Nonlinear Mixed Variational-Like Inequality with Respect to Weakly Relaxed \(\eta -\alpha \) Monotone Mapping in Banach Spaces
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.
KeywordsWeakly 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.
- 4.Ding, X.P.: General algorithm for nonlinear variational-like inequalities in reflexive banach spaces. Indian J. Pure Appl. Math. 29(2), (1998)Google Scholar
- 7.Huang, N.J.: On the generalized implicit quasivariational inequalities. J. Math. Anal. Appl. 216(1), 197–210 (1997)Google Scholar
- 8.Huang, N.J., Deng, C.X.: Auxiliary principle and iterative algorithms for generalized set-valued strongly nonlinear mixed variational-like inequalities. J. Math. Anal. Appl. 256(2), 345–359 (2001)Google Scholar
- 9.Kutbi, M.A., Sintunavarat, W.: On the solution existence of variational-like inequalities problems for weakly relaxed monotone mapping. In: Abstract and Applied Analysis, vol. 2013. Hindawi Publishing Corporation (2013)Google Scholar
- 11.Tremolieres, R., Lions, J.L., Glowinski, R.: Numerical analysis of variational inequalities. Elsevier (2011)Google Scholar