## Journal of Applied Mathematics

• J. Appl. Math.
• Volume 2012, Special Issue (2012), Article ID 341953, 11 pages.

### An Iterative Algorithm on Approximating Fixed Points of Pseudocontractive Mappings

Youli Yu

#### Abstract

Let E be a real reflexive Banach space with a uniformly Gâteaux differentiable norm. Let K be a nonempty bounded closed convex subset of E, and every nonempty closed convex bounded subset of K has the fixed point property for non-expansive self-mappings. Let $f:K\to K$ a contractive mapping and $T:K\to K$ be a uniformly continuous pseudocontractive mapping with $F(T)\ne \varnothing$. Let $\{{\lambda }_{n}\}\subset (0,1/2)$ be a sequence satisfying the following conditions: (i) ${\mathrm{lim}}_{n\to \infty }{\lambda }_{n}=0$; (ii) ${\sum }_{n=0}^{\infty }{\lambda }_{n}=\infty$. Define the sequence $\{{x}_{n}\}$ in K by ${x}_{0}\in K$, ${x}_{n+1}={\lambda }_{n}f({x}_{n})+(1-2{\lambda }_{n}){x}_{n}+{\lambda }_{n}T{x}_{n}$, for all $n\ge 0$. Under some appropriate assumptions, we prove that the sequence $\{{x}_{n}\}$ converges strongly to a fixed point $p\in F(T)$ which is the unique solution of the following variational inequality: $〈f(p)-p,j(z-p)〉\le 0$, for all $z\in F(T)$.

#### Article information

Source
J. Appl. Math., Volume 2012, Special Issue (2012), Article ID 341953, 11 pages.

Dates
First available in Project Euclid: 15 February 2012

https://projecteuclid.org/euclid.jam/1329337705

Digital Object Identifier
doi:10.1155/2012/341953

Mathematical Reviews number (MathSciNet)
MR2846451

Zentralblatt MATH identifier
1295.47104

#### Citation

Yu, Youli. An Iterative Algorithm on Approximating Fixed Points of Pseudocontractive Mappings. J. Appl. Math. 2012, Special Issue (2012), Article ID 341953, 11 pages. doi:10.1155/2012/341953. https://projecteuclid.org/euclid.jam/1329337705