Journal of Applied Mathematics
Volume 2012 (2012), Article ID 695268, 17 pages
http://dx.doi.org/10.1155/2012/695268
Research Article

General Existence Results for Third-Order Nonconvex State-Dependent Sweeping Process with Unbounded Perturbations

M. Bounkhel and B. Al-Senan

Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

Received 19 September 2011; Accepted 2 February 2012

Academic Editor: Mehmet Sezer

Copyright © 2012 M. Bounkhel and B. Al-Senan. 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 prove the existence of solutions for third-order nonconvex state-dependent sweeping process with unbounded perturbations of the form: − 𝐴 ( 𝑥 ( 3 ) ( 𝑡 ) ) ∈ 𝑁 ( 𝐾 ( 𝑡 , ̇ 𝑥 ( 𝑡 ) ) ; 𝐴 ( ̈ 𝑥 ( 𝑡 ) ) ) + 𝐹 ( 𝑡 , 𝑥 ( 𝑡 ) , ̇ 𝑥 ( 𝑡 ) , ̈ 𝑥 ( 𝑡 ) ) + 𝐺 ( 𝑥 ( 𝑡 ) , ̇ 𝑥 ( 𝑡 ) , ̈ 𝑥 ( 𝑡 ) ) a . e . [ 0 , 𝑇 ] , 𝐴 ( ̈ 𝑥 ( 𝑡 ) ) ∈ 𝐾 ( 𝑡 , ̇ 𝑥 ( 𝑡 ) ) , a . e . 𝑡 ∈ [ 0 , 𝑇 ] , 𝑥 ( 0 ) = 𝑥 0 , ̇ 𝑥 ( 0 ) = 𝑢 0 , ̈ 𝑥 ( 0 ) = 𝜐 0 , where 𝑇 > 0 , 𝐾 is a nonconvex Lipschitz set-valued mapping, 𝐹 is an unbounded scalarly upper semicontinuous convex set-valued mapping, and 𝐺 is an unbounded uniformly continuous nonconvex set-valued mapping in a separable Hilbert space ℍ .