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Continuous dependence of solutions of abstract generalized linear differential equations with potential converging uniformly with a weight

Giselle Antunes Monteiro1 and Milan Tvrdý2*

Author Affiliations

1 Mathematical Institute, Academy of Sciences of Czech Republic, Žitná 25, Praha, 115 67, Czech Republic

2 Mathematical Institute, Academy of Sciences of Czech Republic, Žitná 25, Praha, 115 67, Czech Republic

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Boundary Value Problems 2014, 2014:71  doi:10.1186/1687-2770-2014-71


Dedicated to Professor Ivan Kiguradze for his merits in mathematical sciences.

Published: 26 March 2014

Abstract

In this paper we continue our research from (Monteiro and Tvrdý in Discrete Contin. Dyn. Syst. 33(1):283-303, 2013) on continuous dependence on a parameter k of solutions to linear integral equations of the form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M1">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M3">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M4">View MathML</a>, X is a Banach space, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M5">View MathML</a> is the Banach space of linear bounded operators on X, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M7">View MathML</a> have bounded variations on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M9">View MathML</a> are regulated on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M8">View MathML</a>. The integrals are understood as the abstract Kurzweil-Stieltjes integral and the studied equations are usually called generalized linear differential equations (in the sense of Kurzweil, cf. (Kurzweil in Czechoslov. Math. J. 7(82):418-449, 1957) or (Kurzweil in Generalized Ordinary Differential Equations: Not Absolutely Continuous Solutions, 2012)). In particular, we are interested in the situation when the variations <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/71/mathml/M11">View MathML</a> need not be uniformly bounded. Our main goal here is the extension of Theorem 4.2 from (Monteiro and Tvrdý in Discrete Contin. Dyn. Syst. 33(1):283-303, 2013) to the nonhomogeneous case. Applications to second-order systems and to dynamic equations on time scales are included as well.

MSC: 45A05, 34A30, 34N05.

Keywords:
abstract generalized differential equation; continuous dependence; time scale dynamics