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

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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.

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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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References

[1] E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student 63 (1994), 123-145.

[2] P. L. Combettes and A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal. 6 (2005), 117-136.

[3] S. Iemoto and W. Takahashi, Approximating common fixed points of nonexpansive mappings and nonspreading mappings in a Hilbert space, to appear.

[4] T. Igarashi, W. Takahashi and K. Tanaka, Weak convergence theorems for nonspreading mappings and equilibrium problems, to appear.

[5] H. Iiduka and W. Takahashi, Weak convergence theorem by Cesáro means for nonexpansive mappings and inverse-strongly monotone mappings, J. Nonlinear Convex Anal. 7 (2006), 105-113.

[6] F. Kosaka and W. Takahashi, Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces, SIAM. J.Optim. 19 (2008), 824-835.

[7] F. Kosaka and W. Takahashi, Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces., Arch. Math. (Basel) 91 (2008), 166-177.

[8] A. Moudafi, Weak convergence theorems for nonexpansive mappings and equilibrium problems, J. Nonlinear Convex Anal., to appear.

[9] A. Moudafi and M. Th´era, Proximal and dynamical approaches to equilibrium problems, Lecture Notes in Economics and Mathematical Systems, 477, Springer, 1999, pp.187-201.

[10] Z. Opial, Weak covergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967), 591-597.

[11] R. T. Rockafellar, On the maximal monotonicity of subdifferential mappings, Pacific J. Math. 33 (1970), 209-216.

[12] J. Schu, Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bull. Austral. Math. Soc. 43 (1991), 153-159.

[13] A. Tada and W. Takahashi, Strong convergence theorem for an equilibrium problem and a nonexpansive mapping, J. Optim. Theory Appl., in press.

[14] S. Takahashi and W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2007), 506-515.

[15] S. Takahashi, W. Takahashi and M. Toyoda, Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spaces, to appear.

[16] W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, 2000.

[17] W. Takahashi, Convex Analysis and Approximation of Fixed Points (Japanese), Yokohama Publishers, Yokohama, 2000.

[18] W. Takahashi, Introduction to Nonlinear and Convex Analysis (Japanese), Yokohama Publishers, Yokohama, 2005.

[19] W. Takahashi and M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl. 118 (2003), 417-428.

[20] K. K. Tan and H. K. Xu, Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl. 178 (1993), 301-308.

[21] H. K. Xu, Another control condition in an iterative method for nonexpansive mappings, Bull. Austral. Math. Soc. 65 (2002), 109-113.

[22] H. K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. 298 (2004), 279-291.

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Received: June 2009.

Revised: September 2009.