Advances in Difference Equations
Volume 2010 (2010), Article ID 798067, 37 pages
doi:10.1155/2010/798067
Research Article

Some Results for Integral Inclusions of Volterra Type in Banach Spaces

R. P. Agarwal,1,2 M. Benchohra,3 J. J. Nieto,4 and A. Ouahab3

1Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901-6975, USA
2Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
3Laboratoire de Mathématiques, Université de Sidi Bel-Abbès, B.P. 89, Sidi Bel-Abbès 22000, Algeria
4Departamento de Analisis Matematico, Facultad de Matematicas, Universidad de Santiago de Compostela, Santiago de Compostela 15782, Spain

Received 29 July 2010; Revised 16 October 2010; Accepted 29 November 2010

Academic Editor: M. Cecchi

Copyright © 2010 R. P. Agarwal et al. 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 first present several existence results and compactness of solutions set for the following Volterra type integral inclusions of the form: ∫ 𝑦 ( 𝑡 ) ∈ 𝑡 0 𝑎 ( 𝑡 − 𝑠 ) [ 𝐴 𝑦 ( 𝑠 ) + 𝐹 ( 𝑠 , 𝑦 ( 𝑠 ) ) ] 𝑑 𝑠 , a . e . 𝑡 ∈ 𝐽 , where 𝐽 = [ 0 , 𝑏 ] , 𝐴 is the infinitesimal generator of an integral resolvent family on a separable Banach space 𝐸 , and 𝐹 is a set-valued map. Then the Filippov's theorem and a Filippov-Ważewski result are proved.