Auxiliary principle and iterative algorithm for a system of generalized set-valued strongly nonlinear mixed implicit quasi-variational-like inequalities

• Main Authors: Yang-Qing Qiu, Jin-Zuo Chen, Lu-Chuan Ceng
• Format: Article
• Language: English
• Published: SpringerOpen 2016-02-01
• Series: Journal of Inequalities and Applications
• Subjects: system of generalized mixed implicit quasi-variational-like inequalities; auxiliary principle; iterative algorithm; convergence
• Online Access: http://link.springer.com/article/10.1186/s13660-016-0978-3

Abstract In this paper, the auxiliary principle technique is extended to study a system of generalized set-valued strongly nonlinear mixed implicit quasi-variational-like inequalities problem in Hilbert spaces. First, we establish the existence of solutions of the corresponding system of auxiliary variational inequalities problem. Then, using the existence result, we construct a new iterative algorithm. Finally, both the existence of solutions of the original problem and the convergence of iterative sequences generated by the algorithm are proved. We give an affirmative answer to the open problem raised by Noor et al. (Korean J. Comput. Appl. Math. 1:73-89, 1998; J. Comput. Appl. Math. 47:285-312, 1993). Our results improve and extend some known results.