Fixed point problems and a system of generalized nonlinear mixed variational inequalities
- PDF / 317,257 Bytes
- 21 Pages / 595.28 x 793.7 pts Page_size
- 7 Downloads / 202 Views
RESEARCH
Open Access
Fixed point problems and a system of generalized nonlinear mixed variational inequalities Narin Petrot1,2 and Javad Balooee3* *
Correspondence: [email protected] 3 Department of Mathematics, Sari Branch, Islamic Azad University, Sari, Iran Full list of author information is available at the end of the article
Abstract In this paper, we introduce and consider a new system of generalized nonlinear mixed variational inequalities involving six different nonlinear operators and discuss the existence and uniqueness of solution of the aforesaid system. We use three nearly uniformly Lipschitzian mappings Si (i = 1, 2, 3) to suggest and analyze some new three-step resolvent iterative algorithms with mixed errors for finding an element of the set of fixed points of the nearly uniformly Lipschitzian mapping Q = (S1 , S2 , S3 ), which is the unique solution of the system of generalized nonlinear mixed variational inequalities. The convergence analysis of the suggested iterative algorithms under suitable conditions is studied. In the final section, an important remark on a class of some relaxed cocoercive mappings is discussed. MSC: Primary 47H05; secondary 47J20; 49J40; 90C33 Keywords: system of generalized mixed variational inequalities; fixed point problems; nearly uniformly Lipschitzian mapping; three-step resolvent iterative algorithm; convergence
1 Introduction Variational inequality theory, which was initially introduced by Stampacchia [] in , is a branch of applicable mathematics with a wide range of applications in industry, physical, regional, social, pure, and applied sciences. This field is dynamic and is experiencing an explosive growth in both theory and applications; as a consequence, research techniques and problems are drawn from various fields. Variational inequalities have been generalized and extended in different directions using the novel and innovative techniques. An important and useful generalization is called the mixed variational inequality, or the variational inequality of the second kind, involving the nonlinear term. For applications, numerical methods, and other aspects of variational inequalities, see, for example, [–] and the references therein. In recent years, much attention has been given to develop efficient and implementable numerical methods including projection method and its variant forms, Wiener-Hopf (normal) equations, linear approximation, auxiliary principle, and descent framework for solving variational inequalities and related optimization problems. It is well known that the projection method and its variant forms and Wiener-Hopf equations technique cannot be used to suggest and analyze iterative methods for solving mixed variational inequalities due to the presence of the nonlinear term. These facts motivated us to use the technique of resolvent operators, the origin of which can be traced back to © 2013 Petrot and Balooee; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://c
Data Loading...