Abstract and Applied Analysis
Volume 2013 (2013), Article ID 819315, 14 pages
http://dx.doi.org/10.1155/2013/819315
Review Article

Compactness Conditions in the Study of Functional, Differential, and Integral Equations

Józef Banaś1 and Kishin Sadarangani2

1Department of Mathematics, Rzeszów University of Technology, Aleja Powstańców Warszawy 8, 35-959 Rzeszów, Poland
2Departamento de Mathemáticas, Universidad de Las Palmas de Gran Canaria, Campus de Tafira Baja, 35017 Las Palmas de Gran Canaria, Spain

Received 13 December 2012; Accepted 2 January 2013

Academic Editor: Beata Rzepka

Copyright © 2013 Józef Banaś and Kishin Sadarangani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We discuss some existence results for various types of functional, differential, and integral equations which can be obtained with the help of argumentations based on compactness conditions. We restrict ourselves to some classical compactness conditions appearing in fixed point theorems due to Schauder, Krasnosel’skii-Burton, and Schaefer. We present also the technique associated with measures of noncompactness and we illustrate its applicability in proving the solvability of some functional integral equations. Apart from this, we discuss the application of the mentioned technique to the theory of ordinary differential equations in Banach spaces.