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Viscosity approximation methods for monotone mappings and a countable family of nonexpansive mappings

In: Mathematica Slovaca, vol. 61, no. 2
Poom Kumam - Somyot Plubtieng
Detaily:
Rok, strany: 2011, 257 - 274
Kľúčové slová:
nonexpansive mapping, monotone mapping, equilibrium problem, variational inequality, accretive operator
O článku:
We use viscosity approximation methods to obtain strong convergence to common fixed points of monotone mappings and a countable family of nonexpansive mappings. Let $C$ be a nonempty closed convex subset of a Hilbert space $H$ and $PC$ is a metric projection. We consider the iteration process \{xn\}$ of $C$ defined by $x1=x\in C$ is arbitrary and

$$ xn+1nf(xn)+ (1-αn)SnPC(xnnAxn), $$

where $f$ is a contraction on $C$, $\{Sn\}$ is a sequence of nonexpansive self-mappings of a closed convex subset $C$ of $H$, and $A$ is an inverse-strongly-monotone mapping of $C$ into $H$. It is shown that $\{xn\}$ converges strongly to a common element of the set of common fixed points of a countable family of nonexpansive mappings and the set of solutions of the variational inequality for an inverse-strongly-monotone mapping which solves some variational inequality. Finally, the ideas of our results are applied to find a common element of the set of equilibrium problems and the set of solutions of the variational inequality problem, a zero of a maximal monotone operator and a strictly pseudocontractive mapping in a real Hilbert space. The results of this paper extend and improve the results of Chen, Zhang and Fan.
Ako citovať:
ISO 690:
Kumam, P., Plubtieng, S. 2011. Viscosity approximation methods for monotone mappings and a countable family of nonexpansive mappings. In Mathematica Slovaca, vol. 61, no.2, pp. 257-274. 0139-9918. DOI: https://doi.org/10.2478/s12175-011-0010-9

APA:
Kumam, P., Plubtieng, S. (2011). Viscosity approximation methods for monotone mappings and a countable family of nonexpansive mappings. Mathematica Slovaca, 61(2), 257-274. 0139-9918. DOI: https://doi.org/10.2478/s12175-011-0010-9
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