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Tytuł artykułu

Iterative construction of a common fixed point of finite families of nonlinear mappings

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Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Let K be a nonempty closed convex subset of a real reflexive Banach space E with uniformly Gâteuax differentiable norm. Let T1, T2, ..., Tm : K —> K be m Lipschitz mappings (for some m ∈ N) such that (wzór). We construct a new iteration process and prove that the iteration process converges strongly to a common fixed point of these mappings provided at least one of the mappings is pseudocontractive. We also obtain as easy corollaries convergence results for finite families of Lipschitz pseudocontractive mappings and nonexpansive mappings. Furthermore, We prove that a slight modification of our iteration process converges strongly to a common zero of a finite family of Lipschitz accretive operators. Our new iteration process and our method of proof are of independent interest.
Wydawca
Rocznik
Strony
59--77
Opis fizyczny
Bibliogr. 24 poz.
Twórcy
autor
autor
  • Department of Mathematics, Nnamdi Azikiwe University, Awka, Anambra State , Nigeria., euofoedu@yahoo.com
Bibliografia
  • [1] D. Borwein, J. M. Borwein, Fixed point iterations for real functions, J. Math. Anal. Appl. 157(1991), 112-126.
  • [2] F. E. Browder; Nonlinear equations of evolution and nonlinear accretive operators in Banach spaces, Bull. Amer. Math. Soc. 73 (1967), 470-475. Part 2 (1976).
  • [3] F. E. Browder, W. E. Petryshyn; Construction of fixed points of nonlinear mappings in Hilbert spaces, J. Math. Anal. Appl. 20(1967), 197-228.
  • [4] R. E. Bruck, A strong convergent iterative method for the solution of 0 ? Ax for a maximal monotone operator A in a Hilbert space, J. Math. Anal. Appl. 48 (1974), 114-126.
  • [5] W. L. Bynum; Normal structure coefficients for Banach spaces, Pacific J. Math. 86 (l980), 427-436.
  • [6] C. E. Chidume; Iterative approximation of fixed points of Lipschitz pseudocontractive maps, Proc. Amer. Math. Soc. 129 (8) (2001), 2245-2251.
  • [7] C. E. Chidume; On approximation of fixed points of nonexpansive mappings, Houston J. Math. 7 (1981), 345-355. |
  • [8] C. E. Chidume, C. Moore; Fixed point iterations for pseudocontractive maps, Proc. Amer. Math. Soc. 127 (1999), 1163-1170.
  • [9] C. E. Chidume, S. A. Mutangadura; An example on the Mann iteration method for Lipschitz pseudocontractions, Proc. Amer. Math. Soc. 129 (2001), 2359-2363.
  • [10] C. E. Chidume, H. Zegeye; Approximate fixed point sequences and convergence theorems for Lipschitz pseudocontractive maps, Proc. Amer. Math. Soc. 132 (2004), 831-840.
  • [11] T. L. Hicks, J. R. Kubicek, On the Mann iteration process in a Hilbert space, J. Math. Anal. Appl. 59 (l999), 498-504.
  • [12] S. Ishikawa; Fixed point by new iteration method, Proc. Amer. Math. Soc. 44 (1974), 147-150.
  • [13] T. Kato; Nonlinear semigroups and evolution equations, J. Math. Soc. Japan 19 (1967), 508-520.
  • [14] M. A. Krasnosel`skii; Two observations about the method of successive approximaations, Uspehi Math. Nauk. 10(1955), 123-127.
  • [15] W. A. Kirk; Locally nonexpansive mappings in Banach spaces, pp. 178-198, Lecture Notes in Math., 886, Springer-Verlag, Berlin, 1981.
  • [16] T. C. Lim; Characterization of normal structure, Proc. Amer. Math. Soc. 43 (1974), 313-319.
  • [17] T. C. Lim, H. K. Xu; Fixed point theorems for asymptotically nonexpansive mappings, Nonlinear Anal, TMA 2 (1994), 1345-1355.
  • [18] C. Moore, B. V.C. Nnoli; Iterative solution of nonlinear equations involving set-valued uniformly accretive operators, Comput. Math. Appl. 42 (2001), 131-140.
  • [19] W. R. Mann; Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506-510. MR 14: 988f.
  • [20] S. Schaefer; Über die Methode sukzessiver Approximation, Jber. Deutscher Math. Verein 59 (1957), 131-140.
  • [21] N. Shioji, W. Takahashi; Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces, Proc. Amer. Math. Soc. 125 (1997), 3641-3645.
  • [22] J. Schu; Approximation of fixed points of asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 112 (1) (1991), 143-151.
  • [23] W. Takahashi; Nonlinear Functional Analysis, Yokohama publishers, Yokohama, 2000.
  • [24] H. Zhou, L. Wei, Y. J. Cho; Strong convergence theorems on an iterative method for family of finite nonexpansive mappings in reflexive Banach spaces, Appl. Math. Comput. 173 (2006), 196-212.
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-article-LOD6-0017-0007
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