Approximation methods for nonlinear operator equations
Authors:
C. E. Chidume and H. Zegeye
Journal:
Proc. Amer. Math. Soc. 131 (2003), 24672478
MSC (2000):
Primary 47H04, 47H06, 47H30, 47J05, 47J25
Published electronically:
November 13, 2002
MathSciNet review:
1974645
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Let be a real normed linear space and be a uniformly quasiaccretive 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 Manntype iteration process converges strongly to . Furthermore if, in addition, is a uniformly continuous map, it is proved, without the condition on , that the Manntype iteration process converges strongly to . As a consequence, corresponding convergence theorems for fixed points of hemicontractive 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), 3954. MR 95f:47089
 2.
F. E. Browder, Nonlinear mappings of nonexpansive and accetive type in Banach spaces, Bull. Amer. Math. Soc. 73 (1967), 875882. MR 38:581
 3.
S.S. Chang, On Chidume's open questions and approximation solutions of multivalued strongly accretive mapping equations in Banach spaces, J. Math. Anal. Appl. 216 (1997), 94111.
 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), 433445. MR 99e:47083
 5.
C.E. Chidume, Iterative construction of fixed points for multivalued operators of the monotone type, Applicable Analysis 23 (1986), 209218. MR 88e:47111
 6.
C.E. Chidume, Iterative approximation of fixed points of Lipschitzian strictly pseudocontractive mappings, Proc. Amer. Math. Soc. 99 (1987), 283288. MR 87m:47122
 7.
C.E. Chidume, An iterative process for nonlinear Lipschitzian strongly accretive mapping in spaces, J. Math. Anal. Appl. 151 (1990), 453461. MR 91k:47080
 8.
C.E. Chidume, Approximation of fixed points of strongly pseudocontractive mappings, Proc. Amer. Math. Soc. 120 (1994), 545551. MR 94d:47056
 9.
C.E. Chidume, Iterative solution of nonlinear equations of strongly accretive type, Math. Nachr. 189 (1998), 4960. MR 99g:47141
 10.
C.E. Chidume, Convergence theorems for strongly pseudocontractive and strongly accretive nonlinear maps, J. Math. Anal. Appl. 228 (1998), 254264. 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), 142160. 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), 132146. 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), 23592363. 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), 4873. MR 2000e:47080
 15.
L. Deng, On Chidume's open questions, J. Math. Anal. Appl. 174 (1993), 441449. MR 94b:47073
 16.
L. Deng, Iteration processes for nonlinear Lipschitzian strongly accretive mappings in spaces, J. Math. Anal. Appl. 188 (1994), 128140. 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), 981987. MR 96a:47096
 18.
J. C. Dunn, Iterative construction of fixed points for multivalued operators of the monotone type, J. Funct. Ann. 27 (1978), 3850. MR 81f:47056
 19.
S. Ishikawa, Fixed points by a new iteration Method, Proc. Amer. Math. Soc. 44 (1974), 147150. MR 49:1243
 20.
T. Kato, Nonlinear semigroups and evolution equations, J. Math. Soc. Japan 19 (1967), 508520. 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), 114125. MR 97g:47069
 22.
L. W. Liu and Y. Q. Li, On generalized setvalued variational inclusions, J. Math. Anal. Appl. 261 (2001), 231240. MR 2002g:47142
 23.
W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506510. MR 14:998f
 24.
C. Moore and B.V.C. Nnoli, Iterative solution of nonlinear equations involving setvalued uniformly accretive operators, Computers Math. Applic. 42 (2001), 131140. MR 2002f:47145
 25.
M.O. Osilike, Iterative solution of nonlinear equations of the strongly accretive type, J. Math. Anal. Appl. 200 (1996), 259271. MR 97d:65032
 26.
M.O. Osilike, Iterative construction of fixed points of multivalued operators of the accretive type, Soochow J. Math. 22 (1996), 485494. MR 97i:47122
 27.
M.O. Osilike, Iterative construction of fixed points of multivalued operators of the accretive type II, Soochow J. Math. 24 (1998), 141146. MR 99e:47079
 28.
B.E. Rhoades and L. Saliga, Some fixed point iteration procedures II, Nonlinear Analysis forum 6 (2001), 193217. 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), 110. MR 2001c:47080
 30.
X.L. Weng, Iterative construction of fixed points of a dissipative type operator, Tamkang J. Math. 23 (1992), 205215. MR 94c:47077
 31.
