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Cubo (Temuco)

versión On-line ISSN 0719-0646

Cubo vol.13 no.1 Temuco  2011

http://dx.doi.org/10.4067/S0719-06462011000100002 

CUBO A Mathematical Journal Vol.13, N° 01, (11-24). March 2011

CONTENTS

 

Weak Convergence Theorems for Maximal Monotone Operators with Nonspreading mappings in a Hilbert space

 

Hiroko Manaka1 and Wataru Takahashi2

Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Ohokayama, Meguroku, Tokyo 152-8552, Japan. email: hiroko.Manaka@is.titech.ac.jp email: wataru@is.titech.ac.jp


ABSTRACT

Let C be a closed convex subset of a real Hilbert space H. Let T be a nonspreading mapping of C into itself, let A be an α-inverse strongly monotone mapping of C into H and let B be a maximal monotone operator on H such that the domain of B is included in C. We introduce an iterative sequence of finding a point of F(T)∩(A+B)-10, where F(T) is the set of fixed points of T and (A + B)-10 is the set of zero points of A + B. Then, we obtain the main result which is related to the weak convergence of the sequence. Using this result, we get a weak convergence theorem for finding a common fixed point of a nonspreading mapping and a nonexpansive mapping in a Hilbert space. Further, we consider the problem for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonspreading mapping.

Keywords: Nonspreading mapping, maximal monotone operator, inverse strongly-monotone mapping, fixed point, iteration procedure.


RESUMEN

Sea C un subconjunto convexo cerrado de un espacio real de Hilbert H. Sea T una asignación de C en sí mismo, sea A una asignación monótona α-inversa de C en H y sea B un operador monotono máximal en H tal que el dominio de B está incluido en C. Se introduce una secuencia iterativa para encontrar un punto de F(T) n (A + B)-10, donde F(T) es el conjunto de puntos fijos de T y (A + B)-10 es el conjunto de los puntos cero de A + B. Entonces, se obtiene el resultado principal que se relaciona con la convergencia débil de la secuencia.
Utilizando este resultado, obtenemos un teorema de convergencia para encontrar un punto común de una asignación fija y una asignación en un espacio de Hilbert. Además, consideramos el problema para encontrar un elemento común del conjunto de soluciones de un problema de equilibrio y el conjunto de puntos fijos de una asignación.

Mathematics Subject Classification: 46C05.


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Received: June 2009.

Revised: September 2009.

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