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)

 
 

 

Fixed point iteration for pseudocontractive maps


Authors: C. E. Chidume and Chika Moore
Journal: Proc. Amer. Math. Soc. 127 (1999), 1163-1170
MSC (1991): Primary 47H05, 47H06, 47H10, 47H15
DOI: https://doi.org/10.1090/S0002-9939-99-05050-9
MathSciNet review: 1625729
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $K$ be a compact convex subset of a real Hilbert space, $H$; $T:K\rightarrow K$ a continuous pseudocontractive map. Let $\{a_{n}\}, \{b_{n}\}, \{c_{n}\}, \{a_{n}^{'}\}, \{b_{n}^{'}\}$ and $\{c_{n}^{'}\}$ be real sequences in [0,1] satisfying appropriate conditions. For arbitrary $x_{1}\in K,$ define the sequence $\{x_{n}\}_{n=1}^{\infty}$ iteratively by $x_{n+1} = a_{n}x_{n} + b_{n}Ty_{n} + c_{n}u_{n}; y_{n} = a_{n}^{'}x_{n} + b_{n}^{'}Tx_{n} + c_{n}^{'}v_{n},\ n\geq 1,$ where $\{u_{n}\}, \{v_{n}\}$ are arbitrary sequences in $K$. Then, $\{x_{n}\}_{n=1}^{\infty}$ converges strongly to a fixed point of $T$. A related result deals with the convergence of $\{x_{n}\}_{n=1}^{\infty}$ to a fixed point of $T$ when $T$ is Lipschitz and pseudocontractive. Our theorems also hold for the slightly more general class of continuous hemicontractive nonlinear maps.