E. Zeidler, Nonlinear functional analysis and its applications, Part II: Monotone Operators, SpringerVerlag, 1985. MR 91b:47001; MR 91b:47002
 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), 3954. MR 95f:47089
 2.
 F. E. Browder, Nonlinear mappings of nonexpansive and accetive type in Banach spaces, Bull. Amer. Math. Soc. 73 (1967), 875882. MR 38:581
 3.
 S.S. Chang, On Chidume's open questions and approximation solutions of multivalued strongly accretive mapping equations in Banach spaces, J. Math. Anal. Appl. 216 (1997), 94111.
 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), 433445. MR 99e:47083
 5.
 C.E. Chidume, Iterative construction of fixed points for multivalued operators of the monotone type, Applicable Analysis 23 (1986), 209218. MR 88e:47111
 6.
 C.E. Chidume, Iterative approximation of fixed points of Lipschitzian strictly pseudocontractive mappings, Proc. Amer. Math. Soc. 99 (1987), 283288. MR 87m:47122
 7.
 C.E. Chidume, An iterative process for nonlinear Lipschitzian strongly accretive mapping in spaces, J. Math. Anal. Appl. 151 (1990), 453461. MR 91k:47080
 8.
 C.E. Chidume, Approximation of fixed points of strongly pseudocontractive mappings, Proc. Amer. Math. Soc. 120 (1994), 545551. MR 94d:47056
 9.
 C.E. Chidume, Iterative solution of nonlinear equations of strongly accretive type, Math. Nachr. 189 (1998), 4960. MR 99g:47141
 10.
 C.E. Chidume, Convergence theorems for strongly pseudocontractive and strongly accretive nonlinear maps, J. Math. Anal. Appl. 228 (1998), 254264. 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), 142160. 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), 132146. 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), 23592363. 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), 4873. MR 2000e:47080
 15.
 L. Deng, On Chidume's open questions, J. Math. Anal. Appl. 174 (1993), 441449. MR 94b:47073
 16.
 L. Deng, Iteration processes for nonlinear Lipschitzian strongly accretive mappings in spaces, J. Math. Anal. Appl. 188 (1994), 128140. 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), 981987. MR 96a:47096
 18.
 J. C. Dunn, Iterative construction of fixed points for multivalued operators of the monotone type, J. Funct. Ann. 27 (1978), 3850. MR 81f:47056
 19.
 S. Ishikawa, Fixed points by a new iteration Method, Proc. Amer. Math. Soc. 44 (1974), 147150. MR 49:1243
 20.
 T. Kato, Nonlinear semigroups and evolution equations, J. Math. Soc. Japan 19 (1967), 508520. 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), 114125. MR 97g:47069
 22.
 L. W. Liu and Y. Q. Li, On generalized setvalued variational inclusions, J. Math. Anal. Appl. 261 (2001), 231240. MR 2002g:47142
 23.
 W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506510. MR 14:998f
 24.
 C. Moore and B.V.C. Nnoli, Iterative solution of nonlinear equations involving setvalued uniformly accretive operators, Computers Math. Applic. 42 (2001), 131140. MR 2002f:47145
 25.
 M.O. Osilike, Iterative solution of nonlinear equations of the strongly accretive type, J. Math. Anal. Appl. 200 (1996), 259271. MR 97d:65032
 26.
 M.O. Osilike, Iterative construction of fixed points of multivalued operators of the accretive type, Soochow J. Math. 22 (1996), 485494. MR 97i:47122
 27.
 M.O. Osilike, Iterative construction of fixed points of multivalued operators of the accretive type II, Soochow J. Math. 24 (1998), 141146. MR 99e:47079
 28.
 B.E. Rhoades and L. Saliga, Some fixed point iteration procedures II, Nonlinear Analysis forum 6 (2001), 193217. 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), 110. MR 2001c:47080
 30.
 X.L. Weng, Iterative construction of fixed points of a dissipative type operator, Tamkang J. Math. 23 (1992), 205215. MR 94c:47077
 31.
 E. Zeidler, Nonlinear functional analysis and its applications, Part II: Monotone Operators, SpringerVerlag, 1985. MR 91b:47001; MR 91b:47002
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2000):
47H04,
47H06,
47H30,
47J05,
47J25
Retrieve articles in all journals
with MSC (2000):
47H04,
47H06,
47H30,
47J05,
47J25
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:
http://dx.doi.org/10.1090/S0002993902067692
PII:
S 00029939(02)067692
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
