VISCOSITY APPROXIMATION METHODS WITH A SEQUENCE OF CONTRACTIONS

 

Koji Aoyama† and Yasunori Kimura‡

† Department of Economics, Chiba University, Yayoi-cho, Inage-ku, Chiba-shi, Chiba 263-8522, Japan. aoyama@le.chiba-u.ac.jp

‡ Department of Information Science, Toho University, Miyama, Funabashi, Chiba 274-8510, Japan. yasunori@is.sci.toho-u.ac.jp

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ABSTRACT

The aim of this paper is to prove that, in an appropriate setting, every iterative sequence generated by the viscosity approximation method with a sequence of contractions is convergent whenever so is every iterative sequence generated by the Halpern type iterative method. Then, using our results, we show some convergence theorems for variational inequality problems, zero point problems, and fixed point problems.

Keywords and Phrases: Viscosity approximation method, nonexpansive mapping, fixed point, hybrid steepest descent method.

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RESUMEN

La meta de este artículo es probar en un marco de trabajo adecuado que cada sucesión iterativa generada por el método de aproximación de viscosidad con una sucesión cualquiera de contracciones es convergente como lo es cada sucesión iterativa generada por el método iterativo del tipo Halpern. Así, usando nuestro resultado mostramos algunos teoremas de convergencia para problemas de desigualdades variacionales, problemas de punto cero y problemas de punto fijo.

2010 AMS Mathematics Subject Classification: 47H09, 47J20, 47H10.

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Received: March 2012 / Accepted: March 2013.