Algorithms for solutions of extended general mixed variational inequalities and fixed points

Balooee, Javad; Cho, Yeol Je

doi:10.1007/s11590-012-0516-2

Cited by 24 publications

(9 citation statements)

References 44 publications

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“…The existence of a strong connection between the variational inequality problems and the fixed-point problems motivated many investigators to study the problem of finding common elements of the set of solutions of variational inequalities/inclusions and the set of fixed points of given operators. For more details and information, the reader is referred to [4,[38][39][40][41][42][43][44][45][46] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Graph convergence with an application for system of variational inclusions and fixed-point problems

Balooee

Yao

2022

J Inequal Appl

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This paper aims at proposing an iterative algorithm for finding an element in the intersection of the solutions set of a system of variational inclusions and the fixed-points set of a total uniformly L-Lipschitzian mapping. Applying the concepts of graph convergence and the resolvent operator associated with an Ĥ-accretive mapping, a new equivalence relationship between graph convergence and resolvent-operator convergence of a sequence of Ĥ-accretive mappings is established. As an application of the obtained equivalence relationship, the strong convergence of the sequence generated by our proposed iterative algorithm to a common point of the above two sets is proved under some suitable hypotheses imposed on the parameters and mappings. At the same time, the notion of $H(\cdot,\cdot)$ H ( ⋅ , ⋅ ) -accretive mapping that appeared in the literature, where $H(\cdot,\cdot)$ H ( ⋅ , ⋅ ) is an α, β-generalized accretive mapping, is also investigated and analyzed. We show that the notions $H(\cdot,\cdot)$ H ( ⋅ , ⋅ ) -accretive and Ĥ-accretive operators are actually the same, and point out some comments on the results concerning them that are available in the literature.

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Section: Introductionmentioning

confidence: 99%

Graph convergence with an application for system of variational inclusions and fixed-point problems

Balooee

Yao

2022

J Inequal Appl

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“…Many mathematical problems arising in different fields, such as optimization theory, game theory, economic equilibrium, mechanics and social sciences can be formulated as variational inequality / inclusion problems. During the past few decades, significant efforts have been made by many authors to propose and develop several effective methods to find the solutions of these problems, namely, projection method and its variant forms such as resolvent operator method, see, for example, [1][2][3][4][5] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

An iterative method for variational inclusions and fixed points of total uniformly $L$-Lipschitzian mappings

Ansari¹,

BALOOEE²,

Al-Homidan³

2022

CJM

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"The characterizations of $m$-relaxed monotone and maximal $m$-relaxed monotone operators are presented and by defining the resolvent operator associated with a maximal $m$-relaxed monotone operator, its Lipschitz continuity is proved and an estimate of its Lipschitz constant is computed. By using resolvent operator associated with a maximal $m$-relaxed monotone operator, an iterative algorithm is constructed for approximating a common element of the set of fixed points of a total uniformly $L$-Lipschitzian mapping and the set of solutions of a variational inclusion problem involving maximal $m$-relaxed monotone operators. By employing the concept of graph convergence for maximal $m$-relaxed monotone operators, a new equivalence relationship between the graph convergence of a sequence of maximal $m$-relaxed monotone operators and their associated resolvent operators, respectively, to a given maximal $m$-relaxed monotone operator and its associated resolvent operator is established. At the end, we study the strong convergence of the sequence generated by the proposed iterative algorithm to a common element of the above mentioned sets."

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“…Specially, one of the significant generalizations of variational inequality is the general variational inequality which was introduced and investigated by Noor [ 9 ]. Subsequently, Balooee et al [ 10 , 11 ] introduced an algorithm for solving the extended general mixed variational inequalities. However, most of the results related to the existence of solutions and iterative methods for variational inequality problems have been investigated and considered so far to the case where the underlying set is convex.…”

Section: Introductionmentioning

confidence: 99%

Convergence analysis of an iterative algorithm for the extended regularized nonconvex variational inequalities

2017

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In this paper, we suggest and analyze a new system of extended regularized nonconvex variational inequalities and prove the equivalence between the aforesaid system and a fixed point problem. We introduce a new perturbed projection iterative algorithm with mixed errors to find the solution of the system of extended regularized nonconvex variational inequalities. Furthermore, under moderate assumptions, we research the convergence analysis of the suggested iterative algorithm.

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Algorithms for solutions of extended general mixed variational inequalities and fixed points

Cited by 24 publications

References 44 publications

Graph convergence with an application for system of variational inclusions and fixed-point problems

Graph convergence with an application for system of variational inclusions and fixed-point problems

An iterative method for variational inclusions and fixed points of total uniformly $L$-Lipschitzian mappings

Convergence analysis of an iterative algorithm for the extended regularized nonconvex variational inequalities

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