zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Implicit Mann fixed point iterations for pseudo-contractive mappings. (English) Zbl 1219.47108
Summary: Let $K$ be a compact convex subset of a real Hilbert space $H$ and $T:K\rightarrow K$ be a continuous hemi-contractive map. Let $\{a_n\}$, $\{b_n\}$ and $\{c_n\}$ be real sequences in $[0,1]$ such that $a_n+b_n+c_n=1$, and $\{u_n\}$ and $\{v_n\}$ be sequences in $K$. In this paper, we prove that, if $\{b_n\}$, $\{c_n\}$ and $\{v_n\}$ satisfy some appropriate conditions, then, for arbitrary $x_{0}\in K$, the sequence $\{x_n\}$ defined iteratively by $x_n=a_nx_{n - 1}+b_nTv_n+c_nu_n$, $n\geq 1$, converges strongly to a fixed point of $T$.

MSC:
47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties
WorldCat.org
Full Text: DOI
References:
[1] F.E. Browder, Nonlinear operators and nonlinear equations of evolution in Banach spaces, in: Proc. Symposia Pure Math., vol. XVIII, part 2, 1976 · Zbl 0327.47022
[2] Browder, F. E.; Petryshyn, W. V.: Construction of fixed points of nonlinear mappings in Hilbert spaces, J. math. Anal. appl. 20, 197-228 (1967) · Zbl 0153.45701 · doi:10.1016/0022-247X(67)90085-6
[3] Chidume, C. E.; Moore, Chika: Fixed point iteration for pseudocontractive maps, Proc. amer. Math. soc. 127, No. 4, 1163-1170 (1999) · Zbl 0913.47052 · doi:10.1090/S0002-9939-99-05050-9
[4] Chidume, C. E.; Mutangadura, S. A.: An example on the Mann iteration method for Lipschitz pseudocontractions, Proc. amer. Math. soc. 129, No. 8, 2359-2363 (2001) · Zbl 0972.47062 · doi:10.1090/S0002-9939-01-06009-9
[5] Ciric, L. B.; Jesic, S. N.; Milovanovic, M. M.; Ume, J. S.: On the steepest descent approximation method for the zeros of generalized accretive operators, Nonlinear anal. 69, 763-769 (2008) · Zbl 1220.47089 · doi:10.1016/j.na.2007.06.021
[6] Ćirić, L. B.; Ume, J. S.: Ishikawa process with errors for nonlinear equations of generalized monotone type in Banach spaces, Math. nachr. 278, No. 10, 1137-1146 (2005) · Zbl 1092.47054 · doi:10.1002/mana.200310298
[7] Deng, L.: Iteration process for nonlinear Lipschitzian strongly accretive mappings in lp spaces, J. math. Anal. appl. 188, 128-140 (1994) · Zbl 0828.47042 · doi:10.1006/jmaa.1994.1416
[8] Hicks, T. L.; Kubicek, J. R.: On the Mann iteration process in Hilbert space, J. math. Anal. appl. 59, 498-504 (1977) · Zbl 0361.65057 · doi:10.1016/0022-247X(77)90076-2
[9] Huang, Z. Y.: Weak stability of Mann and Ishikawa iterations with errors for phi-hemicontractive operators, Appl. math. Lett. 20, No. 4, 470-475 (2007) · Zbl 1175.47061 · doi:10.1016/j.aml.2006.06.006
[10] Ishikawa, S.: Fixed point by a new iteration method, Proc. amer. Math. soc. 4, No. 1, 147-150 (1974) · Zbl 0286.47036
[11] Kato, T.: Nonlinear semigroups and evolution equations, J. math. Soc. Japan 19, 508-520 (1967) · Zbl 0163.38303 · doi:10.2969/jmsj/01940508
[12] Mann, W. R.: Mean value methods in iteration, Proc. amer. Math. soc. 4, 506-610 (1953) · Zbl 0050.11603 · doi:10.2307/2032162
[13] Osilike, M. O.; Igbokwe, D. I.: Weak and strong convergence theorems for fixed points of pseudocontractions and solutions of monotone type operator equations, Comput. math. Appl. 40, 559-567 (2000) · Zbl 0958.47030 · doi:10.1016/S0898-1221(00)00179-6
[14] Peng, J. W.: Set-valued variational inclusions with T-accretive operators in Banach spaces, App. math. Lett. 19, No. 3, 273-282 (2006) · Zbl 1102.47050 · doi:10.1016/j.aml.2005.04.009
[15] Rafiq, A.: On Mann iteration in Hilbert spaces, Nonlinear anal. 66, No. 10, 2230-2236 (2007) · Zbl 1136.47047 · doi:10.1016/j.na.2006.03.012
[16] Rhoades, B. E.: Comments on two fixed point iteration procedures, J. math. Anal. appl. 56, 741-750 (1976) · Zbl 0353.47029 · doi:10.1016/0022-247X(76)90038-X
[17] Schauder, J.: Der fixpunktsatz in funktionalräumen, Studia math. 2, 171-180 (1930) · Zbl 56.0355.01
[18] Schu, J.: Iterative construction of fixed points of asymptotically nonexpansive mappings, J. math. Anal. appl. 158, 407-413 (1991) · Zbl 0734.47036 · doi:10.1016/0022-247X(91)90245-U
[19] Tan, K. K.; Xu, H. K.: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. math. Anal. appl. 178, 301-308 (1993) · Zbl 0895.47048 · doi:10.1006/jmaa.1993.1309
[20] Xu, Y.: Ishikawa and Mann iterative processes with errors for nonlinear strongly accretive operator equations, J. math. Anal. appl. 224, 91-101 (1998) · Zbl 0936.47041 · doi:10.1006/jmaa.1998.5987