Nonlinear Mixed Variational-Like Inequality with Respect to Weakly Relaxed \(\eta -\alpha \) Monotone Mapping in Banach Spaces

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 143)


In this paper, we have studied Nonlinear Mixed Variational-like inequality with respect to weakly relaxed \(\eta -\alpha \) monotone mapping, involving a nonlinear bifunction, in Banach space. Significance of weakly relaxed \(\eta -\alpha \) monotonicity is illustrated through an example. Existence of the solution to the problem is established using KKM (Knaster, Kuratowski and Mazurkiewicz) technique. Also we have proposed an iterative algorithm using auxiliary principle technique, which involves formulation of an auxiliary minimizing problem and then characterizing it by an auxiliary variational inequality problem. Solvability of the auxiliary variational inequality problem is established. Finally convergence of the iterates to the exact solution is proved.


Weakly relaxed \(\eta -\alpha \) monotone mapping KKM technique Auxiliary principle technique Iterative algorithm 



The first author wishes to thank DST-INSPIRE (Code No. IF110762) for the grant of research fellowship and to IIT Bhubaneswar for providing the research facilities.


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© Springer India 2015

Authors and Affiliations

  1. 1.IIT BhubaneswarOdishaIndia

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