• Title/Summary/Keyword: resolvent operator

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STRONG CONVERGENCE OF AN ITERATIVE METHOD FOR FINDING COMMON ZEROS OF A FINITE FAMILY OF ACCRETIVE OPERATORS

  • Jung, Jong-Soo
    • Communications of the Korean Mathematical Society
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    • v.24 no.3
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    • pp.381-393
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    • 2009
  • Strong convergence theorems on viscosity approximation methods for finding a common zero of a finite family accretive operators are established in a reflexive and strictly Banach space having a uniformly G$\hat{a}$teaux differentiable norm. The main theorems supplement the recent corresponding results of Wong et al. [29] and Zegeye and Shahzad [32] to the viscosity method together with different control conditions. Our results also improve the corresponding results of [9, 16, 18, 19, 25] for finite nonexpansive mappings to the case of finite pseudocontractive mappings.

CONVERGENCE AND STABILITY OF THREE-STEP ITERATIVE SCHEME WITH ERRORS FOR COMPLETELY GENERALIZED STRONGLY NONLINEAR QUASIVARIATIONAL INEQUALITIES

  • ZHANG FENGRONG;GAO HAIYAN;LIU ZEQING;KANG SHIN MIN
    • Journal of applied mathematics & informatics
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    • v.20 no.1_2
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    • pp.465-478
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    • 2006
  • In this paper, we introduce a new class of completely generalized strongly nonlinear quasivariational inequalities and establish its equivalence with a class of fixed point problems by using the resolvent operator technique. Utilizing this equivalence, we develop a three-step iterative scheme with errors, obtain a few existence theorems of solutions for the completely generalized non-linear strongly quasivariational inequality involving relaxed monotone, relaxed Lipschitz, strongly monotone and generalized pseudocontractive mappings and prove some convergence and stability results of the sequence generated by the three-step iterative scheme with errors. Our results include several previously known results as special cases.

RESOLVENT INEQUALITY OF LAPLACIAN IN BESOV SPACES

  • Han, Hyuk;Pak, Hee Chul
    • Journal of the Chungcheong Mathematical Society
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    • v.22 no.1
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    • pp.117-121
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    • 2009
  • For $1{\leq}p$, $q{\leq}{\infty}$ and $s{\in}\mathbb{R}$, it is proved that there exists a constant C > 0 such that for any $f{\in}B^{s+2}_{p,q}(\mathbb{R}^n)$ $${\parallel}f{\parallel}_{B^{s+2}_{p,q}(\mathbb{R}^n)}{\leq}C{\parallel}f\;-\;{\Delta}f{\parallel}_{B^{s}_{p,q}(\mathbb{R}^n)}$$, which tells us that the operator $I-\Delta$ is $B^{s+2}_{p,q}$-coercive on the Besov space $B^s_{p,q}$.

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GENERALIZED MULTIVALUED QUASIVARIATIONAL INCLUSIONS FOR FUZZY MAPPINGS

  • Liu, Zeqing;Ume, Jeong-Sheok;Kang, Shin-Min
    • The Pure and Applied Mathematics
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    • v.14 no.1 s.35
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    • pp.37-48
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    • 2007
  • In this paper, we introduce and study a class of generalized multivalued quasivariational inclusions for fuzzy mappings, and establish its equivalence with a class of fuzzy fixed-point problems by using the resolvent operator technique. We suggest a new iterative algorithm for the generalized multivalued quasivariational inclusions. Further, we establish a few existence results of solutions for the generalized multivalued quasivariational inclusions involving $F_r$-relaxed Lipschitz and $F_r$-strongly monotone mappings, and discuss the convergence criteria for the algorithm.

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RELAXED PROXIMAL POINT ALGORITHMS BASED ON A-AXIMAL RELAXED MONOTONICITY FRAMEWORKS WITH APPLICATIONS

  • Agarwal, Ravi P.;Verma, Ram U.
    • East Asian mathematical journal
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    • v.27 no.5
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    • pp.545-555
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    • 2011
  • Based on the A-maximal(m)-relaxed monotonicity frameworks, the approximation solvability of a general class of variational inclusion problems using the relaxed proximal point algorithm is explored, while generalizing most of the investigations, especially of Xu (2002) on strong convergence of modified version of the relaxed proximal point algorithm, Eckstein and Bertsekas (1992) on weak convergence using the relaxed proximal point algorithm to the context of the Douglas-Rachford splitting method, and Rockafellar (1976) on weak as well as strong convergence results on proximal point algorithms in real Hilbert space settings. Furthermore, the main result has been applied to the context of the H-maximal monotonicity frameworks for solving a general class of variational inclusion problems. It seems the obtained results can be used to generalize the Yosida approximation that, in turn, can be applied to first- order evolution inclusions, and can also be applied to Douglas-Rachford splitting methods for finding the zero of the sum of two A-maximal (m)-relaxed monotone mappings.

ITERATIVE ALGORITHM FOR RANDOM GENERALIZED NONLINEAR MIXED VARIATIONAL INCLUSIONS WITH RANDOM FUZZY MAPPINGS

  • Faizan Ahmad, Khan;Eid Musallam, Aljohani;Javid, Ali
    • Nonlinear Functional Analysis and Applications
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    • v.27 no.4
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    • pp.881-894
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    • 2022
  • In this paper, we consider a class of random generalized nonlinear mixed variational inclusions with random fuzzy mappings and random relaxed cocoercive mappings in real Hilbert spaces. We suggest and analyze an iterative algorithm for finding the approximate solution of this class of inclusions. Further, we discuss the convergence analysis of the iterative algorithm under some appropriate conditions. Our results can be viewed as a refinement and improvement of some known results in the literature.