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)
Viscosity approximation methods for pseudocontractive mappings in Banach spaces. (English) Zbl 1178.47049
Let $K$ be a closed convex subset of a Banach space $E$ and let $T : K \to E$ be a continuous weakly inward pseudocontractive mapping. Then for $t\in(0,1)$, there exists a sequence $\{y_t\} \subset K$ satisfying $y_t = (1 - t)f(y_t) + tT(y_t)$, where $f \in \Pi_K := \{f : K\to K,\text{ a contraction with a suitable contractive constant}\}$. Suppose further that $F(T) \ne \emptyset$ and $E$ is reflexive and strictly convex which has uniformly Gâteaux differentiable norm. Then it is proved that $\{y_t\}$ converges strongly to a fixed point of $T$ which is also a solution of certain variational inequality. Moreover, an explicit iteration process which converges strongly to a fixed point of $T$ and hence to a solution of certain variational inequality is constructed, provided that $T$ is Lipschitzian.

MSC:
47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties
65J15Equations with nonlinear operators (numerical methods)
WorldCat.org
Full Text: DOI
References:
[1] Barbu, V.; Precupanu, Th.: Convexity and optimization in Banach spaces. (1978) · Zbl 0379.49010
[2] Browder, F. E.: Convergence of approximants to fixed points of nonexpansive nonlinear mappings in Banach spaces. Arch. ration. Mech. anal. 24, 82-90 (1967) · Zbl 0148.13601
[3] Chidume, C. E.; Moore, C.: The solution by iteration of nonlinear equations in uniformly smooth Banach spaces. J. math. Anal. appl. 215, 132-146 (1997) · Zbl 0906.47050
[4] Chidume, C. E.; Mutangadura, S.: An example on the Mann iteration methods for Lipschitzian pseudocontractions. Proc. am. Math. soc. 129, 2359-2363 (2001) · Zbl 0972.47062
[5] Chidume, C. E.; Zegeye, H.: Approximate fixed point sequences and convergence theorems for Lipschitz pseudocontractive maps. Proc. am. Math. soc. 132, 831-840 (2004) · Zbl 1051.47041
[6] Chidume, C. E.; Zegeye, H.; Aneke, S. J.: Approximation of fixed points of weakly contractive non-self maps in Banach spaces. J. math. Anal. appl. 270, 189-199 (2002) · Zbl 1005.47053
[7] Cioranescu, I.: Geometry of Banach spaces, duality mapping and nonlinear problems. (1990) · Zbl 0712.47043
[8] Deimling, K.: Zeros of accretive operators. Manuscr. math. 13, 365-374 (1974) · Zbl 0288.47047
[9] Ishikawa, S.: Fixed points by a new iteration method. Proc. am. Math. soc. 44, 147-150 (1974) · Zbl 0286.47036
[10] Ishikawa, S.: Fixed points and iteration of a nonexpansive mapping in a Banach space. Proc. am. Math. soc. 59, 65-71 (1976) · Zbl 0352.47024
[11] Kato, T.: Nonlinear semi-groups and evolution equations. J. math. Soc. jpn. 19, 508-520 (1967) · Zbl 0163.38303
[12] Lim, T. C.; Xu, H. K.: Fixed point theorems for asymptotically nonexpansive mappings. Nonlinear anal. 22, 1345-1355 (1994) · Zbl 0812.47058
[13] Martin, R. H.: Differential equations on closed subsets of a Banach space. Trans. am. Math. soc. 179, 399-414 (1973) · Zbl 0293.34092
[14] Megginson, R. E.: An introduction to Banach space theory. (1998) · Zbl 0910.46008
[15] Morales, C. H.: Strong convergence theorems for pseudocontractive mappings in Banach spaces. Houston J. Math. 16, 549-557 (1990) · Zbl 0728.47037
[16] Morales, C. H.; Jung, J. S.: Convergence of paths for pseudo-contractive mappings in Banach spaces. Proc. am. Math. soc. 128, 3411-3419 (2000) · Zbl 0970.47039
[17] Moudafi, A.: Viscosity approximation methods for fixed point problems. J. math. Anal. appl. 241, 46-55 (2000) · Zbl 0957.47039
[18] Osilike, M. O.: Iterative solution of nonlinear equations of the $\psi $-strongly accretive type. J. math. Anal. appl. 200, 259-271 (1996) · Zbl 0860.65039
[19] Petryshyn, W. V.: Construction of fixed points of demicompact mappings in Hilbert spaces. J. math. Anal. appl. 14, 276-284 (1966) · Zbl 0138.39802
[20] Qihou, L.: The convergence theorems of the sequence of Ishikawa iterates for hemicontractive mappings. J. math. Anal. appl. 148, 55-62 (1990) · Zbl 0729.47052
[21] Reich, S.: Strong convergence theorems for resolvents of accretive operators in Banach spaces. J. math. Anal. appl. 75, 287-292 (1980) · Zbl 0437.47047
[22] Y. Song, R. Chen, Strong convergence theorems on an iterative method for a family of finite nonexpansive mappings, Appl. Math. Comput., in press, doi:10.1016/j.amc.2005.12.013. · Zbl 1139.47050
[23] Takahashi, W.; Ueda, Y.: On reich’s strong convergence theorems for resolvents of accretive operators. J. math. Anal. appl. 104, 546-553 (1984) · Zbl 0599.47084
[24] Takahashi, W.: Nonlinear functional analysis. (2000) · Zbl 0997.47002
[25] Xu, H. K.: Viscosity approximation methods for nonexpansive mappings. J. math. Anal. appl. 298, 279-281 (2004) · Zbl 1061.47060
[26] Zeidler, E.: Nonlinear functional analysis and its applications, part II: Monotone operators. (1985) · Zbl 0583.47051
[27] Zhang, S.: On the convergence problems of Ishikawa and Mann iteration process with errors for $\psi $-pseudo contractive type mappings. Appl. math. Mech. 21, 1-10 (2000)