Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Approximation of fixed points of strongly pseudocontractive mappings


Author: C. E. Chidume
Journal: Proc. Amer. Math. Soc. 120 (1994), 545-551
MSC: Primary 47H10; Secondary 47H09, 47H15
DOI: https://doi.org/10.1090/S0002-9939-1994-1165050-6
MathSciNet review: 1165050
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ E$ be a real Banach space with a uniformly convex dual, and let $ K$ be a nonempty closed convex and bounded subset of $ E$. Let $ T:K \to K$ be a continuous strongly pseudocontractive mapping of $ K$ into itself. Let $ \{ {c_n}\} _{n = 1}^\infty $ be a real sequence satisfying: (i) $ 0 < {c_n} < 1$ for all $ n \geqslant 1$; (ii) $ \sum\nolimits_{n = 1}^\infty {{c_n} = \infty } $; and (iii) $ \sum\nolimits_{n = 1}^\infty {{c_n}b({c_n}) < \infty } $, where $ b:[0,\infty ) \to [0,\infty )$ is some continuous nondecreasing function satisfying $ b(0) = 0,\,b(ct) \leqslant cb(t)$ for all $ c \geqslant 1$. Then the sequence $ \{ {x_n}\} _{n = 1}^\infty $ generated by $ {x_1} \in K$,

$\displaystyle {x_{n + 1}} = (1 - {c_n}){x_n} + {c_n}T{x_n},\qquad n \geqslant 1,$

converges strongly to the unique fixed point of $ T$. A related result deals with the Ishikawa iteration scheme when $ T$ is Lipschitzian and strongly pseudocontractive.

References [Enhancements On Off] (What's this?)

  • [1] N. A. Assad and W. A. Kirk, Fixed point theorems for set-valued mappings of contractive type, Pacific J. Math. 43 (1972), 553-562. MR 0341459 (49:6210)
  • [2] J. Bogin, On strict pseudo-contractions and a fixed point theorem, Technion Preprint Series No. MT-219, Haifa, Israel, 1974.
  • [3] F. E. Browder, The solvability of nonlinear functional equations, Duke Math. J. 30 (1963), 557-566. MR 0156204 (27:6133)
  • [4] -, Nonlinear monotone and accretive operators in Banach spaces, Proc. Nat. Acad. Sci. U.S.A. 61 (1968), 388-393. MR 0290205 (44:7389)
  • [5] -, Nonlinear mappings of nonexpansive and accretive type in Banach spaces, Bull. Amer. Math. Soc. 73 (1967), 875-882. MR 0232255 (38:581)
  • [6] F. E. Browder and W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl. 20 (1967), 197-228. MR 0217658 (36:747)
  • [7] C. E. Chidume, Iterative approximation of fixed points of Lipschitzian strictly pseudocontractive mappings, Proc. Amer. Math. Soc. 99 (1987), 283-288. MR 870786 (87m:47122)
  • [8] -, Iterative solution of nonlinear equations of the monotone type in Banach spaces, Bull. Austral. Math. Soc. 42 (1990), 21-31. MR 1066356 (91i:47080)
  • [9] K. Deimling, Zeros of accretive operators, Manuscripta Math. 13 (1974), 283-288. MR 0350538 (50:3030)
  • [10] J. A. Gatica and W. A. Kirk, Fixed point theorems for Lipschitzian pseudo-contractive mappings, Proc. Amer. Math. Soc. 36 (1972), 111-115. MR 0306993 (46:6114)
  • [11] J. Gwinner, On the convergence of some iteration processes in uniformly convex Banach spaces, Proc. Amer. Math. Soc. 81 (1978), 29-35. MR 0477899 (57:17399)
  • [12] T. L. Hicks and J. R. Kubicek, On the Mann iteration process in Hilbert space, J. Math. Anal. Appl. 59 (1977), 498-504. MR 0513062 (58:23802)
  • [13] S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc. 149 (1944), 147-150. MR 0336469 (49:1243)
  • [14] -, Fixed points and iteration of a nonexpansive mappings in a Banach space, Proc. Amer. Math. Soc. 73 (1976), 65-71. MR 0412909 (54:1030)
  • [15] T. Kato, Nonlinear semigroups and evolution equations, J. Math. Soc. Japan 19 (1967), 508-520. MR 0226230 (37:1820)
  • [16] W. A. Kirk, A fixed point theorem for local pseudo-contraction in uniformly convex spaces, Manuscripta Math. 30 (1979), 89-102. MR 552364 (80k:47064)
  • [17] -, Remarks on pseudocontractive mappings, Proc. Amer. Math. Soc. 25 (1970), 820-823. MR 0264481 (41:9074)
  • [18] W. A. Kirk and C. Morales, On the approximation of fixed points of locally nonexpansive mappings, Canad. Math. Bull. 24 (1981), 441-445. MR 644533 (83h:47040)
  • [19] W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506-510. MR 0054846 (14:988f)
  • [20] R. H. Martin, Jr., A global existence theorem for autonomous differential equations in Banach spaces, Proc. Amer. Math. Soc. 26 (1970), 307-314. MR 0264195 (41:8791)
  • [21] C. Morales, Surjectivity theorems for multi-valued mappings of accretive type, Comment. Math. Univ. Carolin. 26 (1985). MR 803937 (87c:47074)
  • [22] R. N. Mukerjee, Construction of fixed points of strictly pseudocontractive mappings in generalized Hilbert spaces and related applications, Indian J. Pure Appl. Math. 15 (1966), 276-284.
  • [23] G. Nevanlinna and S. Reich, Strong convergence of contraction semigroups and of iterative methods for accretive operators in Banach spaces, Israel J. Math. 32 (1979), 44-58. MR 531600 (80e:47057)
  • [24] S. Reich, An iterative procedure for constructing zeros of accretive sets in Banach spaces, Nonlinear Anal. 2 (1978), 85-92. MR 512657 (81b:47065)
  • [25] -, Constructing zeros of accretive operators. II, Appl. Anal. 9 (1979), 159-163. MR 547355 (82d:65052b)
  • [26] -, Constructive techniques for accretive and monotone operators, Applied Nonlinear Analysis (V. Lakshmikantham, ed.), Academic Press, New York, 1979, pp. 335-345. MR 537545 (80g:47059)
  • [27] -, Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl. 85 (1980), 287-292. MR 576291 (82a:47050)
  • [28] B. E. Rhoades, Comments on two fixed point iteration methods, J. Math. Anal. Appl. 56 (1976), 741-750. MR 0430880 (55:3885)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 47H10, 47H09, 47H15

Retrieve articles in all journals with MSC: 47H10, 47H09, 47H15


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1994-1165050-6
Article copyright: © Copyright 1994 American Mathematical Society

American Mathematical Society