Abstract
By using the new parametric resolvent operator technique associated with (A,η,m)-monotone operators, the purpose of this paper is to analyze and establish an existence theorem for a new class of generalized nonlinear parametric (A,η,m)-proximal operator system of equations with non-monotone multi-valued operators in Hilbert spaces. The results presented in this paper generalize the sensitivity analysis results of recent work on strongly monotone quasi-variational inclusions, nonlinear implicit quasi-variational inclusions, and nonlinear mixed quasi-variational inclusion systems in Hilbert spaces.MSC:49J40, 47H05, 90C33.
Highlights
Since the study of the sensitivity of solutions for variational inclusion problems involving strongly monotone and relaxed cocoercive mappings under suitable second order and regularity assumptions is an increasing interest, there are many motivated researchers basing their work on the generalized resolvent operator techniques, which is used to develop powerful and efficient numerical techniques for solving variational inequalities, related optimization, control theory, operations research, transportation network modeling, and mathematical programming problems
It is well known that the project technique and the resolvent operator technique can be used to establish an equivalence between variational inequalities, variational inclusions, and resolvent equations
∈ E(x, y, ω) + M(x, x, ω), ∈ F(u, y, λ) + N(y, y, λ). It follows from the definition of Q(ω, λ) that (x, y) ∈ Q(ω, λ) is a solution of problem ( . ) if and only if there exist (x, y) ∈ H × H, and u ∈ S(x, ω) such that equation ( . ) holds
Summary
Since the study of the sensitivity (analysis) of solutions for variational inclusion (operator equation) problems involving strongly monotone and relaxed cocoercive mappings under suitable second order and regularity assumptions is an increasing interest, there are many motivated researchers basing their work on the generalized resolvent operator (equation) techniques, which is used to develop powerful and efficient numerical techniques for solving (mixed) variational inequalities, related optimization, control theory, operations research, transportation network modeling, and mathematical programming problems. It is well known that the project technique and the resolvent operator technique can be used to establish an equivalence between (mixed) variational inequalities, variational inclusions, and resolvent equations. For example, [ – ] and the references therein. We consider the following system of (A, η, m)-proximal operator equations: For each fixed (ω, λ) ∈ × , find (z, t), (x, y) ∈ H × H such that u ∈ S(x, ω) and. If S : H × → H is a single-valued operator, for each fixed (ω, λ) ∈ × , problem ) reduces to the following problem of finding (x, y), (z, t) ∈ H × H such that:. ) is equivalent to the following nonlinear equation:.
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