Weerayuth Nilsrakoo - Academia.edu (original) (raw)

Papers by Weerayuth Nilsrakoo

Research paper thumbnail of Weak convergence theorems for a countable family of relatively nonexpansive mappings

Journal of Nonlinear and Convex Analysis, 2009

This paper is concerned with convergence of an approximating common fixed point sequence of count... more This paper is concerned with convergence of an approximating common fixed point sequence of countable Lipschitzian mappings in a uniformly convex Banach space. We also establish weak convergence theorems for finding a common element of the set of fixed points, the set of solutions of an equilibrium problem, and the set of solutions of a variational inequality. With an appropriate setting, we obtain and improve the corresponding results recently proved by Moudafi [A. Moudafi, Weak convergence theorems for nonexpansive mappings and equilibrium problems. J. Nonlinear Convex Anal. 9 (2008) 37-43], Tada-Takahashi [A. Tada and W. Takahashi, Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem.

Research paper thumbnail of A new three-step fixed point iteration scheme for asymptotically nonexpansive mappings

Applied Mathematics and Computation, Oct 1, 2006

In the present paper, we define and study a new three-step iterative scheme inspired by Suantai [... more In the present paper, we define and study a new three-step iterative scheme inspired by Suantai [J. Math. Anal. Appl. 311 (2005) 506-517]. This scheme includes many well-known iterations, for examples, modified Mann-type, modified Ishikawa-type iterative schemes, and the three-step iterative scheme of Xu and Noor. Several convergence theorems of this scheme are established for asymptotically nonexpansive mappings. Our results extend and improve the recent ones announced by Schu [

Research paper thumbnail of Convergence theorems for a countable family of Lipschitzian mappings

Applied Mathematics and Computation, Aug 15, 2009

This paper is concerned with convergence of an approximating common fixed point sequence of count... more This paper is concerned with convergence of an approximating common fixed point sequence of countable Lipschitzian mappings in a uniformly convex Banach space. We introduce a new condition for a class of mappings to obtain several weak and strong convergence theorems. This new condition is implied by many previous known conditions introduced by many authors. We also apply our results for a class of nonexpansive mappings and asymptotically nonexpansive mappings and we immediately obtain convergence theorems proved by Song-Chen, Kimura-Takahashi, Tan-Xu, and many others.

Research paper thumbnail of A common minimum-norm solution of a generalized equilibrium problem and a fixed point problem for a countable family of nonexpansive mappings

Thai Journal of Mathematics, May 31, 2014

Research paper thumbnail of Strong convergence theorems by Halpern-Mann iterations for relatively nonexpansive mappings in Banach spaces

Applied Mathematics and Computation, Mar 1, 2011

Research paper thumbnail of The James constant of normalized norms on R2

J Inequal Appl, 2006

We introduce a new class of normalized norms on R 2 which properly contains all absolute normaliz... more We introduce a new class of normalized norms on R 2 which properly contains all absolute normalized norms. We also give a criterion for deciding whether a given norm in this class is uniformly nonsquare. Moreover, an estimate for the James constant is presented and the exact value of some certain norms is computed. This gives a partial answer to the question raised by Kato et al.

Research paper thumbnail of Generalized Noor iterations with errors for asymptotically nonexpansive mappings

In the present paper, we define and study a new three-step iterative schemes with errors. Several... more In the present paper, we define and study a new three-step iterative schemes with errors. Several strong convergence theorems of this scheme are established for asymptotically nonexpansive mappings. Our results extend and improve the recent ones announced by M. O. Osilike and S. C. Aniagbosor [Math. Comput. Modelling 32, No. 10, 1181–1191 (2000; Zbl 0971.47038)], Y. J. Cho, H. Zhou and G. Guo [Comput. Math. Appl. 47, No. 4-5, 707-717 (2004; Zbl 1081.47063)], Z. Liu and S. M. Kang [Acta Math. Sin., Engl. Ser. 20, No. 6, 1009–1018 (2004; Zbl 1098.47059)], K. Nammanee, M. A. Noor and S. Suantai [J. Math. Anal. Appl. 314, No. 1, 320–334 (2006; Zbl 1087.47054)], and many others.

