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)
Strong convergence of path for continuous pseudo-contractive mappings. (English) Zbl 1121.47055

The path (or viscosity) fixed point approximation originates in a series of papers of Browder, starting from the seminal paper [F. E. Browder, Proc. Nat. Acad. Sci. USA 56, 1080–1086 (1966; Zbl 0148.13502)]. The interest for these kind of fixed point iterative methods seems to have been reawakened by two recent papers, namely, [C. H. Morales and J. S. Jung, Proc. Am. Math. Soc. 128, 3411–3419 (2000; Zbl 0970.47039)], devoted to the study of path convergence for pseudo-contractive mappings and [A. Moudafi, J. Math. Anal. Appl. 241, 46–55 (2000; Zbl 0957.47039)], devoted to viscosity approximation of fixed points for nonexpansive mappings.

The main idea of viscosity (path) methods is to approximate fixed points of a mapping T that has a “rich” set of fixed points, by means of a path defined by a convex combination U λ of T and of a certain contractive type function f which has a unique fixed point (e.g., a strict contraction, or a strongly pseudo-contraction). As the resulting mapping U λ , defined by a parameter λ(0,1), is itself a strict contraction (or a strongly pseudo-contraction, respectively) and therefore has a unique fixed point, the desired path is obtained as this unique fixed point, {x λ }.

The paper under review is intended to obtain path convergence for approximating fixed points of a pseudo-contraction T by means of a path defined by the convex combination of T and of a certain strongly pseudo-contraction h.

MSC:
47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties
47H10Fixed point theorems for nonlinear operators on topological linear spaces
65J15Equations with nonlinear operators (numerical methods)