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Approximating fixed points of a countable family of strict pseudocontractions in Banach spaces

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Języki publikacji
EN
Abstrakty
EN
We prove the strong convergence of the modified Mann-type iterative scheme for a countable family of strict pseudocontractions in q-uniformly smooth Banach spaces. Our results mainly improve and extend the results announced in [Y. Yao, H. Zhou, Y.-C. Liou, Strong convergence of a modified Krasnoselski-Mann iterative algorithm for non-expansive mappings, J. Appl. Math. Comput. 29 (2009), 383–389].
Rocznik
Strony
67--79
Opis fizyczny
Bibliogr. 37 poz.
Twórcy
  • University of Phayao School of Science Phayao 56 000, Thailand
Bibliografia
  • [1] R.P. Agarwal, D. O’Regan, D.R. Sahu, Fixed Point Theory for Lipschitzian-type Mappings with Applications, Springer, 2009.
  • [2] K. Aoyama, Y. Kimura, W. Takahashi, M. Toyoda, Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space, Nonlinear Anal. 67 (2007), 2350–2360.
  • [3] D. Boonchari, S. Saejung,Weak and strong convergence theorems of an implicit iteration for a countable family of continuous pseudocontractive mappings, J. Comput. Appl. Math. 233 (2009), 1108–1116.
  • [4] D. Boonchari, S. Saejung, Construction of common fixed points of a countable family of _-demicontractiove mappings in arbitrary Banach spaces, Appl. Math. Comput. 216 (2010), 173–178.
  • [5] F.E. Browder, W.V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl. 20 (1967), 197–228.
  • [6] R.E. Bruck, Nonexpansive projections on subsets of Banach spaces, Pacific J. Math. 47 (1973), 341–355.
  • [7] L.C. Ceng, D.S. Shyu, J.C. Yao, Relaxed composite implicit iteration process for common fixed points of a finite family of strictly pseudocontractive mappings, Fixed Point Theory Appl. 2009 (2009), Art. ID 402602, 16 pages.
  • [8] L.C. Ceng, A. Petrusel, J.C. Yao, Strong convergence of modified implicit iterative algorithms with perturbed mappings for continuous pseudocontractive mappings, Appl. Math. Comput. 209 (2009), 162–176.
  • [9] L.C. Ceng, S. Al-Homidan, Q.H. Ansari, J.C. Yao, An iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings, J. Comput. Appl. Math. 223 (2009), 967–974.
  • [10] R. Chen, Y. Song, H. Zhou, Convergence theorems for implicit iteration process for a finite family of continuous pseudocontractive mappings, J. Math. Anal. Appl. 314 (2006), 701–709.
  • [11] C.E. Chidume, Geometric Properties of Banach Spaces and Nonlinear Iterations, [in:] Springer Lecture Notes Series, 2009.
  • [12] C.E. Chidume, C.O. Chidume, Iterative approximation of fixed points of nonexpansive mappings, J. Math. Anal. Appl. 318 (2006), 288–295.
  • [13] K. Goebel, S. Reich, Uniform convexity, hyperbolic geometry, and nonexpansive mappings Marcel Dekker, New York, 1984.
  • [14] J.P. Gossez, D.E. Lami, Some geometric properties related to the fixed point theory for nonexpansive mappings, Pacific J. Math. 40 (1972), 565–573.
  • [15] B. Halpern, Fixed points of nonexpanding maps, Bull. Amer. Math. Soc. 73 (1967), 957–961.
  • [16] T.H. Kim, H.K. Xu, Strong convergence of modified Mann iterations, Nonlinear Anal. 61 (2005), 51–60.
  • [17] P.L. Lions, Approximation de points fixes de contractions, C.R. Acad. Sci. Paris Ser. A-B 284 (1977), A1357–A1359.
  • [18] W.R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506–510.
  • [19] G. Marino, H.K. Xu, Weak and strong convergence theorems for strictly pseudocontractions in Hilbert spaces, J. Math. Anal. Appl. 339 (2007), 336–349.
  • [20] Z. Opial, Weak convergence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967), 591–597.
  • [21] M.O. Osilike, A. Udomene, Demiclosedness principle and convergence results for strictly pseudocontractive mappings of Browder-Petryshyn type, J. Math. Anal. Appl. 256 (2001), 431–445.
  • [22] J.W. Peng, J.C. Yao, Ishikawa iterative algorithms for a generalized equilibrium problem and fixed point problems of a pseudo-contraction mapping, J. Glob. Optim. 46 (2010), 331–345.
  • [23] S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 67 (1979), 274–276.
  • [24] S. Reich, Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl. 75 (1980), 287–292.
  • [25] S. Reich, Approximating fixed points of nonexpansive mappings, PanAmer. Math. J. 4 (1994) 2, 23–28.
  • [26] N. Shioji, W. Takahashi, Strong convergence of approximated sequences for nonexpansive mapping in Banach spaces, Proc. Amer. Math. Soc. 125 (1997), 3641–3645.
  • [27] R. Wittmann, Approximation of fixed points of nonexpansive mappings, Arch. Math. (Basel) 58 (1992), 486–491.
  • [28] H.K. Xu, Another control condition in iterative method for nonexpansive mappings, Bull. Astral. Math. Soc. 65 (2002), 109–113.
  • [29] H.K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc. 66 (2002), 240–256.
  • [30] H.K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal. 16 (1991), 1127–1138.
  • [31] Y. Yao, H. Zhou, Y.-C. Liou, Strong convergence of a modified Krasnoselski-Mann iterative algorithm for non-expansive mappings, J. Appl. Math. Comput. 29 (2009), 383–389.
  • [32] Y. Zhang, Y. Guo, Weak convergence theorems of three iterative methods for strictly pseudocontractive mappings of Browder-Petryshyn type, Fixed Point Theory Appl. 2008Q (2008), Art. ID 672301, 13 pages.
  • [33] H. Zhang, Y. Su, Convergence theorems for strict pseudo-contractions in q-uniformly smooth Banach spaces, Nonlinear Anal. 71 (2009), 4572–4580.
  • [34] H. Zhou, Convergence theorems of fixed points for Lipschitz pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl. 343 (2008), 546–556.
  • [35] H. Zhou, Convergence theorems of common fixed points for a finite family of Lipschitz pseudocontractions in Banach spaces, Nonlinear Anal. 68 (2008), 2977–2983.
  • [36] H. Zhou, Convergence teorems for λ-strict pseudocontractions in 2-uniformly smooth Banach spaces, Nonlinear Anal. 69 (2008), 3160–3173.
  • [37] H. Zhou, Convergence theorems for λ-strict pseudo-contractions in q-uniformly smooth Banach spaces, Acta Math. Sin., Engl. Ser. 26 (2010), 743–758.
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Bibliografia
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