Abstract
In order to unify some variational inequality problems, in this paper, a new system of generalized quasivariational inclusion (for short, (SGQVI)) is introduced. By using Banach contraction principle, some existence and uniqueness theorems of solutions for (SGQVI) are obtained in real Banach spaces. Two new iterative algorithms to find the common element of the solutions set for (SGQVI) and the fixed points set for Lipschitz mappings are proposed. Convergence theorems of these iterative algorithms are established under suitable conditions. Further, convergence rates of the convergence sequences are also proved in real Banach spaces. The main results in this paper extend and improve the corresponding results in the current literature. 2000 MSC: 47H04; 49J40.
Highlights
Variational inclusion problems, which are generalizations of variational inequalities introduced by Stampacchia [1] in the early sixties, are among the most interesting and intensively studied classes of mathematics problems and have wide applications in the fields of optimization and control, economics, electrical networks, game theory, engineering science, and transportation equilibria
Qin et al [15] studied the approximation of solutions to a system of variational inclusions in Banach spaces and established a strong convergence theorem in uniformly convex and 2-uniformly smooth Banach spaces
Kamraksa and Wangkeeree [16] introduced a general iterative method for a general system of variational inclusions and proved a strong convergence theorem in strictly convex and 2-uniformly smooth Banach spaces
Summary
Variational inclusion problems, which are generalizations of variational inequalities introduced by Stampacchia [1] in the early sixties, are among the most interesting and intensively studied classes of mathematics problems and have wide applications in the fields of optimization and control, economics, electrical networks, game theory, engineering science, and transportation equilibria. Petrot [18] applied the resolvent operator technique to find the common solutions for a generalized system of relaxed cocoercive mixed variational inequality problems and fixed point problems for Lipschitz mappings in Hilbert spaces.
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