• Title/Summary/Keyword: generalized resolvent operator

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SYSTEM OF GENERALIZED MULTI-VALUED RESOLVENT EQUATIONS: ALGORITHMIC AND ANALYTICAL APPROACH

  • Javad Balooee;Shih-sen Chang;Jinfang Tang
    • Bulletin of the Korean Mathematical Society
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    • v.60 no.3
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    • pp.785-827
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    • 2023
  • In this paper, under some new appropriate conditions imposed on the parameter and mappings involved in the resolvent operator associated with a P-accretive mapping, its Lipschitz continuity is proved and an estimate of its Lipschitz constant is computed. This paper is also concerned with the construction of a new iterative algorithm using the resolvent operator technique and Nadler's technique for solving a new system of generalized multi-valued resolvent equations in a Banach space setting. The convergence analysis of the sequences generated by our proposed iterative algorithm under some appropriate conditions is studied. The final section deals with the investigation and analysis of the notion of H(·, ·)-co-accretive mapping which has been recently introduced and studied in the literature. We verify that under the conditions considered in the literature, every H(·, ·)-co-accretive mapping is actually P-accretive and is not a new one. In the meanwhile, some important comments on H(·, ·)-co-accretive mappings and the results related to them appeared in the literature are pointed out.

Sensitivity Analysis for Generalized Nonlinear Implicit Quasi-variational Inclusions

  • Jeong, Jae Ug
    • Kyungpook Mathematical Journal
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    • v.46 no.3
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    • pp.345-356
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    • 2006
  • In this paper, by using the concept of the resolvent operator, we study the behavior and sensitivity analysis of the solution set for a new class of parametric generalized nonlinear implicit quasi-variational inclusion problem in $L_p(p{\geq}2)$ spaces. The results presented in this paper are new and generalize many known results in this field.

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ON THE GENERALIZED SET-VALUED MIXED VARIATIONAL INEQUALITIES

  • Zhao, Yali;Liu, Zeqing;Kang, Shin-Min
    • Communications of the Korean Mathematical Society
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    • v.18 no.3
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    • pp.459-468
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    • 2003
  • In this paper, we introduce and study a new class of the generalized set-valued mixed variational inequalities. Using the resolvent operator technique, we construct a new iterative algorithm for solving this class of the generalized set-valued mixed variational inequalities. We prove the existence of solutions for the generalized set-valued mixed variational inequalities and the convergence of the iterative sequences generated by the algorithm.

A MODIFIED PROXIMAL POINT ALGORITHM FOR SOLVING A CLASS OF VARIATIONAL INCLUSIONS IN BANACH SPACES

  • LIU, YING
    • Journal of applied mathematics & informatics
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    • v.33 no.3_4
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    • pp.401-415
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    • 2015
  • In this paper, we propose a modified proximal point algorithm which consists of a resolvent operator technique step followed by a generalized projection onto a moving half-space for approximating a solution of a variational inclusion involving a maximal monotone mapping and a monotone, bounded and continuous operator in Banach spaces. The weak convergence of the iterative sequence generated by the algorithm is also proved.

A PROXIMAL POINT ALGORITHM FOR SOLVING THE GENERAL VARIATIONAL INCLUSIONS WITH M(·, ·)-MONOTONE OPERATORS IN BANACH SPACES

  • Chen, Junmin;Wang, Xian;He, Zhen
    • East Asian mathematical journal
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    • v.29 no.3
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    • pp.315-326
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    • 2013
  • In this paper, a new monotonicity, $M({\cdot},{\cdot})$-monotonicity, is introduced in Banach spaces, and the resolvent operator of an $M({\cdot},{\cdot})$-monotone operator is proved to be single valued and Lipschitz continuous. By using the resolvent operator technique associated with $M({\cdot},{\cdot})$-monotone operators, we construct a proximal point algorithm for solving a class of variational inclusions. And we prove the convergence of the sequences generated by the proximal point algorithms in Banach spaces. The results in this paper extend and improve some known results in the literature.

