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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 ( ) 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 , 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:47H05, 47J20, 49J40, 90C33.
Highlights
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
Applying nearly uniformly Lipschitzian mappings Si (i =, ) and the aforesaid equivalent alternative formulation, we suggest and analyze some new three-step resolvent iterative algorithms with mixed errors for finding the element of the set of fixed points of the nearly uniformly Lipschitzian mapping Q = (S, S, S ), which is the unique solution of the SGNMVID
Applying nearly uniformly Lipschitzian mappings Si (i =, ) and by using the equivalent alternative formulation ( . ), we suggest and analyze some new three-step resolvent iterative algorithms with mixed errors for finding an element of the set of fixed points of Q = (S, S, S ), which is the unique solution of SGNMVID ( . )
Summary
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. We introduce and consider a new system of generalized nonlinear mixed variational inequalities involving six different nonlinear operators (SGNMVID). Applying nearly uniformly Lipschitzian mappings Si (i = , , ) and the aforesaid equivalent alternative formulation, we suggest and analyze some new three-step resolvent iterative algorithms with mixed errors for finding the element of the set of fixed points of the nearly uniformly Lipschitzian mapping Q = (S , S , S ), which is the unique solution of the SGNMVID. If φ is a proper, convex and lower-semicontinuous function, its subdifferential ∂φ is a maximal monotone operator, see Theorem in [ ] In this case, we can define the resolvent operator associated with the subdifferential ∂φ of parameter λ as follows: Jφλ(u) = (I + λ∂φ)– (u), ∀u ∈ H.
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