34A12 Initial value problems for ODE, existence, uniqueness, etc. of solutions 45N05 Abstract integral equations, integral equations in abstract spaces 47H10 Fixed point theorems for nonlinear operators on topological linear spaces