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

  • 1. F.E. Browder, Nonlinear operators and nonlinear equations of evolution in Banach spaces, Proc. of Symposia in Pure Math., Vol. XVIII, Part 2, 1976. MR 53:8982
  • 2. F.E. Browder and W.V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert spaces, J. Math. Anal. Appl. 20 (1967), 197-228. MR 36:747
  • 3. C.E. Chidume, Iterative approximation of Lipschitz strictly pseudocontractive mappings, Proc. Amer. Math. Soc. 99 (2), (1987), 283-288. MR 87m:47122
  • 4. C.E. Chidume, Approximation of fixed points of strongly pseudocontractive mappings, Proc. Amer. Math. Soc. 120 (2), (1994), 545-551. MR 94d:47056
  • 5. C.E. Chidume, Global iteration schemes for strongly pseudocontractive maps, to appear, Proc. Amer. Math. Soc. (1998). CMP 97:13
  • 6. C.E. Chidume, Iterative solution of nonlinear equations of strongly accretive type, Math. Nachr. 189 (1998), 49-60. CMP 98:07
  • 7. C.E. Chidume and Chika Moore, The solution by iteration of nonlinear equations in uniformly smooth Banach spaces, J. Math. Anal. Appl. 215 (1), (1997), 132-146. CMP 98:03
  • 8. C.E. Chidume and M.O. Osilike, Ishikawa iteration process for nonlinear Lipschitz strongly accretive mappings, J. Math. Anal. Appl. 192 (1995), 727-741. MR 96i:47099
  • 9. C.E. Chidume and M.O. Osilike, Nonlinear accretive and pseudocontractive operator equations in Banach spaces, Nonlinear Anal. TM 31 (7), (1998), 779-789. CMP 98:06
  • 10. M.G. Crandall and A. Pazy, On the range of accretive operators, Israel J. Math. 27 (1977), 235-246. MR 56:1142
  • 11. L. Deng, On Chidume's open problems, J. Math. Anal. Appl. 174 (2), (1993), 441-449. MR 94b:47073
  • 12. L. Deng, Iteration process for nonlinear Lipschitzian strongly accretive mappings in $L_{p}$ spaces, J. Math. Anal. Appl. 188 (1994), 128-140. MR 96f:47124
  • 13. L. Deng and X.P. Ding, Iterative approximation of Lipschitz strictly pseudocontractive mappings in uniformly smooth Banach spaces, Nonlinear Anal. TMA 24 (7), (1995), 981-987. MR 96a:47096
  • 14. T.L. Hicks and J.R. Kubicek, On the Mann iteration process in Hilbert space, J. Math. Anal. Appl. 59 (1977), 498-504. MR 58:23802
  • 15. S. Ishikawa, Fixed point by a new iteration method, Proc. Amer. Math. Soc. 4 (1), (1974), 147-150. MR 49:1243
  • 16. L.S. Liu, Ishikawa and Mann iteration process with errors for nonlinear strongly accretive mappings in Banach spaces, J. Math. Anal. Appl. 194 (1995), 114-125. MR 97g:47069
  • 17. W.R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506-610. MR 14:988f
  • 18. Liu Qihou, On Naimpally and Singh's open questions, J. Math. Anal. Appl. 124 (1987), 157-164. MR 88j:47078
  • 19. Liu Qihou, The convergence theorems of the sequence of Ishikawa iterates for hemicontractive mappings, J. Math. Anal. Appl. 148 (1990), 55-62. MR 92b:47094
  • 20. Liu Qihou, Convergence theorems of the sequence of iterates for asymptotically demicontractive and hemicontractive mappings, Nonlinear Anal. TMA 26 (11), (1996), 1835-1842. MR 97d:47069
  • 21. S. Reich, An iterative procedure for constructing zeros of accretive sets in Banach spaces, Nonlinear Anal. 2 (1978), 85-92. MR 81b:47065
  • 22. S. Reich, Constructive techniques for accretive and monotone operators in Applied Nonlinear Analysis, Academic Press, New York, (1979), 335-345. MR 80g:47059
  • 23. S. Reich, Constructing zeros of accretive operators, I II, Applicable Analysis 9 (1979), 159-163. MR 82d:65052b
  • 24. S. Reich, Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl. 75 (1980), 287-292. MR 82a:47050
  • 25. B.E. Rhoades, Comments on two fixed point iteration procedures, J. Math. Anal. Appl. 56 (1976), 741-750. MR 55:3885
  • 26. J. Schu, On a theorem of C.E. Chidume concerning the iterative approximation of fixed points, Math. Nachr. 153 (1991), 313-319. MR 93b:47123
  • 27. J. Schu, Iterative construction of fixed points of strictly pseudocontractive mappings, Applicable Analysis 40 (1991), 67-72. MR 92c:47072
  • 28. J. Schu, Iterative construction of fixed points of asymptotically nonexpansive mappings, J. Math. Anal. Appl. 158 (1991), 407-413. MR 92d:47072
  • 29. Kok-Keong Tan and H.K. Xu, Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl. 178 (1993), 301-308. MR 94g:47076
  • 30. Yuguang Xu, Ishikawa ana Mann iterative processes with errors for nonlinear strongly accretive operator equations, J. Math. Anal. Appl., (to appear).
  • 31. Z.B. Xu and G.F. Roach, A necessary and sufficient condition for convergence of steepest descent approximation to accretive operator equations, J. Math. Anal. Appl. 167 (1992), 340-354. MR 93e:47086
  • 32. X.L. Weng, Fixed point iteration for local strictly pseudocontractive mappings, Proc. Amer. Math. Soc. 113 (1991), 727-731. MR 92b:47099

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 47H05, 47H06, 47H10, 47H15

Retrieve articles in all journals with MSC (1991): 47H05, 47H06, 47H10, 47H15


Additional Information

C. E. Chidume
Affiliation: The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy
Email: chidume@ictp.trieste.it

Chika Moore
Affiliation: The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy

DOI: https://doi.org/10.1090/S0002-9939-99-05050-9
Received by editor(s): August 1, 1997
Communicated by: David R. Larson
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society