Abstract
AbstractIn this paper, we introduce and discuss a new system of generalized nonlinear mixed quasivariational inclusions with(Hi,ηi)-monotone operators in Hilbert spaces, which includes several systems of variational inequalities and variational inclusions as special cases. By employing the resolvent operator technique associated with(Hi,ηi)-monotone operators, we suggest two iterative algorithms for computing the approximate solutions of the system of generalized nonlinear mixed quasivariational inclusions. Under certain conditions, we obtain the existence of solutions for the system of generalized nonlinear mixed quasivariational inclusions and prove the convergence of the iterative sequences generated by the iterative algorithms. The results presented in this paper extend, improve and unify many known results in recent literature.MSC:47J20, 49J40.
Highlights
1 Introduction Variational inclusions, as important extensions of the classical variational inequalities, provide us with simple, natural, general and unified frameworks in the study of many fields including mechanics, physics, optimization and control, nonlinear programming, economics and transportation equilibrium, as well as engineering sciences. Owing to their wide applications, a lot of existence results and iterative algorithms of solutions for various variational inclusions have been studied in recent years
We prove the existence of solutions for the system of generalized nonlinear mixed quasivariational inclusions and show the convergence of the iterative sequences generated by the iterative algorithms in Hilbert spaces
Based on Lemma . and Lemma . , we suggest the following two sorts of iterative algorithms for the system of generalized nonlinear mixed quasivariational inclusions ( . )
Summary
Variational inclusions, as important extensions of the classical variational inequalities, provide us with simple, natural, general and unified frameworks in the study of many fields including mechanics, physics, optimization and control, nonlinear programming, economics and transportation equilibrium, as well as engineering sciences. A new system of generalized nonlinear mixed quasivariational inclusions with (Hi, ηi)monotone operators is introduced and studied in Hilbert spaces.
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