Research paper thumbnail of A new strong convergence theorem for non-Lipschitzian mappings in a uniformly convex Banach space

In the present paper, we establish new strong convergence theorems of the modified Mann and the m... more In the present paper, we establish new strong convergence theorems of the modified Mann and the modified Ishikawa iterative scheme with errors for a mapping which is asymptotically nonexpansive in the intermediate sense in a uniformly convex Banach space. Our theorems significantly extend and improve Kim and Kim’s results. The results in the paper even in the case of asymptotically nonexpansive mappings are new.

Research paper thumbnail of Equilibrium problems and Moudafi’s viscosity approximation methods in Hilbert spaces

Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis

We establish an iterative scheme by means of Mann’s method and Moudafi’s method to find a common ... more We establish an iterative scheme by means of Mann’s method and Moudafi’s method to find a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space. We prove a convergence theorem of our iteration under a weaker assumption than in [S. Takahashi and W. Takahashi, J. Math. Anal. Appl. 331, No. 1, 506–515 (2007; Zbl 1122.47056)]. The new iteration considered in the paper is applied to find a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of a variational inequality problem for continuous monotone mappings. Consequently, the corresponding results for α-inverse-strongly monotone mappings, r-strongly monotone mappings and relaxed (γ,r)-cocoercive mappings are obtained, respectively. We also propose a slightly modified Mann-type iteration to obtain a strong convergence theorem for continuous pseudocontractive mappings.

Research paper thumbnail of Strong Convergence Theorems of a Generalized Three-Step Iteration for a Nonexpansive Nonself-Mapping

The purpose of this paper is to establish several strong convergence theorems of a generalized th... more The purpose of this paper is to establish several strong convergence theorems of a generalized three-step fixed point iteration with errors for a nonexpansive nonself-mapping in a uniformly convex Banach space. Our results not only significantly improve and extend the recent ones announced by Thianwan and Suantai (14), Shahzad (12), and many oth- ers, but also unify many known results for a nonexpansive self-mapping.

Research paper thumbnail of Weak and strong convergence theorems for countable Lipschitzian mappings and its applications

Nonlinear Analysis: Theory, Methods & Applications, 2008

We use Mann’s iteration and the hybrid method in mathematical programming to obtain weak and stro... more We use Mann’s iteration and the hybrid method in mathematical programming to obtain weak and strong convergence to common fixed points of a countable family of Lipschitzian mappings. Finally, we apply our results to solve the equilibrium problems and variational inequalities for continuous monotone mappings.

Research paper thumbnail of The James constant of normalized norms on ℝ2

Journal of Inequalities and Applications, 2006

We introduce a new class of normalized norms on R 2 which properly contains all absolute normaliz... more We introduce a new class of normalized norms on R 2 which properly contains all absolute normalized norms. We also give a criterion for deciding whether a given norm in this class is uniformly nonsquare. Moreover, an estimate for the James constant is presented and the exact value of some certain norms is computed. This gives a partial answer to the question raised by Kato et al.

Research paper thumbnail of Strong Convergence to Common Fixed Points of Countable Relatively Quasi-Nonexpansive Mappings

Fixed Point Theory and Applications, 2008

We prove that a sequence generated by the monotone CQ-method converges strongly to a common fixed... more We prove that a sequence generated by the monotone CQ-method converges strongly to a common fixed point of a countable family of relatively quasi-nonexpansive mappings in a uniformly convex and uniformly smooth Banach space. Our result is applicable to a wide class of mappings.

