Generalized linear differential equations in a Banach space: Continuous dependence on a parameter

Giselle A. Monteiro; Milan Tvrdý

doi:10.3934/dcds.2013.33.283

Article Contents

2013, Volume 33, Issue 1: 283-303. Doi: 10.3934/dcds.2013.33.283

This issue Previous Article On the periodic solutions of a class of Duffing differential equations Next Article Existence, regularity and boundary behaviour of bounded variation solutions of a one-dimensional capillarity equation

Generalized linear differential equations in a Banach space: Continuous dependence on a parameter

Giselle A. Monteiro^1, and
Milan Tvrdý^2,

1.
Instituto de Ciências Matemáticas e Computação, Universidade de São Paulo, Caixa Postal 668, 13560-970, São Carlos, SP
2.
Institute of Mathematics, Academy of Sciences of Czech Republic, Žitná 25, CZ 115 67 Praha 1

Received: June 30, 2011

Revised: October 31, 2011

Published: December 2012

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Abstract

This paper deals with integral equations of the form \begin{eqnarray*} x(t)=\tilde{x}+∫_a^td[A]x+f(t)-f(a), t∈[a,b], \end{eqnarray*} in a Banach space $X,$ where $-\infty\ < a < b < \infty$, $\tilde{x}∈ X,$ $f:[a,b]→X$ is regulated on [a,b] and $A(t)$ is for each $t∈[a,b], $ a linear bounded operator on $X,$ while the mapping $A:[a,b]→L(X)$ has a bounded variation on [a,b] Such equations are called generalized linear differential equations. Our aim is to present new results on the continuous dependence of solutions of such equations on a parameter. Furthermore, an application of these results to dynamic equations on time scales is given.

Keywords:

Generalized differential equations in Banach space,
continuous dependence,
time scale dynamics,
Kurzweil-Stieljes integral.

Mathematics Subject Classification: Primary: 45A05, 34A30; Secondary: 34N05.

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