Generalized Mann Viscosity Implicit Rules for Solving Systems of Variational Inequalities with Constraints of Variational Inclusions and Fixed Point Problems

• Main Authors: Lu-Chuan Ceng, Meijuan Shang
• Format: Article
• Language: English
• Published: MDPI AG 2019-10-01
• Series: Mathematics
• Subjects: generalized mann-like viscosity rule; system of variational inequalities; variational inclusions; nonexpansive mappings; strong convergence; uniform convexity; uniform smoothness
• Online Access: https://www.mdpi.com/2227-7390/7/10/933

In this work, let <i>X</i> be Banach space with a uniformly convex and <i>q</i>-uniformly smooth structure, where <inline-formula> <math display="inline"> <semantics> <mrow> <mn>1</mn> <mo>&lt;</mo> <mi>q</mi> <mo>&#8804;</mo> <mn>2</mn> </mrow> </semantics> </math> </inline-formula>. We introduce and consider a generalized Mann-like viscosity implicit rule for treating a general optimization system of variational inequalities, a variational inclusion and a common fixed point problem of a countable family of nonexpansive mappings in <i>X</i>. The generalized Mann-like viscosity implicit rule investigated in this work is based on the Korpelevich&#8217;s extragradient technique, the implicit viscosity iterative method and the Mann&#8217;s iteration method. We show that the iterative sequences governed by our generalized Mann-like viscosity implicit rule converges strongly to a solution of the general optimization system.