# American Institute of Mathematical Sciences

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January  2013, 33(1): 283-303. doi: 10.3934/dcds.2013.33.283

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

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

Received  July 2011 Revised  November 2011 Published  September 2012

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.
Citation: Giselle A. Monteiro, Milan Tvrdý. Generalized linear differential equations in a Banach space: Continuous dependence on a parameter. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 283-303. doi: 10.3934/dcds.2013.33.283
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