• Title, Summary, Keyword: resolvent operator

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A RESOLVENT APPROACH FOR SOLVING A SET-VALUED VARIATIONAL INCLUSION PROBLEM USING WEAK-RRD SET-VALUED MAPPING

  • Ahmad, Iqbal;Ahmad, Rais;Iqbal, Javid
    • Korean Journal of Mathematics
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    • v.24 no.2
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    • pp.199-213
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    • 2016
  • The resolvent operator approach of [2] is applied to solve a set-valued variational inclusion problem in ordered Hilbert spaces. The resolvent operator under consideration is called relaxed resolvent operator and we demonstrate some of its properties. To obtain the solution of a set-valued variational inclusion problem, an iterative algorithm is developed and weak-RRD set-valued mapping is used. The problem as well as main result of this paper are more general than many previous problems and results available in the literature.

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.

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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VARIATIONAL-LIKE INCLUSION SYSTEMS VIA GENERAL MONOTONE OPERATORS WITH CONVERGENCE ANALYSIS

  • Dadashi, Vahid;Roohi, Mehdi
    • East Asian mathematical journal
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    • v.26 no.1
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    • pp.95-103
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    • 2010
  • In this paper using Lipschitz continuity of the resolvent operator associated with general H-maximal m-relaxed $\eta$-monotone operators, existence and uniqueness of the solution of a variational inclusion system is proved. Also, an iterative algorithm and its convergence analysis is given.

S-ASYMPTOTICALLY ω-PERIODIC MILD SOLUTIONS FOR THE SYSTEMS OF DIFFERENTIAL EQUATIONS WITH PIECEWISE CONSTANT ARGUMENT IN BANACH SPACES

  • Lee, Hyun Mork;Jang, Hyun Ho;Yun, Chan Mi
    • Journal of the Chungcheong Mathematical Society
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    • v.31 no.1
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    • pp.13-27
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    • 2018
  • By using of the Banach fixed point theorem, the theory of a strongly continuous semigroup of operators and resolvent operator, we investigate the existence and uniqueness of S-asymptotically ${\omega}-periodic$ mild solutions for some differential (integrodifferential) equations with piecewise constant argument when specially ${\omega}$ is an integer.

A SYSTEM OF NONLINEAR VARIATIONAL INCLUSIONS WITH GENERAL H-MONOTONE OPERATORS IN BANACH SPACES

  • Li, Jinsong;Wang, Wei;Cho, Min-Hyung;Kang, Shin-Min
    • East Asian mathematical journal
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    • v.26 no.5
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    • pp.671-680
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    • 2010
  • A system of nonlinear variational inclusions involving general H-monotone operators in Banach spaces is introduced. Using the resolvent operator technique, we suggest an iterative algorithm for finding approximate solutions to the system of nonlinear variational inclusions, and establish the existence of solutions and convergence of the iterative algorithm for the system of nonlinear variational inclusions.

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 SYSTEM OF NONLINEAR SET-VALUED IMPLICIT VARIATIONAL INCLUSIONS IN REAL BANACH SPACES

  • Bai, Chuanzhi;Yang, Qing
    • Communications of the Korean Mathematical Society
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    • v.25 no.1
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    • pp.129-137
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    • 2010
  • In this paper, we introduce and study a system of nonlinear set-valued implicit variational inclusions (SNSIVI) with relaxed cocoercive mappings in real Banach spaces. By using resolvent operator technique for M-accretive mapping, we construct a new class of iterative algorithms for solving this class of system of set-valued implicit variational inclusions. The convergence of iterative algorithms is proved in q-uniformly smooth Banach spaces. Our results generalize and improve the corresponding results of recent works.

AN ITERATIVE METHOD FOR NONLINEAR MIXED IMPLICIT VARIATIONAL INEQUALITIES

  • JEONG, JAE UG
    • Honam Mathematical Journal
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    • v.26 no.4
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    • pp.391-399
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    • 2004
  • In this paper, we develop an iterative algorithm for solving a class of nonlinear mixed implicit variational inequalities in Hilbert spaces. The resolvent operator technique is used to establish the equivalence between variational inequalities and fixed point problems. This equivalence is used to study the existence of a solution of nonlinear mixed implicit variational inequalities and to suggest an iterative algorithm for solving variational inequalities. In our results, we do not assume that the mapping is strongly monotone.

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