SOLVABILITY FOR A SYSTEM OF GENERALIZED NONLINEAR ORDERED VARIATIONAL INCLUSIONS IN ORDERED BANACH SPACES

  • Salahuddin, Salahuddin
    • Korean Journal of Mathematics
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    • v.25 no.3
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    • pp.359-377
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    • 2017
  • In this paper, we consider a system of generalized nonlinear ordered variational inclusions in real ordered Banach spaces and define an iterative algorithm for a solution of our problems. By using the resolvent operator techniques to prove an existence result for the solution of the system of generalized nonlinear ordered variational inclusions and discuss convergence of sequences suggested by the algorithms.

PARAMETRIC GENERALIZED MIXED IMPLICIT QUASI-VARIATIONAL INCLUSIONS

  • Park, Jong-Yeoul;Jeong, Jae-Ug
    • Journal of the Korean Mathematical Society
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    • v.44 no.4
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    • pp.889-902
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    • 2007
  • An existence theorem for a new class of parametric generalized mixed implicit quasi-variational inclusion problems is established in Hilbert spaces. Further, we study the behavior and sensitivity analysis of the solution set in this class of parametric variational inclusion problems.

PARAMETRIC GENERALIZED MULTI-VALUED NONLINEAR QUASI-VARIATIONAL INCLUSION PROBLEM

  • Khan, F.A.;Alanazi, A.M.;Ali, Javid;Alanazi, Dalal J.
    • Nonlinear Functional Analysis and Applications
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    • v.26 no.5
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    • pp.917-933
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    • 2021
  • In this paper, we investigate the behavior and sensitivity analysis of a solution set for a parametric generalized multi-valued nonlinear quasi-variational inclusion problem in a real Hilbert space. For this study, we utilize the technique of resolvent operator and the property of a fixed-point set of a multi-valued contractive mapping. We also examine Lipschitz continuity of the solution set with respect to the parameter under some appropriate conditions.

GENERAL FRAMEWORK FOR PROXIMAL POINT ALGORITHMS ON (A, η)-MAXIMAL MONOTONICIT FOR NONLINEAR VARIATIONAL INCLUSIONS

  • Verma, Ram U.
    • Communications of the Korean Mathematical Society
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    • v.26 no.4
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    • pp.685-693
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    • 2011
  • General framework for proximal point algorithms based on the notion of (A, ${\eta}$)-maximal monotonicity (also referred to as (A, ${\eta}$)-monotonicity in literature) is developed. Linear convergence analysis for this class of algorithms to the context of solving a general class of nonlinear variational inclusion problems is successfully achieved along with some results on the generalized resolvent corresponding to (A, ${\eta}$)-monotonicity. The obtained results generalize and unify a wide range of investigations readily available in literature.

GRAPH CONVERGENCE AND GENERALIZED CAYLEY OPERATOR WITH AN APPLICATION TO A SYSTEM OF CAYLEY INCLUSIONS IN SEMI-INNER PRODUCT SPACES

  • Mudasir A. Malik;Mohd Iqbal Bhat;Ho Geun Hyun
    • Nonlinear Functional Analysis and Applications
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    • v.28 no.1
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    • pp.265-286
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    • 2023
  • In this paper, we introduce and study a generalized Cayley operator associated to H(·, ·)-monotone operator in semi-inner product spaces. Using the notion of graph convergence, we give the equivalence result between graph convergence and convergence of generalized Cayley operator for the H(·, ·)-monotone operator without using the convergence of the associated resolvent operator. To support our claim, we construct a numerical example. As an application, we consider a system of generalized Cayley inclusions involving H(·, ·)-monotone operators and give the existence and uniqueness of the solution for this system. Finally, we propose a perturbed iterative algorithm for finding the approximate solution and discuss the convergence of iterative sequences generated by the perturbed iterative algorithm.