Research paper thumbnail of On the Fixed-Point Set of a Family of Relatively Nonexpansive and Generalized Nonexpansive Mappings

Fixed Point Theory and Applications, 2010

We prove that the set of common fixed points of a given countable family of relatively nonexpansi... more We prove that the set of common fixed points of a given countable family of relatively nonexpansive mappings is identical to the fixed-point set of a single strongly relatively nonexpansive mapping. This answers Kohsaka and Takahashi's question in positive. We also introduce the concept of strongly generalized nonexpansive mappings and prove the analogue version of the result above for Ibaraki-Takahashi's generalized nonexpansive mappings. The duality theorem for two classes of strongly relatively nonexpansive mappings and of strongly generalized nonexpansive mappings is proved.

Research paper thumbnail of A New Strong Convergence Theorem for Equilibrium Problems and Fixed Point Problems in Banach Spaces

Fixed Point Theory and Applications, 2011

We introduce a new iterative sequence for finding a common element of the set of fixed points of ... more We introduce a new iterative sequence for finding a common element of the set of fixed points of a relatively nonexpansive mapping and the set of solutions of an equilibrium problem in a Banach space. Then, we study the strong convergence of the sequences. With an appropriate setting, we obtain the corresponding results due to Takahashi-Takahashi and Takahashi-Zembayashi. Some of our results are established with weaker assumptions.

Research paper thumbnail of An Implicit Iteration for a Countable Family of Nonexpansive Mappings in Banach Spaces

Numerical Functional Analysis and Optimization - NUMER FUNC ANAL OPTIMIZ, 2009

We introduce an implicit sequence for an infinite family of nonexpansive mappings in a uniformly ... more We introduce an implicit sequence for an infinite family of nonexpansive mappings in a uniformly convex Banach space. We prove weak and strong convergence theorems for finding a common fixed point of the mappings. Our results not only include Plubtieng et al. (Numer. Funct. Anal. Optim. 2007; 28:737–749), Kikkawa and Takahashi (Ann. Univ. Mariae Curie-Skłodowska Sect. A 2004; 58:69–78), Kimura and Takahashi (Set-Valued Anal. 2008; 16:597–619) as special cases but also are established under the weaker assumptions.

Research paper thumbnail of Weak convergence theorems for a countable family of relatively nonexpansive mappings

Journal of Nonlinear and Convex Analysis, 2009

This paper is concerned with convergence of an approximating common fixed point sequence of count... more This paper is concerned with convergence of an approximating common fixed point sequence of countable Lipschitzian mappings in a uniformly convex Banach space. We also establish weak convergence theorems for finding a common element of the set of fixed points, the set of solutions of an equilibrium problem, and the set of solutions of a variational inequality. With an appropriate setting, we obtain and improve the corresponding results recently proved by Moudafi [A. Moudafi, Weak convergence theorems for nonexpansive mappings and equilibrium problems. J. Nonlinear Convex Anal. 9 (2008) 37-43], Tada-Takahashi [A. Tada and W. Takahashi, Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem.

Research paper thumbnail of A new three-step fixed point iteration scheme for asymptotically nonexpansive mappings

Applied Mathematics and Computation, Oct 1, 2006

In the present paper, we define and study a new three-step iterative scheme inspired by Suantai [... more In the present paper, we define and study a new three-step iterative scheme inspired by Suantai [J. Math. Anal. Appl. 311 (2005) 506-517]. This scheme includes many well-known iterations, for examples, modified Mann-type, modified Ishikawa-type iterative schemes, and the three-step iterative scheme of Xu and Noor. Several convergence theorems of this scheme are established for asymptotically nonexpansive mappings. Our results extend and improve the recent ones announced by Schu [

Research paper thumbnail of Convergence theorems for a countable family of Lipschitzian mappings

Applied Mathematics and Computation, Aug 15, 2009

This paper is concerned with convergence of an approximating common fixed point sequence of count... more This paper is concerned with convergence of an approximating common fixed point sequence of countable Lipschitzian mappings in a uniformly convex Banach space. We introduce a new condition for a class of mappings to obtain several weak and strong convergence theorems. This new condition is implied by many previous known conditions introduced by many authors. We also apply our results for a class of nonexpansive mappings and asymptotically nonexpansive mappings and we immediately obtain convergence theorems proved by Song-Chen, Kimura-Takahashi, Tan-Xu, and many others.

