Phayap Katchang

Fixed point result and applications on a b-metric space endowed with an arbitrary binary relation

In this paper, we introduce the concept of q-set-valued α-quasi-contraction mapping and establish the existence of a fixed point theorem for this mapping in b-metric spaces. Our results are generalizations and extensions of the result of Aydi et al. (Fixed Point Theory Appl. 2012:88, 2012) and some recent results. We also state some illustrative examples to claim that our results properly generalize some results in the literature. Further, by applying the main results, we investigate a fixed point theorem in a b-metric space endowed with an arbitrary binary relation. At the end of this paper, we give open problems for further investigation.MSC:47H10, 54H25.

Mizoguchi-Takahashi’s Fixed Point Theorem with , Functions

We introduce the notion of generalized -admissible mappings. By using this notion, we prove a fixed point theorem. Our result generalizes Mizoguchi-Takahashi’s fixed point theorem. We also provide some examples to show the generality of our work.

Common Solutions of Generalized Mixed Equilibrium Problems, Variational Inclusions, and Common Fixed Points for Nonexpansive Semigroups and Strictly Pseudocontractive Mappings

We introduce a new iterative scheme by shrinking projection method for finding a common element of the set of solutions of generalized mixed equilibrium problems, the set of common solutions of variational inclusion problems with set-valued maximal monotone mappings and inverse-strongly monotone mappings, the set of solutions of fixed points for nonexpansive semigroups, and the set of common fixed points for an infinite family of strictly pseudocontractive mappings in a real Hilbert space. We prove that the sequence converges strongly to a common element of the above four sets under some mind conditions. Furthermore, by using the above result, an iterative algorithm for solution of an optimization problem was obtained. Our results improve and extend the corresponding results of Martinez-Yanes and Xu (2006), Shehu (2011), Zhang et al. (2008), and many authors.

A General Iterative Method of Fixed Points for Mixed Equilibrium Problems and Variational Inclusion Problems

The purpose of this paper is to investigate the problem of finding a common element of the set of solutions for mixed equilibrium problems, the set of solutions of the variational inclusions with set-valued maximal monotone mappings and inverse-strongly monotone mappings, and the set of fixed points of a family of finitely nonexpansive mappings in the setting of Hilbert spaces. We propose a new iterative scheme for finding the common element of the above three sets. Our results improve and extend the corresponding results of the works by Zhang et al. (2008), Peng et al. (2008), Peng and Yao (2009), as well as Plubtieng and Sriprad (2009) and some well-known results in the literature.

A System of Mixed Equilibrium Problems, a General System of Variational Inequality Problems for Relaxed Cocoercive, and Fixed Point Problems for Nonexpansive Semigroup and Strictly Pseudocontractive Mappings

We introduce an iterative algorithm for finding a common element of the set of solutions of a system of mixed equilibrium problems, the set of solutions of a general system of variational inequalities for Lipschitz continuous and relaxed cocoercive mappings, the set of common fixed points for nonexpansive semigroups, and the set of common fixed points for an infinite family of strictly pseudocontractive mappings in Hilbert spaces. Furthermore, we prove a strong convergence theorem of the iterative sequence generated by the proposed iterative algorithm under some suitable conditions which solves some optimization problems. Our results extend and improve the recent results of Chang et al. (2010) and many others.

A Viscosity of Cesàro Mean Approximation Methods for a Mixed Equilibrium, Variational Inequalities, and Fixed Point Problems

We introduce a new iterative method for finding a common element of the set of solutions for mixed equilibrium problem, the set of solutions of the variational inequality for a -inverse-strongly monotone mapping, and the set of fixed points of a family of finitely nonexpansive mappings in a real Hilbert space by using the viscosity and Cesàro mean approximation method. We prove that the sequence converges strongly to a common element of the above three sets under some mind conditions. Our results improve and extend the corresponding results of Kumam and Katchang (2009), Peng and Yao (2009), Shimizu and Takahashi (1997), and some authors.

Strong convergence of the modified Ishikawa iterative method for infinitely many nonexpansive mappings in Banach spaces

In this paper, we introduce a new modified Ishikawa iterative process for computing fixed points of an infinite family nonexpansive mapping in the framework of Banach spaces. Then, we establish the strong convergence theorem of the proposed iterative scheme under some mild conditions which solves a variational inequality. The results obtained in this paper extend and improve on the recent results of Qin et al. [Strong convergence theorems for an infinite family of nonexpansive mappings in Banach spaces, Journal of Computational and Applied Mathematics 230 (1) (2009) 121-127], Cho et al. [Approximation of common fixed points of an infinite family of nonexpansive mappings in Banach spaces, Computers and Mathematics with Applications 56 (2008) 2058-2064] and many others.

Viscosity approximation to a common solution of variational inequality problems and fixed point problems for Lipschitzian semigroup in Banach spaces

We introduce a new iterative algorithm based on a viscosity approximation method for finding the common solution of variational inequality problems for an inverse strongly accretive operator and the solution of fixed point problems for Lipschitzian semigroup mappings in Banach spaces. In controlling suitable conditions, strong convergence theorems are proven. Our results extend and improve the recent results of some authors in the literature in this field.MSC 200047H10; Secondary 47H09, 43A07, 47H20, 47J20

Modified Mann iterative algorithms by hybrid projection methods for nonexpansive semigroups and mixed equilibrium problems

Abstract. In this paper, we propose a modified Mann iterative algorithm by two hybrid projection methods for finding a common element of the set of fixed points of nonexpansive semigroups and the set of solutions of a mixed equilibrium problem in a real Hilbert space. Then, we obtain interesting and new strong convergence theorems for the sequences generated by these processes by using the hybrid projection methods in the mathematical programming. The results presented in this paper extend and improve the corresponding one by Nakajo and Takahashi [J. Math. Anal. Appl. 279 (2003), 372–379].

Strong convergence theorems for solving a general system of finite variational inequalities for finite accretive operators and fixed points of nonexpansive semigroups with weak contraction mappings

In this paper, we prove a strong convergence theorem for finding a common solution of a general system of finite variational inequalities for finite different inverse-strongly accretive operators and solutions of fixed point problems for a nonexpansive semigroup in a Banach space based on a viscosity approximation method by using weak contraction mappings. Moreover, we can apply the above results to find the solutions of the class of k-strictly pseudocontractive mappings and apply a general system of finite variational inequalities into a Hilbert space. The results presented in this paper extend and improve the corresponding results of Ceng et al. (2008), Katchang and Kumam (2011), Wangkeeree and Preechasilp (2012), Yao et al. (2010) and many other authors.MSC:47H05, 47H10, 47J25.