Approximation methods for nonlinear operator equations
Authors:
C. E. Chidume and H. Zegeye
Journal:
Proc. Amer. Math. Soc. 131 (2003), 2467-2478
MSC (2000):
Primary 47H04, 47H06, 47H30, 47J05, 47J25
DOI:
https://doi.org/10.1090/S0002-9939-02-06769-2
Published electronically:
November 13, 2002
MathSciNet review:
1974645
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: Let be a real normed linear space and
be a uniformly quasi-accretive map. For arbitrary
define the sequence
by
where
is a positve real sequence satisfying the following conditions: (i)
; (ii)
. For
, assume that
and that
, where
(the set of all positive integers):
and
is a strictly increasing function with
. It is proved that a Mann-type iteration process converges strongly to
. Furthermore if, in addition,
is a uniformly continuous map, it is proved, without the condition on
, that the Mann-type iteration process converges strongly to
. As a consequence, corresponding convergence theorems for fixed points of hemi-contractive maps are proved.
- 1. Alber, Ya., Reich, S.: An iterative method for solving a class of nonlinear operator equations in Banach spaces, Panamerican Math. J., Vol. 4, No. 2 (1994), 39-54. MR 95f:47089
- 2. F. E. Browder, Nonlinear mappings of nonexpansive and accetive type in Banach spaces, Bull. Amer. Math. Soc. 73 (1967), 875-882. MR 38:581
- 3.
S.-S. Chang, On Chidume's open questions and approximation solutions of multi-valued strongly accretive mapping equations in Banach spaces, J. Math. Anal. Appl. 216 (1997), 94
111.
- 4. S.-S. Chang and K. K. Tan, Iteration process of fixed point for operators of monotone type in Banach spaces, Bull. Austral. Math. Soc. 57 (1998), 433-445. MR 99e:47083
- 5. C.E. Chidume, Iterative construction of fixed points for multi-valued operators of the monotone type, Applicable Analysis 23 (1986), 209-218. MR 88e:47111
- 6. C.E. Chidume, Iterative approximation of fixed points of Lipschitzian strictly pseudo-contractive mappings, Proc. Amer. Math. Soc. 99 (1987), 283-288. MR 87m:47122
- 7.
C.E. Chidume, An iterative process for nonlinear Lipschitzian strongly accretive mapping in
spaces, J. Math. Anal. Appl. 151 (1990), 453-461. MR 91k:47080
- 8. C.E. Chidume, Approximation of fixed points of strongly pseudo-contractive mappings, Proc. Amer. Math. Soc. 120 (1994), 545-551. MR 94d:47056
- 9. C.E. Chidume, Iterative solution of nonlinear equations of strongly accretive type, Math. Nachr. 189 (1998), 49-60. MR 99g:47141
- 10. C.E. Chidume, Convergence theorems for strongly pseudocontractive and strongly accretive nonlinear maps, J. Math. Anal. Appl. 228 (1998), 254-264. MR 99h:47065
- 11. C.E. Chidume and C. Moore, Steepest descent method for equilibrium points of nonlinear systems with accretive operators, J. Math. Anal. Appl. 245 (2000), 142-160. MR 2001a:47075
- 12. C.E. Chidume and C. Moore. The solution by iteration of nonlinear equations in uniformly smooth Banach spaces, J. Math. Anal. Appl. 215 (1997), 132-146. MR 98m:47107
- 13. C. E. Chidume and S. Mutangadura, An example on the Mann iteration methods for Lipschitzian pseudocontractions, Proc. Amer. Math. Soc. 129 (2001), 2359-2363. MR 2002f:47104
- 14.
C. E. Chidume, H. Zegeye and B. Ntatin, A generalized steepest descent approximation for the zeros of
-accretive operators, J. Math. Anal. Appl. 261 (1999), 48
73. MR 2000e:47080
- 15.
L. Deng, On Chidume's open questions, J. Math. Anal. Appl. 174 (1993), 441
449. MR 94b:47073
- 16.
L. Deng, Iteration processes for nonlinear Lipschitzian strongly accretive mappings in
spaces, J. Math. Anal. Appl. 188 (1994), 128
140. MR 96f:47124
- 17.
L. Deng and X. P. Ding, Iterative approximation of Lipschitz strictly pseudocontractive mappings in uniformly smooth Banach spaces, Nonlinear Analysis 24 (1995), 981
987. MR 96a:47096
- 18.
J. C. Dunn, Iterative construction of fixed points for multivalued operators of the monotone type, J. Funct. Ann. 27 (1978), 38
50. MR 81f:47056
- 19.
S. Ishikawa, Fixed points by a new iteration Method, Proc. Amer. Math. Soc. 44 (1974), 147
150. MR 49:1243
- 20.
T. Kato, Nonlinear semi-groups and evolution equations, J. Math. Soc. Japan 19 (1967), 508
520. MR 30:1820
- 21.
L. Liu, Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces, J. Math. Anal. Appl. 194 (1995), 114
125. MR 97g:47069
- 22. L. W. Liu and Y. Q. Li, On generalized set-valued variational inclusions, J. Math. Anal. Appl. 261 (2001), 231-240. MR 2002g:47142
- 23. W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506-510. MR 14:998f
- 24. C. Moore and B.V.C. Nnoli, Iterative solution of nonlinear equations involving set-valued uniformly accretive operators, Computers Math. Applic. 42 (2001), 131-140. MR 2002f:47145
- 25.
M.O. Osilike, Iterative solution of nonlinear equations of the
-strongly accretive type, J. Math. Anal. Appl. 200 (1996), 259-271. MR 97d:65032
- 26. M.O. Osilike, Iterative construction of fixed points of multi-valued operators of the accretive type, Soochow J. Math. 22 (1996), 485-494. MR 97i:47122
- 27. M.O. Osilike, Iterative construction of fixed points of multi-valued operators of the accretive type II, Soochow J. Math. 24 (1998), 141-146. MR 99e:47079
- 28. B.E. Rhoades and L. Saliga, Some fixed point iteration procedures II, Nonlinear Analysis forum 6 (2001), 193-217. MR 2002f:47115
- 29.
S. Zhang, On the convergence problems of Ishikawa and Mann iteration process with errors for
-pseudo contractive type mappings. Appl. Math. Mechanics 21 (2000), 1-10. MR 2001c:47080
- 30. X.L. Weng, Iterative construction of fixed points of a dissipative type operator, Tamkang J. Math. 23 (1992), 205-215. MR 94c:47077
- 31.
E. Zeidler, Nonlinear functional analysis and its applications, Part II: Monotone Operators, Springer
Verlag, 1985. MR 91b:47001; MR 91b:47002
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Additional Information
C. E. Chidume
Affiliation:
The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy
Email:
chidume@ictp.trieste.it
H. Zegeye
Affiliation:
The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy
Email:
habz@ictp.trieste.it
DOI:
https://doi.org/10.1090/S0002-9939-02-06769-2
Keywords:
Bounded operators,
nonexpansive retraction,
uniformly accretive maps,
uniformly pseudocontractive maps,
uniformly smooth Banach spaces
Received by editor(s):
December 8, 2001
Received by editor(s) in revised form:
March 18, 2002
Published electronically:
November 13, 2002
Additional Notes:
The second author undertook this work with the support of the “ICTP Programme for Training and Research in Italian Laboratories, Trieste, Italy".
Communicated by:
Joseph A. Ball
Article copyright:
© Copyright 2002
American Mathematical Society