Ludwik Byszewski,

Teresa Winiarska


The aim of this paper is to prove two theorems on the existence and uniqueness of mild and classical solutions of a nonlocal semilinear functional-differential evolution Cauchy problem in a Banach space. The method of semigroups, the Banach fixed-point theorem and the Bochenek theorem (see [3]) about the existence and uniqueness of the classical solution of the first order differential evolution problem in a not necessarily reflexive Banach space are used to prove the existence and uniqueness of the solutions of the considered problem. The results are based on publications [1 — 8].

Słowa kluczowe: evolution problem, functional-differential problem, nonlocal problem

Balachandran K., Ilamaran S., Existence and uniqueness of mild and strong solutions of a semilinear evolution equation with nonlocal conditions, Indian J. Pure Appl. Math., 25.4, 1994, 411—418.

Balasubramaniam, P. Chandrasekaran, M. Existence of solutions of nonlinear integrodifferential equation with nonlocal boundary conditions in Banach space, Atti Sem. Mat. Fis. Univ. Modena, 46, 1998, 1—13.

Bochenek J., The existence of a solution of a semilinear first–order differential equation in a Banach space, Univ. Iag. Acta Math., 31 1994, 61—68.

Byszewski L., Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, J. Math. Anal. Appl., 162.2 1991, 494—505.

Kato T., Perturbation Theory for Linear Operators, Springer-Verlag, New York, Berlin, Heidelberg 1966.

Kołodziej K., Existence and uniqueness of solutions of a semilinear functionaldifferential evolution nonlocal Cauchy problem, JAMSA, 13.2 2000, 171–179.

Pazy A., Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer–Verlag, New York, Berlin, Heidelberg, Tokyo, 1983.

Winiarska T., Differential Equations with Parameters, Monograph 68, Cracow University of Technology 1988.