Viscosity Approximation Methods for Multivalued Nonexpansive Mappings

Multivalued $$*$$ ∗ -nonexpansive mappings are studied in Banach spaces. The demiclosedness principle is established. Here we focus on the problem of solving a variational inequality which is defined on the set of fixed points of a multivalued $$*$$ ∗ -nonexpansive mapping. For this purpose, we introduce two algorithms approximating the unique solution of the variational inequality.

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“…These problems are widely studied as for the single-valued as for the multivalued case; for details one should refer to [5,17,19,25,29].…”

Section: Introductionmentioning

confidence: 99%

Some Results on the Approximation of Solutions of Variational Inequalities for Multivalued Maps on Banach Spaces

Muglia

Marino

2021

Mediterr. J. Math.

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“…Let us note that, in their paper, the authors assumed that T is single valued on F ix(T ). Wu and Zhao [23] studied a viscosity method in the setting of uniformly convex and smooth Banach spaces; Panyanak and Suantai [24] considered the viscosity approximation method in geodesic spaces.…”

Section: Introductionmentioning

confidence: 99%

On solving variational inequalities defined on fixed point sets of multivalued mappings in Banach spaces

Muglia

2020

J. Fixed Point Theory Appl.

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We are concerned with the problem of solving variational inequalities which are defined on the set of fixed points of a multivalued nonexpansive mapping in a reflexive Banach space. Both implicit and explicit approaches are studied. Strong convergence of the implicit method is proved if the space satisfies Opial’s condition and has a duality map weakly continuous at zero, and the strong convergence of the explicit method is proved if the space has a weakly continuous duality map. An essential assumption on the multivalued nonexpansive mapping is that the mapping be single valued on its nonempty set of fixed points.

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“…Variational inequality theory and fixed point theory are two important fields in non-linear analysis and optimization. Much attention has been given to developing implementable viscosity iterative methods for solving variational inequality problems, due to their applications in many real world problems, such as signal processing, saddle point problems, equilibrium problems, and game theory, in the frameworks of Hilbert spaces or Banach spaces; see [1][2][3][4][5][6][7][8][9] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Generalized Viscosity Implicit Iterative Process for Asymptotically Non-Expansive Mappings in Banach Spaces

Pan

Wang

2019

Mathematics

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In this paper, we propose a generalized viscosity implicit iterative method for asymptotically non-expansive mappings in Banach spaces. The strong convergence theorem of this algorithm is proved, which solves the variational inequality problem. Moreover, we provide some applications to zero-point problems and equilibrium problems. Further, a numerical example is given to illustrate our convergence analysis. The results generalize and improve corresponding results in the literature.

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Viscosity Approximation Methods for Multivalued Nonexpansive Mappings

Abstract: Based on some iteration schemes, we study the viscosity approximation results for multivalued nonexpansive mappings in Hilbert space and Banach space. For that mapping, we obtain a fixed point to solve its related variational inequality.

Cited by 5 publications

References 20 publications

Some Results on the Approximation of Solutions of Variational Inequalities for Multivalued Maps on Banach Spaces

Some Results on the Approximation of Solutions of Variational Inequalities for Multivalued Maps on Banach Spaces

On solving variational inequalities defined on fixed point sets of multivalued mappings in Banach spaces

Generalized Viscosity Implicit Iterative Process for Asymptotically Non-Expansive Mappings in Banach Spaces

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