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We study, in Carathéodory assumptions, existence, continuation and continuous dependence of extremal solutions for an abstract and rather general class of hereditary differential equations. By some examples we prove that, unlike the nonfunctional case, solved Cauchy problems for hereditary differential equations may not have local extremal solutions.

Finite time stability and relative controllability of second order linear differential systems with pure delay

Mengmeng Li, Michal Fečkan, JinRong Wang (2023)

Applications of Mathematics

We first consider the finite time stability of second order linear differential systems with pure delay via giving a number of properties of delayed matrix functions. We secondly give sufficient and necessary conditions to examine that a linear delay system is relatively controllable. Further, we apply the fixed-point theorem to derive a relatively controllable result for a semilinear system. Finally, some examples are presented to illustrate the validity of the main theorems.

First order integro-differential equations in Banach algebras involving Carathéodory and discontinuous nonlinearities.

Dhage, B.C., Karande, B.D. (2005)

Electronic Journal of Qualitative Theory of Differential Equations [electronic only]

Frequency analysis of preconditioned waveform relaxation iterations

Andrzej Augustynowicz, Zdzisław Jackiewicz (1999)

Applicationes Mathematicae

The error analysis of preconditioned waveform relaxation iterations for differential systems is presented. This analysis extends and refines previous results by Burrage, Jackiewicz, Nørsett and Renaut by incorporating all terms in the expansion of the error of waveform relaxation iterations in the Laplace transform domain. Lower bounds for the size of the window of rapid convergence are also obtained. The theory is illustrated for waveform relaxation methods applied to differential systems resulting...

Functional differential equations

Tadeusz Jankowski (2002)

Czechoslovak Mathematical Journal

The method of quasilinearization is a well-known technique for obtaining approximate solutions of nonlinear differential equations. In this paper we apply this technique to functional differential problems. It is shown that linear iterations converge to the unique solution and this convergence is superlinear.

Functional differential equations - a reciprocity principle.

Williams, Lloyd K. (1986)

International Journal of Mathematics and Mathematical Sciences

Functional differential equations in Banach spaces: Growth and decay of solutions.

Hans Engler (1981)

Journal für die reine und angewandte Mathematik

Functional-differential equations with Riemann-Liouville integrals in the nonlinearities

Milan Medveď (2014)

Mathematica Bohemica

A sufficient condition for the nonexistence of blowing-up solutions to evolution functional-differential equations in Banach spaces with the Riemann-Liouville integrals in their right-hand sides is proved. The linear part of such type of equations is an infinitesimal generator of a strongly continuous semigroup of linear bounded operators. The proof of the main result is based on a desingularization method applied by the author in his papers on integral inequalities with weakly singular kernels....

Fuzzy solutions for neutral functional differential equations with nonlocal conditions.

Arara, Amaria, Benchohra, Mouffak (2004)

Georgian Mathematical Journal

Generalized functional-differential equations of neutral type

Michał Kisielewicz (1983)

Annales Polonici Mathematici

Generalized solutions of functional differential inclusions.

Machina, Anna, Bulgakov, Aleksander, Grigorenko, Anna (2008)

Abstract and Applied Analysis

Gleichmäßig asymptotische Stabilität stationäre Punkte bei Differential-Differenzengleichungen.

Klaus Doden (1977)

Journal für die reine und angewandte Mathematik

Global dynamics of a delay differential system of a two-patch SIS-model with transport-related infections

Yukihiko Nakata, Gergely Röst (2015)

Mathematica Bohemica

We describe the global dynamics of a disease transmission model between two regions which are connected via bidirectional or unidirectional transportation, where infection occurs during the travel as well as within the regions. We define the regional reproduction numbers and the basic reproduction number by constructing a next generation matrix. If the two regions are connected via bidirectional transportation, the basic reproduction number $R_{0}$ characterizes the existence of equilibria as well as...

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