On nonlocal evolution problem for the equation of the first order

Ludwik Byszewski,

Teresa Winiarska

Abstrakt

The aim of the paper is to prove theorems about the existence and uniqueness of mild
and classical solutions of a nonlocal semilinear functional-differential evolution Cauchy
problem. The method of semigroups, the Banach fixed-point theorem and theorems (see [2])
about the existence and uniqueness of the classical solutions of the first-order differential
evolution problems in a not necessarily reflexive Banach space are used to prove the existence
and uniqueness of the solutions of the problems considered. The results obtained are based
on publications [1–6].

Słowa kluczowe: evolution Cauchy problem, existence and uniqueness of the solutions, nonlocal conditions
References

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.

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., Application of properties of the right-hand sides of evolution equations to an investigation of nonlocal evolution problems, Nonlinear Analysis, 33, 1998, 413-426.

Kato T., Perturbation Theory for Linear Operators, Springer–Verlag, New York, Berlin, Heidelberg, 1996.

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 Parameter, Monograph 68, Technical University of Cracow, Cracow 1988.