Arif Rafiq

Implicit fixed point iterations

The paper is published: Rostocker Mathematisches Kolloquium, Rostock. Math. Kolloq. 62, 21 - 39 (2007)

MSC: 47H10   Fixed-point theorems
  47H17  
  54H25   Fixed-point and coincidence theorems [See also 47H10, 55M20]

Abstract:   Let $K$ be a compact convex subset of a real Hilbert space $H$; $% T:K\rightarrow K$ a continuous hemicontractive map. Let $\{\alpha _{n}\}$ be a real sequence in $[0,1]$ satisfying appropriate conditions, then for arbitrary $x_{0}\in K$ and $\{v_{n}\}$ in $K,$ the sequence $\{x_{n}\}$ defined iteratively by $x_{n}=\alpha _{n}x_{n-1}+(1-\alpha _{n})Tv_{n},$ $% n\geq 1$ converges strongly to a fixed point of $T$.
We also establish a strong convergence of an implicit iteration process to a common fixed point for a finite family of $\psi -$uniformly pseudocontractive and $\psi -$uniformly accretive mappings in real Banach spaces.
The results presented in this paper extend and improve the corresponding results of Refs. \cite{ari4, ari9, ari19,ari20, ari22, ari25, ari44}.

Keywords:   Implicit iteration process, Mann iteration, $\psi -$uniformly pseudocontractive and $\psi -$uniformly accretive mappings, Common fixed point, Banach space, Hilbert Space


karin.martin@uni-rostock.de
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