Byszewski, Ludwik; Lakshmikantham, V. Theorem about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space. (English) Zbl 0694.34001 Appl. Anal. 40, No. 1, 11-19 (1991). 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: \[ x'=f(t,x),\quad t\in I,\quad x(t_ 0)+g(t_ 1,...,t_ p,x(\cdot))=x_ 0, \] where \(I=[t_ 0,T]\), \(t_ 0<t_ 1<...<t_ p\leq T(p\in {\mathbb{N}})\), \(x=(x_ 1,...,x_ n)\in \Omega\), \(x_ 0=(x_{10},...,x_{no})\in \Omega\), \(f=(f_ 1,...,f_ n)\in C(I\times \Omega,E)\), \(g=(g_ 1,...,g_ n):\) \(I^ p\times \Omega \to E\), \(g(t_ 1,...,t_ p,\cdot)\in C(\Omega,E)\), \(\Omega\) \(\subset E\) and \(E=E_ 1\times...\times E_ n\), where \(E_ i(i=1,...,n)\) are Banach spaces with norms \(\| \cdot \|\). 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 Cited in 4 ReviewsCited in 237 Documents MSC: 34G20 Nonlinear differential equations in abstract spaces 34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations 45N05 Abstract integral equations, integral equations in abstract spaces 47H10 Fixed-point theorems Keywords:existence; uniqueness; nonlocal abstract Cauchy problem; Banach space Citations:Zbl 0456.34002; Zbl 0209.362 PDFBibTeX XMLCite \textit{L. Byszewski} and \textit{V. Lakshmikantham}, Appl. Anal. 40, No. 1, 11--19 (1991; Zbl 0694.34001) Full Text: DOI References: [1] DOI: 10.1002/zamm.19900700312 · Zbl 0709.35018 · doi:10.1002/zamm.19900700312 [2] DOI: 10.1155/S1048953390000065 · Zbl 0726.35023 · doi:10.1155/S1048953390000065 [3] Byszewski L., Journal of Mathematical Analysis and Applications 3 (1990) [4] Byszewski L., Journal of Applied Mathematics and Stochastic Analysis 3 (1990) [5] Byszewski L., Applicable Analysis 3 (1990) [6] Kolodziej W., Mathematical Analysis 3 (1978) [7] Lakshmikantham V., Nonlinear Differential Equations in Abstract Spaces (1981) · Zbl 0456.34002 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.