Abstract and Applied Analysis

Convergence of a Viscosity Iterative Method for Multivalued Nonself-Mappings in Banach Spaces

Jong Soo Jung

Abstract

Let $E$ be a reflexive Banach space having a weakly sequentially continuous duality mapping ${J}_{\phi }$ with gauge function $\phi$, $C$ a nonempty closed convex subset of $E$, and $T:C\to \mathrm{𝒦}\left(E\right)$ a multivalued nonself-mapping such that ${P}_{T}$ is nonexpansive, where ${P}_{T}\left(x\right)=\left\{{u}_{x}\in Tx:\parallel x-{u}_{x}\parallel =d\left(x,Tx\right)\right\}$. Let $f:C\to C$ be a contraction with constant $k$. Suppose that, for each $v\in C$ and $t\in \left(0,1\right)$, the contraction defined by ${S}_{t}x=t{P}_{T}x+\left(1-t\right)v$ has a fixed point ${x}_{t}\in C$. Let $\left\{{\alpha }_{n}\right\}$, $\left\{{\beta }_{n}\right\},$ and $\left\{{\gamma }_{n}\right\}$ be three sequences in $\left(0,1\right)$ satisfying approximate conditions. Then, for arbitrary ${x}_{0}\in C$, the sequence $\left\{{x}_{n}\right\}$ generated by ${x}_{n}\in {\alpha }_{n}f\left({x}_{n-1}\right)+{\beta }_{n}{x}_{n-1}+{\gamma }_{n}{P}_{T}\left({x}_{n}\right)\mathrm{ }$ for all $\mathrm{ }n\ge 1$ converges strongly to a fixed point of $T$.

Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 369412, 7 pages.

Dates
First available in Project Euclid: 18 April 2013

https://projecteuclid.org/euclid.aaa/1366306750

Digital Object Identifier
doi:10.1155/2013/369412

Mathematical Reviews number (MathSciNet)
MR3034954

Zentralblatt MATH identifier
1275.47124

Citation

Jung, Jong Soo. Convergence of a Viscosity Iterative Method for Multivalued Nonself-Mappings in Banach Spaces. Abstr. Appl. Anal. 2013 (2013), Article ID 369412, 7 pages. doi:10.1155/2013/369412. https://projecteuclid.org/euclid.aaa/1366306750

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