Research paper thumbnail of A common minimum-norm solution of a generalized equilibrium problem and a fixed point problem for a countable family of nonexpansive mappings

Thai Journal of Mathematics, May 31, 2014

Research paper thumbnail of Strong convergence theorems by Halpern-Mann iterations for relatively nonexpansive mappings in Banach spaces

Applied Mathematics and Computation, Mar 1, 2011

Research paper thumbnail of The James constant of normalized norms on R2

J Inequal Appl, 2006

We introduce a new class of normalized norms on R 2 which properly contains all absolute normaliz... more We introduce a new class of normalized norms on R 2 which properly contains all absolute normalized norms. We also give a criterion for deciding whether a given norm in this class is uniformly nonsquare. Moreover, an estimate for the James constant is presented and the exact value of some certain norms is computed. This gives a partial answer to the question raised by Kato et al.

Research paper thumbnail of Generalized Noor iterations with errors for asymptotically nonexpansive mappings

In the present paper, we define and study a new three-step iterative schemes with errors. Several... more In the present paper, we define and study a new three-step iterative schemes with errors. Several strong convergence theorems of this scheme are established for asymptotically nonexpansive mappings. Our results extend and improve the recent ones announced by M. O. Osilike and S. C. Aniagbosor [Math. Comput. Modelling 32, No. 10, 1181–1191 (2000; Zbl 0971.47038)], Y. J. Cho, H. Zhou and G. Guo [Comput. Math. Appl. 47, No. 4-5, 707-717 (2004; Zbl 1081.47063)], Z. Liu and S. M. Kang [Acta Math. Sin., Engl. Ser. 20, No. 6, 1009–1018 (2004; Zbl 1098.47059)], K. Nammanee, M. A. Noor and S. Suantai [J. Math. Anal. Appl. 314, No. 1, 320–334 (2006; Zbl 1087.47054)], and many others.

Research paper thumbnail of A new strong convergence theorem for non-Lipschitzian mappings in a uniformly convex Banach space

In the present paper, we establish new strong convergence theorems of the modified Mann and the m... more In the present paper, we establish new strong convergence theorems of the modified Mann and the modified Ishikawa iterative scheme with errors for a mapping which is asymptotically nonexpansive in the intermediate sense in a uniformly convex Banach space. Our theorems significantly extend and improve Kim and Kim’s results. The results in the paper even in the case of asymptotically nonexpansive mappings are new.

Research paper thumbnail of Equilibrium problems and Moudafi’s viscosity approximation methods in Hilbert spaces

Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis

We establish an iterative scheme by means of Mann’s method and Moudafi’s method to find a common ... more We establish an iterative scheme by means of Mann’s method and Moudafi’s method to find a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space. We prove a convergence theorem of our iteration under a weaker assumption than in [S. Takahashi and W. Takahashi, J. Math. Anal. Appl. 331, No. 1, 506–515 (2007; Zbl 1122.47056)]. The new iteration considered in the paper is applied to find a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of a variational inequality problem for continuous monotone mappings. Consequently, the corresponding results for α-inverse-strongly monotone mappings, r-strongly monotone mappings and relaxed (γ,r)-cocoercive mappings are obtained, respectively. We also propose a slightly modified Mann-type iteration to obtain a strong convergence theorem for continuous pseudocontractive mappings.

Research paper thumbnail of Strong Convergence Theorems of a Generalized Three-Step Iteration for a Nonexpansive Nonself-Mapping

The purpose of this paper is to establish several strong convergence theorems of a generalized th... more The purpose of this paper is to establish several strong convergence theorems of a generalized three-step fixed point iteration with errors for a nonexpansive nonself-mapping in a uniformly convex Banach space. Our results not only significantly improve and extend the recent ones announced by Thianwan and Suantai (14), Shahzad (12), and many oth- ers, but also unify many known results for a nonexpansive self-mapping.

