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

Author

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  • Shamshad Husain
  • Sanjeev Gupta
  • Vishnu Narayan Mishra

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.

Suggested Citation

  • Shamshad Husain & Sanjeev Gupta & Vishnu Narayan Mishra, 2013. "Generalized -Cocoercive Operators and Generalized Set-Valued Variational-Like Inclusions," Journal of Mathematics, Hindawi, vol. 2013, pages 1-10, June.
    • Handle: RePEc:hin:jjmath:738491
    DOI: 10.1155/2013/738491
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    References listed on IDEAS

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    1. X. P. Ding, 2005. "Existence of Solutions and an Algorithm for Mixed Variational-Like Inequalities in Banach Spaces," Journal of Optimization Theory and Applications, Springer, vol. 127(2), pages 285-302, November.
    2. C. E. Chidume & K. R. Kazmi & H. Zegeye, 2004. "Iterative approximation of a solution of a general variational-like inclusion in Banach spaces," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2004, pages 1-10, January.
    3. Q. H. Ansari & J. C. Yao, 2001. "Iterative Schemes for Solving Mixed Variational-Like Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 108(3), pages 527-541, March.
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