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2007 | 5 | 2 | 345-357
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Approximating common fixed points of asymptotically nonexpansive mappings by composite algorithm in Banach spaces

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EN
Let E be a uniformly convex Banach space and K a nonempty convex closed subset which is also a nonexpansive retract of E. Let T 1, T 2 and T 3: K → E be asymptotically nonexpansive mappings with {k n}, {l n} and {j n}. [1, ∞) such that Σn=1∞(k n − 1) < ∞, Σn=1∞(l n − 1) < ∞ and Σn=1∞(j n − 1) < ∞, respectively and F nonempty, where F = {x ∈ K: T 1x = T 2x = T 3 x} = x} denotes the common fixed points set of T 1, T 2 and T 3. Let {α n}, {α′ n} and {α″ n} be real sequences in (0, 1) and ∈ ≤ {α n}, {α′ n}, {α″ n} ≤ 1 − ∈ for all n ∈ N and some ∈ > 0. Starting from arbitrary x 1 ∈ K define the sequence {x n} by $$\left\{ \begin{gathered} z_n = P(\alpha ''_n T_3 (PT_3 )^{n - 1} x_n + (1 - \alpha ''_n )x_n ), \hfill \\ y_n = P(\alpha '_n T_2 (PT_2 )^{n - 1} z_n + (1 - \alpha '_n )x_n ), \hfill \\ x_{n + 1} = P(\alpha _n T_1 (PT_1 )^{n - 1} y_n + (1 - \alpha _n )x_n ). \hfill \\ \end{gathered} \right.$$ (i) If the dual E* of E has the Kadec-Klee property then {x n} converges weakly to a common fixed point p ∈ F; (ii) If T satisfies condition (A′) then {x n} converges strongly to a common fixed point p ∈ F.
Wydawca
Czasopismo
Rocznik
Tom
5
Numer
2
Strony
345-357
Opis fizyczny
Daty
wydano
2007-06-01
online
2007-06-01
Twórcy
autor
autor
Bibliografia
  • [1] C.E. Chidume, E.U. Ofoedu and H. Zegeye: “Strong and weak convergence theorems for asymptotically nonexpansive mappings”, J. Math., Anal. Appl., Vol. 280, (2003), pp. 364–374. http://dx.doi.org/10.1016/S0022-247X(03)00061-1
  • [2] W.J. Davis and P. Enflo: Contractive projections on l p -spaces, Analysis at Urbana 1, Cambridge University Press, New York, 1989, pp. 151–161.
  • [3] J.G. Falset, W. Kaczor, T. Kuczumow and S. Reich:, “Weak convergence theorems for asymptotically nonexpansive mappings and semigroups”, Nonlinear Anal., Vol. 43, (2001), pp. 377–401. http://dx.doi.org/10.1016/S0362-546X(99)00200-X
  • [4] K. Goebel and W.A. Kirk: “A fixed point theorem for asymptotically nonexpansive mappings”, Proc. Amer. Math. Soc., Vol. 35, (1972), pp. 171–174. http://dx.doi.org/10.2307/2038462
  • [5] W. Kaczor: “Weak convergence of almost orbits of asymptotically nonexpansive commutative semigroups”, J. Math. Anal. Appl., Vol. 272, (2002), pp. 565–574. http://dx.doi.org/10.1016/S0022-247X(02)00175-0
  • [6] M. Maiti and M.K. Gosh: “Approximating fixed points by Ishikawa iterates”, Bull. Austral. Math. Soc., Vol. 40, (1989), pp. 113–117.
  • [7] M.O. Osilike and S.C. Aniagbosor: “Weak and strong convergence theorems for fixed points for asymptotically nonexpansive mappings”, Math. Comput. Modelling, Vol. 32, (2000), pp. 1181–1191. http://dx.doi.org/10.1016/S0895-7177(00)00199-0
  • [8] B.E. Rhoades: “Fixed point iterations for certain nonlinear mappings”, J. Math. Anal. Appl., Vol. 183, (1994), pp. 118–120. http://dx.doi.org/10.1006/jmaa.1994.1135
  • [9] H.F. Senter and W.G. Doston: “Approximating fixed points of nonexpansive mapping”, Proc. Amer. Math. Soc., Vol. 44(2), (1974), pp. 375–380. http://dx.doi.org/10.2307/2040440
  • [10] J. Schu: “Iterative construction of fixed points of asymptotically nonexpansive mappings”, J. Math. Anal. Appl., Vol. 158, (1991), pp. 407–413. http://dx.doi.org/10.1016/0022-247X(91)90245-U
  • [11] J. Schu: “Weak and strong convergence to fixed points of asymptotically nonexpansive mappings”, Bull. Austral. Math. Soc., Vol. 43, (1991), pp. 153–159. http://dx.doi.org/10.1017/S0004972700028884
  • [12] N. Shahzad: “Approximating fixed points of non-self nonexpansive mappings in Banach spaces”, Nonlinear Anal., Vol. 61, (2005), pp. 1031–1039. http://dx.doi.org/10.1016/j.na.2005.01.092
  • [13] K.K. Tan and H.K. Xu: “Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process”, J. Math. Anal. Appl., Vol. 178, (1993), pp. 301–308. http://dx.doi.org/10.1006/jmaa.1993.1309
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Bibliografia
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