Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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WEAK AND STRONG CONVERGENCE FOR QUASI-NONEXPANSIVE MAPPINGS IN BANACH SPACES

  • Kim, Gang-Eun (Department of Applied Mathematics Pukyong National University)
  • Received : 2011.04.19
  • Published : 2012.07.31
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Abstract

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].

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References

  1. F. E. Browder, Semicontractive and semiaccretive nonlinear mappings in Banach spaces, Bull. Amer. Math. Soc. 74 (1968), 660-665. https://doi.org/10.1090/S0002-9904-1968-11983-4
  2. W. G. Dotson, On the Mann iterative process, Trans. Amer. Math. Soc. 149 (1970), 65-73. https://doi.org/10.1090/S0002-9947-1970-0257828-6
  3. M. K. Ghosh and L. Debnath, Approximating common fixed points of families of quasinonexpansive mappings, Internat. J. Math. Math. Sci. 18 (1995), no. 2, 287-292. https://doi.org/10.1155/S0161171295000354
  4. C. W. Groetsch, A note on segmenting Mann iterates, J. Math. Anal. Appl. 40 (1972), 369-372. https://doi.org/10.1016/0022-247X(72)90056-X
  5. S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44 (1974), 147-150. https://doi.org/10.1090/S0002-9939-1974-0336469-5
  6. M. Maiti and M. K. Ghosh, Approximating fixed points by Ishikawa iterates, Bull. Austral. Math. Soc. 40 (1989), no. 1, 113-117. https://doi.org/10.1017/S0004972700003555
  7. W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506-510. https://doi.org/10.1090/S0002-9939-1953-0054846-3
  8. Z. Opial, Weak convergence of the sequence of successive approximations for nonexpan- sive mappings, Bull. Amer. Math. Soc. 73 (1967), 591-597. https://doi.org/10.1090/S0002-9904-1967-11761-0
  9. J. Schu, Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bull. Austral. Math. Soc. 43 (1991), no. 1, 153-159. https://doi.org/10.1017/S0004972700028884
  10. H. F. Senter and W. G. Dotson, Approximating fixed points of nonexpansive mappings, Proc. Amer. Math. Soc. 44 (1974), 375-380. https://doi.org/10.1090/S0002-9939-1974-0346608-8
  11. N. Shahzad, Approximating fixed points of non-self nonexpansive mappings in Banach spaces, Nonlinear Anal. 61 (2005), no. 6, 1031-1039. https://doi.org/10.1016/j.na.2005.01.092
  12. K. K. Tan and H. K. Xu, Approximating fixed points of nonexpansive mappings by the Ishikawa Iteration process, J. Math. Anal. Appl. 178 (1993), no. 2, 301-308. https://doi.org/10.1006/jmaa.1993.1309
  13. Y. Xu, Ishikawa and Mann iterative processes with errors for nonlinear strongly accretive operator equations, J. Math. Anal. Appl. 224 (1998), no. 1, 91-101. https://doi.org/10.1006/jmaa.1998.5987

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