Ludwik Byszewski

The aim of the paper is to prove two theorems on the existence and uniqueness of mild and classical solutions of a semilinear functional-differential evolution second order equation together with nonlocal conditions. The theory of strongly continuous cosine families of linear operators in a Banach space is applied. The paper is based on publications [1–9] and is a generalization of paper [6].

The aim of the paper is to prove two theorems on continuous dependence of mild solutions,
on initial nonlocal data, of the nonlocal Cauchy problems. For this purpose, the method
of semigroups and the theory of cosine family in Banach spaces are applied. The paper is based
on publications [1–5].

W artykule udowodniono dwa twierdzenia o ciągłej zależności rozwiązań całkowych od nielokalnych warunków początkowych, semiliniowych nielokalnych zagadnień Cauchy’ego. W tym celu zastosowano
metodę półgrup i teorię rodziny cosinus w przestrzeniach Banacha. Artykuł bazuje na publikacjach [1–6]
i jest pewnym uogólnieniem publikacji [3].

The aim of this paper is to prove the existence and uniqueness of solutions of the Dirichlet nonlocal problem with nonlocal initial condition. The considerations are extensions of results by E. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. J. Toledo-Melero [1].

The aim of this paper is to give two theorems on the existence and uniqueness of mild and classical solutions of a nonlocal semilinear integro-differential evolution Cauchy problem for the first order equation. The method of semigroups, the Banach fixed-point theorem and the Bochenek theorem are applied to prove the existence and uniqueness of the solutions of the considered problem.

The aim of this paper is to prove the theorem on the existence and uniqueness of the classical solution of the initial-boundary value problem for a nonlinear second-order delay differential equation. For this purpose, we apply the Banach contraction principle and the Bielecki norm. The paper is based on publications [1–7] and is a generalisation of publication [6]. 

Streszczenie

Nieliniowe równanie różniczkowe rzędu drugiego z opóźnieniem

W artykule udowodniono twierdzenie o istnieniu i jednoznaczności klasycznego rozwiązania zagadnienia początkowo-brzegowego dla nieliniowego równania różniczkowego rzędu drugiego z opóźnieniem. W tym celu stosowane jest twierdzenie Banacha o punkcie stałym i norma Bieleckiego.  Artykuł bazuje na publikacjach [1–7] i jest uogólnieniem publikacji [6]. 

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].

In this paper, we investigate the existence and uniqueness of the classical solution to an abstract
nonlocal Cauchy problem. For this purpose, we apply a notion of mild solution and the Banach
contraction theorem.

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].

The uniqueness of classical solutions to parabolic semilinear problems together with nonlocal initial
conditions with integrals, for the operator

The result requires that the nonlocal conditions with integrals be introduced.