Let C be a closed convex subset of a real Hilbert space H and assume that T is a κ-strict pseudocontraction on C with a fixed point, for some 0 κ < 1. Given an initial guess x0 ∈ C and given also a real sequence {αn} in (0, 1). The Mann’s algorithm generates a sequence {xn} by the formula: xn+1 = αnxn + (1 − αn)T xn, n 0. It is proved that if the control sequence {αn} is chosen so that κ < αn < 1 and ∞n=0(αn − κ)(1 − αn)=∞, then {xn} converges weakly to a fixed point of T. However this convergence is in general not strong. We then modify Mann’s algorithm by applying projections onto suitably constructed closed convex sets to get an algorithm which generates a strong convergent sequence. This result extends a recent result of Nakajo and Takahashi [K. Nakajo, W. Takahashi, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl. 279 (2003) 372–379] from nonexpansive mappings to strict pseudo-contractions. Keywords: Strict pseudo-contraction; Mann’s algorithm; Weak (strong) convergence; Fixed point; Projection

WEAK AND STRONG CONVERGENCE THEOREMS FOR STRICT PSEUDO-CONTRACTIONS IN HILBERT SPACES

MARINO, Giuseppe;
2007

Abstract

Let C be a closed convex subset of a real Hilbert space H and assume that T is a κ-strict pseudocontraction on C with a fixed point, for some 0 κ < 1. Given an initial guess x0 ∈ C and given also a real sequence {αn} in (0, 1). The Mann’s algorithm generates a sequence {xn} by the formula: xn+1 = αnxn + (1 − αn)T xn, n 0. It is proved that if the control sequence {αn} is chosen so that κ < αn < 1 and ∞n=0(αn − κ)(1 − αn)=∞, then {xn} converges weakly to a fixed point of T. However this convergence is in general not strong. We then modify Mann’s algorithm by applying projections onto suitably constructed closed convex sets to get an algorithm which generates a strong convergent sequence. This result extends a recent result of Nakajo and Takahashi [K. Nakajo, W. Takahashi, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl. 279 (2003) 372–379] from nonexpansive mappings to strict pseudo-contractions. Keywords: Strict pseudo-contraction; Mann’s algorithm; Weak (strong) convergence; Fixed point; Projection
strict pseudo-contraction; Mann's algorithm; weak (strong) convergence
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/20.500.11770/149494
 Attenzione

Attenzione! I dati visualizzati non sono stati sottoposti a validazione da parte dell'ateneo

Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 509
  • ???jsp.display-item.citation.isi??? 490
social impact