Research paper thumbnail of Weak and strong convergence theorems for countable Lipschitzian mappings and its applications

Nonlinear Analysis: Theory, Methods & Applications, 2008

We use Mann’s iteration and the hybrid method in mathematical programming to obtain weak and stro... more We use Mann’s iteration and the hybrid method in mathematical programming to obtain weak and strong convergence to common fixed points of a countable family of Lipschitzian mappings. Finally, we apply our results to solve the equilibrium problems and variational inequalities for continuous monotone mappings.

Research paper thumbnail of The James constant of normalized norms on ℝ2

Journal of Inequalities and Applications, 2006

We introduce a new class of normalized norms on R 2 which properly contains all absolute normaliz... more We introduce a new class of normalized norms on R 2 which properly contains all absolute normalized norms. We also give a criterion for deciding whether a given norm in this class is uniformly nonsquare. Moreover, an estimate for the James constant is presented and the exact value of some certain norms is computed. This gives a partial answer to the question raised by Kato et al.

Research paper thumbnail of Strong Convergence to Common Fixed Points of Countable Relatively Quasi-Nonexpansive Mappings

Fixed Point Theory and Applications, 2008

We prove that a sequence generated by the monotone CQ-method converges strongly to a common fixed... more We prove that a sequence generated by the monotone CQ-method converges strongly to a common fixed point of a countable family of relatively quasi-nonexpansive mappings in a uniformly convex and uniformly smooth Banach space. Our result is applicable to a wide class of mappings.

Research paper thumbnail of On the Fixed-Point Set of a Family of Relatively Nonexpansive and Generalized Nonexpansive Mappings

Fixed Point Theory and Applications, 2010

We prove that the set of common fixed points of a given countable family of relatively nonexpansi... more We prove that the set of common fixed points of a given countable family of relatively nonexpansive mappings is identical to the fixed-point set of a single strongly relatively nonexpansive mapping. This answers Kohsaka and Takahashi's question in positive. We also introduce the concept of strongly generalized nonexpansive mappings and prove the analogue version of the result above for Ibaraki-Takahashi's generalized nonexpansive mappings. The duality theorem for two classes of strongly relatively nonexpansive mappings and of strongly generalized nonexpansive mappings is proved.

Research paper thumbnail of A New Strong Convergence Theorem for Equilibrium Problems and Fixed Point Problems in Banach Spaces

Fixed Point Theory and Applications, 2011

We introduce a new iterative sequence for finding a common element of the set of fixed points of ... more We introduce a new iterative sequence for finding a common element of the set of fixed points of a relatively nonexpansive mapping and the set of solutions of an equilibrium problem in a Banach space. Then, we study the strong convergence of the sequences. With an appropriate setting, we obtain the corresponding results due to Takahashi-Takahashi and Takahashi-Zembayashi. Some of our results are established with weaker assumptions.

Research paper thumbnail of An Implicit Iteration for a Countable Family of Nonexpansive Mappings in Banach Spaces

Numerical Functional Analysis and Optimization - NUMER FUNC ANAL OPTIMIZ, 2009

We introduce an implicit sequence for an infinite family of nonexpansive mappings in a uniformly ... more We introduce an implicit sequence for an infinite family of nonexpansive mappings in a uniformly convex Banach space. We prove weak and strong convergence theorems for finding a common fixed point of the mappings. Our results not only include Plubtieng et al. (Numer. Funct. Anal. Optim. 2007; 28:737–749), Kikkawa and Takahashi (Ann. Univ. Mariae Curie-Skłodowska Sect. A 2004; 58:69–78), Kimura and Takahashi (Set-Valued Anal. 2008; 16:597–619) as special cases but also are established under the weaker assumptions.