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Journal of Mathematics
Volume 2013 (2013), Article ID 738491, 10 pages
http://dx.doi.org/10.1155/2013/738491
Research Article

Generalized -Cocoercive Operators and Generalized Set-Valued Variational-Like Inclusions

1Department of Applied Mathematics, Faculty of Engineering & Technology, Aligarh Muslim University, Aligarh 202002, India
2Department of Applied Mathematics & Humanities, Sardar Vallabhbhai National Institute of Technology, Ichchhanath Mahadev Road, Surat 395 007, India

Received 18 March 2013; Revised 4 May 2013; Accepted 8 May 2013

Academic Editor: Kaleem R. Kazmi

Copyright © 2013 Shamshad Husain 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 investigate a new class of cocoercive operators named generalized -cocoercive operators in Hilbert spaces. We prove that generalized -cocoercive operator is single-valued and Lipschitz continuous and extends the concept of resolvent operators associated with -cocoercive operators to the generalized -cocoercive operators. Some examples are given to justify the definition of generalized -cocoercive operators. Further, we consider a generalized set-valued variational-like inclusion problem involving generalized -cocoercive operator. In terms of the new resolvent operator technique, we give the approximate solution and suggest an iterative algorithm for the generalized set-valued variational-like inclusions. Furthermore, we discuss the convergence criteria of iterative algorithm under some suitable conditions. Our results can be viewed as a generalization of some known results in the literature.