zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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 theorems for strictly pseudocontractive mappings of Browder–Petryshyn type. (English) Zbl 1159.47054

This article deals with the iteration scheme

x n+1 =(1-α n )x n +α n Ty n +u n ,y n =(1-β n )x n +β n Tx n ,n=1,2,,

where T:KK is a strictly pseudocontractive mapping with FixT, K is a nonempty closed convex subset of a real q-uniformly smooth Banach space E (q>1), K+KK, u n is a bounded sequence in K, (α n ),(β n ) are sequences in [0,1], and the following conditions are satisfied: n=1 u n <, α n λ(q/c q ) 1/(q-1) , n=1 β n τ < (τ=min{1,(q-1)}).

Recall that a mapping T with domain D(T) is called strictly pseudocontractive if there exists λ>0 such that for all x,yD(T) there exists jJ(x-y) satisfying

(I-T)x-(I-T)y,jλ(I-T)x-(I-T)y 2 ;

here J is the normalized duality (multi)mapping. The constant c q is a minimal constant for which the inequality x+y q x q +qy,J q (x)+c q y q holds.

The main result is that the iterates (x n ) converge strongly to a fixed point of T if and only if (x n 0 is bounded and lim inf n d(x n ,FixT)=0.

47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties
47H10Fixed point theorems for nonlinear operators on topological linear spaces