Mediterranean Journal of Mathematics

, Volume 11, Issue 2, pp 687–701 | Cite as

A Family of Measures of Noncompactness in the Space \({L^1_{\text{loc}}(\mathbb{R}_+)}\) and its Application to Some Nonlinear Volterra Integral Equation

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Article

Abstract

The aim of this paper is to study a new family of measures of noncompactness in the space \({L^1_{\text{loc}}(\mathbb{R}_+)}\) consisting of all real functions locally integrable on \({\mathbb{R}_+}\) , equipped with a suitable topology. As an example of applications of the technique associated with that family of measures of noncompactness, we study the existence of solutions of a nonlinear Volterra integral equation in the space \({L^1_{\text{loc}}(\mathbb{R}_+)}\) . The obtained result generalizes several ones obtained earlier with help of other methods.

Mathematics Subject Classification (2010)

Primary 47H10 Secondary 47H30 

Keywords

Carathéodory condition Lebesgue locally integrable function limit of an inverse system Fréchet space measure of noncompactness weak topology 

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© The Author(s) 2013

Open AccessThis article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

Authors and Affiliations

  1. 1.Department of MathematicsRzeszów University of TechnologyRzeszowPoland

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