Kodai Mathematical Journal

Some convergence theorems for asymptotically pseudocontractive mappings

Arif Rafiq

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Abstract

Let K be a nonempty closed convex subset of a real Banach space E,T : KK a uniformly L-Lipschitzian asymptotically pseudocontractive mapping with sequence {kn}n ≥ 0 $\subset$ [1, ∞), limn → ∞ kn = 1 such that p $\in$ F(T) = {x $\in$ K : Tx = x}. Let {αn}n ≥ 0 $\subset$ [0,1] be such that ∑n ≥ 0 αn = ∞ and limn → ∞ αn = 0. For arbitrary x0 $\in$ K and {vn}n ≥ 0 in K let {xn}n ≥ 0 be iteratively defined by

xn + 1 = (1 - αn)xn + αn Tnvn, n ≥ 0,

satisfying limn → ∞ ||vn - xn|| = 0. Suppose there exists a strictly increasing function φ : [0, ∞) → [0, ∞), φ (0) = 0 such that

<Tnx - p, j (x - p)> ≤ kn ||x - p||2 - φ (||x - p||), ∀x $\in$ K.

Then {xn}n ≥ 0 converges strongly to p $\in$ F (T).

The remark at the end is important.

Article information

Source
Kodai Math. J. Volume 30, Number 1 (2007), 74-84.

Dates
First available in Project Euclid: 30 March 2007

Permanent link to this document
https://projecteuclid.org/euclid.kmj/1175287623

Digital Object Identifier
doi:10.2996/kmj/1175287623

Mathematical Reviews number (MathSciNet)
MR2319078

Zentralblatt MATH identifier
1138.47048

Citation

Rafiq, Arif. Some convergence theorems for asymptotically pseudocontractive mappings. Kodai Math. J. 30 (2007), no. 1, 74--84. doi:10.2996/kmj/1175287623. https://projecteuclid.org/euclid.kmj/1175287623


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