• Title/Summary/Keyword: Fixed point theorems

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INCLUSION AND INTERSECTION THEOREMS WITH APPLICATIONS IN EQUILIBRIUM THEORY IN G-CONVEX SPACES

  • Balaj, Mircea;O'Regan, Donal
    • Journal of the Korean Mathematical Society
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    • v.47 no.5
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    • pp.1017-1029
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    • 2010
  • In this paper we obtain a very general theorem of $\rho$-compatibility for three multivalued mappings, one of them from the class $\mathfrak{B}$. More exactly, we show that given a G-convex space Y, two topological spaces X and Z, a (binary) relation $\rho$ on $2^Z$ and three mappings P : X $\multimap$ Z, Q : Y $\multimap$ Z and $T\;{\in}\;\mathfrak{B}$(Y,X) satisfying a set of conditions we can find ($\widetilde{x},\;\widetilde{y}$) ${\in}$ $X\;{\times}\;Y$ such that $\widetilde{x}\;{\in}\;T(\widetilde{y})$ and $P(\widetilde{x}){\rho}\;Q(\widetilde{y})$. Two particular cases of this general result will be then used to establish existence theorems for the solutions of some general equilibrium problems.

CONVERGENCE THEOREMS OF A FINITE FAMILY OF ASYMPTOTICALLY QUASI-NONEXPANSIVE TYPE MAPPINGS IN BANACH SPACES

  • Saluja, Gurucharan Singh
    • East Asian mathematical journal
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    • v.27 no.1
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    • pp.35-49
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    • 2011
  • In this paper, we study multi-step iterative algorithm with errors and give the necessary and sufficient condition to converge to com mon fixed points for a finite family of asymptotically quasi-nonexpansive type mappings in Banach spaces. Also we have proved a strong convergence theorem to converge to common fixed points for a finite family said mappings on a nonempty compact convex subset of a uniformly convex Banach spaces. Our results extend and improve the corresponding results of [2, 4, 7, 8, 9, 10, 12, 15, 20].

STRONG CONVERGENCE THEOREMS FOR ASYMPTOTICALLY QUASI-NONEXPANSIVE MAPPINGS AND INVERSE-STRONGLY MONOTONE MAPPINGS

  • He, Xin-Feng;Xu, Yong-Chun;He, Zhen
    • East Asian mathematical journal
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    • v.27 no.1
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    • pp.1-9
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    • 2011
  • In this paper, we consider an iterative scheme for finding a common element of the set of fixed points of a asymptotically quasi nonexpansive mapping and the set of solutions of the variational inequality for an inverse strongly monotone mapping in a Hilbert space. Then we show that the sequence converges strongly to a common element of two sets. Using this result, we consider the problem of finding a common fixed point of a asymptotically quasi-nonexpansive mapping and strictly pseudocontractive mapping and the problem of finding a common element of the set of fixed points of a asymptotically quasi-nonexpansive mapping and the set of zeros of an inverse-strongly monotone mapping.

APPROXIMATING COMMON FIXED POINTS OF ONE-STEP ITERATIVE SCHEME WITH ERROR FOR NON-SELF ASYMPTOTICALLY NONEXPANSIVE IN THE INTERMEDIATE SENSE MAPPINGS

  • Saluja, Gurucharan Singh;Nashine, Hemant Kumar
    • East Asian mathematical journal
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    • v.26 no.3
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    • pp.429-440
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    • 2010
  • In this paper, we study a new one-step iterative scheme with error for approximating common fixed points of non-self asymptotically nonexpansive in the intermediate sense mappings in uniformly convex Banach spaces. Also we have proved weak and strong convergence theorems for above said scheme. The results obtained in this paper extend and improve the recent ones, announced by Zhou et al. [27] and many others.

STRONG CONVERGENCE THEOREMS FOR NONEXPANSIVE MAPPINGS AND INVERSE-STRONGLY-MONOTONE MAPPINGS IN A BANACH SPACE

  • Liu, Ying
    • East Asian mathematical journal
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    • v.26 no.5
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    • pp.627-639
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    • 2010
  • In this paper, we introduce a new iterative sequence finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for an inverse-strongly-monotone mapping in a Banach space. Then we show that the sequence converges strongly to a common element of two sets. Using this result, we consider the problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of zeros of an inverse-strongly-monotone mapping, the fixed point problem and the classical variational inequality problem. Our results improve and extend the corresponding results announced by many others.

DEMI-CLOSED PRINCIPLE AND WEAK CONVERGENCE PROBLEMS FOR ASYMPTOTICALLY NONEXPANSIVE MAPPINGS

  • Chang, Shih-Sen;Cho, Yeol-Je;Haiyun Zhou
    • Journal of the Korean Mathematical Society
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    • v.38 no.6
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    • pp.1245-1260
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    • 2001
  • A demi-closed theorem and some new weak convergence theorems of iterative sequences for asymptotically nonexpansive and nonexpansive mappings in Banach spaces are obtained. The results presented in this paper improve and extend the corresponding results of [1],[8]-[10],[12],[13],[15],[16], and [18].

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INVESTIGATION OF SOME FIXED POINT THEOREMS IN HYPERBOLIC SPACES FOR A THREE STEP ITERATION PROCESS

  • Atalan, Yunus;Karakaya, Vatan
    • Korean Journal of Mathematics
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    • v.27 no.4
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    • pp.929-947
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    • 2019
  • In the present paper, we investigate the convergence, equivalence of convergence, rate of convergence and data dependence results using a three step iteration process for mappings satisfying certain contractive condition in hyperbolic spaces. Also we give nontrivial examples for the rate of convergence and data dependence results to show effciency of three step iteration process. The results obtained in this paper may be interpreted as a refinement and improvement of the previously known results.

A HYBRID PROJECTION METHOD FOR COMMON ZERO OF MONOTONE OPERATORS IN HILBERT SPACES

  • Truong, Minh Tuyen
    • Communications of the Korean Mathematical Society
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    • v.32 no.2
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    • pp.447-456
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    • 2017
  • The purpose of this paper is to introduce some strong convergence theorems for the problem of finding a common zero of a finite family of monotone operators and the problem of finding a common fixed point of a finite family of nonexpansive in Hilbert spaces by hybrid projection method.

HAUSDORFF TOPOLOGY INDUCED BY THE FUZZY METRIC AND THE FIXED POINT THEOREMS IN FUZZY METRIC SPACES

  • WU, HSIEN-CHUNG
    • Journal of the Korean Mathematical Society
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    • v.52 no.6
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    • pp.1287-1303
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    • 2015
  • The Hausdorff topology induced by a fuzzy metric space under more weak assumptions is investigated in this paper. Another purpose of this paper is to obtain the Banach contraction theorem in fuzzy metric space based on a natural concept of Cauchy sequence in fuzzy metric space.

STRONG CONVERGENCE OF AN ITERATIVE ALGORITHM FOR SYSTEMS OF VARIATIONAL INEQUALITIES AND FIXED POINT PROBLEMS IN q-UNIFORMLY SMOOTH BANACH SPACES

  • Jeong, Jae Ug
    • Korean Journal of Mathematics
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    • v.20 no.2
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    • pp.225-237
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    • 2012
  • In this paper, we introduce a new iterative scheme to investigate the problem of nding a common element of nonexpansive mappings and the set of solutions of generalized variational inequalities for a $k$-strict pseudo-contraction by relaxed extra-gradient methods. Strong convergence theorems are established in $q$-uniformly smooth Banach spaces.