• Title/Summary/Keyword: Opial’s condition

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WEAK CONVERGENCE THEOREMS FOR ALMOST-ORBITS OF AN ASYMPTOTICALLY NONEXPANSIVE SEMIGROUP IN BANACH SPACES

  • Kim, J.K.;Nam, Y.M.;Jin, B.J.
    • Communications of the Korean Mathematical Society
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    • v.13 no.3
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    • pp.501-513
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    • 1998
  • In this paper, we deal with the asymptotic behavior for the almost-orbits {u(t)} of an asymptotically nonexpansive semigroup S = {S(t) : t $\in$ G} for a right reversible semitopological semigroup G, defined on a suitable subset C of Banach spaces with the Opial's condition, locally uniform Opial condition, or uniform Opial condition.

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FIXED POINTS OF SUMS OF NONEXPANSIVE MAPS AND COMPACT MAPS

  • Bae, Jongsook;An, Daejong
    • Korean Journal of Mathematics
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    • v.10 no.1
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    • pp.19-23
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    • 2002
  • Let X be a Banach space satisfying Opial's condition, C a weakly compact convex subset of $X,F:C{\rightarrow}X$ a nonexpansive map, and let $G:C{\rightarrow}X$ be a compact and demiclosed map. We prove that F + G has a fixed point in C if $F+G:C{\rightarrow}X$ is a weakly inward map.

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APPROXIMATING COMMON FIXED POINTS FOR TOTAL ASYMPTOTICALLY NONEXPANSIVE MAPPINGS

  • Kim, Gang-Eun
    • Journal of applied mathematics & informatics
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    • v.30 no.1_2
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    • pp.71-82
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    • 2012
  • In this paper, we first show the weak convergence of the modified Ishikawa iteration process with errors of two total asymptotically nonexpansive mappings, which generalizes the result due to Khan and Fukhar-ud-din [1]. Next, we show the strong convergence of the modified Ishikawa iteration process with errors of two total asymptotically nonexpansive mappings satisfying Condition ($\mathbf{A}^{\prime}$), which generalizes the result due to Fukhar-ud-din and Khan [2].

WEAK AND STRONG CONVERGENCE OF THREE-STEP ITERATIONS WITH ERRORS FOR TWO ASYMPTOTICALLY NONEXPANSIVE MAPPINGS

  • Jeong, Jae-Ug
    • Journal of applied mathematics & informatics
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    • v.26 no.1_2
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    • pp.325-336
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    • 2008
  • In this paper, we prove the weak and strong convergence of the three-step iterative scheme with errors to a common fixed point for two asymptotically nonexpansive mappings in a uniformly convex Banach space under a condition weaker than compactness. Our theorems improve and generalize some previous results.

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Fixed Point Theorems for Multivalued Mappings in Banach Spaces

  • Bae, Jong Sook;Park, Myoung Sook
    • Journal of the Chungcheong Mathematical Society
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    • v.3 no.1
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    • pp.103-110
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    • 1990
  • Let K be a nonempty weakly compact convex subset of a Banach space X and T : K ${\rightarrow}$ C(X) a nonexpansive mapping satisfying $P_T(x){\cap}clI_K(x){\neq}{\emptyset}$. We first show that if I - T is semiconvex type then T has a fixed point. Also we obtain the same result without the condition that I - T is semiconvex type in a Banach space satisfying Opial's condition. Lastly we extend the result of [5] to the case, that T is an 1-set contraction mapping.

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WEAK AND STRONG CONVERGENCE FOR QUASI-NONEXPANSIVE MAPPINGS IN BANACH SPACES

  • Kim, Gang-Eun
    • Bulletin of the Korean Mathematical Society
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    • v.49 no.4
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    • pp.799-813
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    • 2012
  • In this paper, we first show that the iteration {$x_n$} defined by $x_{n+1}=P((1-{\alpha}_n)x_n +{\alpha}_nTP[{\beta}_nTx_n+(1-{\beta}_n)x_n])$ converges strongly to some fixed point of T when E is a real uniformly convex Banach space and T is a quasi-nonexpansive non-self mapping satisfying Condition A, which generalizes the result due to Shahzad [11]. Next, we show the strong convergence of the Mann iteration process with errors when E is a real uniformly convex Banach space and T is a quasi-nonexpansive self-mapping satisfying Condition A, which generalizes the result due to Senter-Dotson [10]. Finally, we show that the iteration {$x_n$} defined by $x_{n+1}={\alpha}_nSx_n+{\beta}_nT[{\alpha}^{\prime}_nSx_n+{\beta}^{\prime}_nTx_n+{\gamma}^{\prime}_n{\upsilon}_n]+{\gamma}_nu_n$ converges strongly to a common fixed point of T and S when E is a real uniformly convex Banach space and T, S are two quasi-nonexpansive self-mappings satisfying Condition D, which generalizes the result due to Ghosh-Debnath [3].