Publications (39)6.85 Total impact
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Article: Generalized variational-like inequalities with compositely monotone multifunctions
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ABSTRACT: In this paper, we introduce two classes of generalized vari-ational-like inequalities with compositely monotone multifunctions in Ba-nach spaces. Using the KKM-Fan lemma and the Nadler's result, we prove the existence of solutions for generalized variational-like inequalities with compositely relaxed η − α monotone multifunctions in reflexive Banach spaces. On the other hand we also derive the solvability of generalized variational-like inequalities with compositely relaxed η − α semimonotone multifunctions in arbitrary Banach spaces by virtue of the Kakutani-Fan-Glicksberg fixed-point theorem. The results presented in this paper ex-tend and improve some earlier and recent results in the literature.⃝2008 The Korean Mathematical Society J. Korean Math. Soc. 04/2008; 45(45):841-858. -
Article: Strong Convergence of an Iterative Method with Perturbed Mappings for Nonexpansive and Accretive Operators
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ABSTRACT: A new iterative method for finding a zero of m-accretive operators is proposed. This method, involving a so-called perturbed mapping, provides a way to construct sunny nonexpansive retractions. Several strong convergence theorems for this method are established in a Banach space that is either uniformly smooth or reflexive with a weakly continuous duality map.Numerical Functional Analysis and Optimization 03/2008; 29(3-4):324-345. · 0.71 Impact Factor -
Article: Approximate proximal algorithms for generalized variational inequalities with paramonotonicity and pseudomonotonicity.
Computers & Mathematics with Applications. 01/2008; 55:1262-1269. -
Article: Iterative approximation of solutions for a class of completely generalized set-valued quasi-variational inclusions.
Computers & Mathematics with Applications. 01/2008; 56:978-987. -
Article: Iterative algorithm for finding approximate solutions of mixed quasi-variational-like inclusions.
Computers & Mathematics with Applications. 01/2008; 56:942-952. -
Article: Approximate proximal methods in vector optimization
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ABSTRACT: This paper studies the vector optimization problem of finding weakly efficient points for mappings in a Banach space Y, with respect to the partial order induced by a closed, convex, and pointed cone C ⊂ Y with a nonempty interior. The proximal method in vector optimization is extended to develop an approximate proximal method for this problem by virtue of the approximate proximal point method for finding a root of a maximal monotone operator. In this approximate proximal method, the subproblems consist of finding weakly efficient points for suitable regularizations of the original mapping. We present both an absolute and a relative version, in which the subproblems are solved only approximately. Weak convergence of the generated sequence to a weak efficient point is established. In addition, we also discuss an extension to Bregman-function-based proximal algorithms for finding weakly efficient points for mappings.European Journal of Operational Research 02/2007; · 1.82 Impact Factor -
Article: An extragradient-like approximation method for variational inequality problems and fixed point problems
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ABSTRACT: The purpose of this paper is to investigate the problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of a variational inequality problem for a monotone, Lipschitz continuous mapping. We introduce an extragradient-like approximation method which is based on so-called extragradient method and viscosity approximation method. We establish a strong convergence theorem for two iterative sequences generated by this method.Applied Mathematics and Computation. 01/2007; -
Article: Convergence and certain control conditions for hybrid viscosity approximation methods
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ABSTRACT: Very recently, Yao, Chen and Yao [20] proposed a hybrid viscosity approximation method, which combines the viscosity approximation method and the Mann iteration method. Under the convergence of one parameter sequence to zero, they derived a strong convergence theorem in a uniformly smooth Banach space. In this paper, under the convergence of no parameter sequence to zero, we prove the strong convergence of the sequence generated by their method to a fixed point of a nonexpansive mapping, which solves a variational inequality. An appropriate example such that all conditions of this result are satisfied and their condition βn→0 is not satisfied is provided. Furthermore, we also give a weak convergence theorem for their method involving a nonexpansive mapping in a Hilbert space.Nonlinear Analysis: Theory, Methods & Applications. 73(7):2078-2087. -
Article: Iterative algorithms for a general system of generalized nonlinear mixed composite-type equilibria
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ABSTRACT: In this paper, we consider and study a general system of generalized nonlinear mixed composite-type equilibria in Hilbert spaces. First, we prove the existence and uniqueness of the solution for this system of generalized nonlinear mixed composite-type equilibria. Second, the Mann iterative method with errors is extended to develop some new iterative algorithms for finding approximate solutions for this system of generalized nonlinear mixed composite-type equilibria. We also derive the strong convergence of the sequences generated by these iterative algorithms in Hilbert spaces.Computers & Mathematics with Applications. -
Article: Hybrid viscosity approximation schemes for equilibrium problems and fixed point problems of infinitely many nonexpansive mappings
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ABSTRACT: Recently, Takahashi and Takahashi [S. Takahashi, W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl., 2006, doi:10.1016/j.jmaa.2006.08.036] suggested and analyzed an iterative scheme by the viscosity approximation method for finding 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. In this paper, we introduce a new iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of an equilibrium problem and the set of common fixed points of infinitely many nonexpansive mappings in a Hilbert space. Then, we prove a strong convergence theorem which is the improvements and extension of Takahashi and Takahashi’s (2006) corresponding result. Using this theorem, we obtain two corollaries which improve and extend their corresponding results.Applied Mathematics and Computation. -
Article: Iterative algorithm for solving mixed quasi-variational-like inequalities with skew-symmetric terms in Banach spaces
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Article: Relaxed viscosity approximation methods for fixed point problems and variational inequality problems
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ABSTRACT: Let X be a real strictly convex and reflexive Banach space with a uniformly Gâteaux differentiable norm and C be a nonempty closed convex subset of X. Let be a sequence of nonexpansive self-mappings on C such that the common fixed point set and f:C→C be a given contractive mapping, and {λn} be a sequence of nonnegative numbers in [0,1]. Consider the following relaxed viscosity approximation method where Wn is the W-mapping generated by Tn,Tn−1,…,T1 and λn,λn−1,…,λ1 for each n≥1. It is proven that under very mild conditions on the parameters, the sequence {xn} of approximate solutions generated by the proposed method converges strongly to some p∈F where p is the unique solution in F to the following variational inequality:Nonlinear Analysis: Theory, Methods & Applications. -
Article: Convergence Analysis of a Hybrid Relaxed-Extragradient Method for Monotone Variational Inequalities and Fixed Point Problems
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ABSTRACT: In this paper we introduce a hybrid relaxed-extragradient method for finding a common element of the set of common fixed points of N nonexpansive mappings and the set of solutions of the variational inequality problem for a monotone, Lipschitz-continuous mapping. The hybrid relaxed-extragradient method is based on two well-known methods: hybrid and extragradient. We derive a strong convergence theorem for three sequences generated by this method. Based on this theorem, we also construct an iterative process for finding a common fixed point of N + 1 mappings, such that one of these mappings is taken from the more general class of Lipschitz pseudocontractive mappings and the rest N mappings are nonexpansive. -
Article: Strong convergence of an iterative algorithm for nonself multimaps in Banach spaces
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ABSTRACT: Let E be a uniformly convex Banach space having a uniformly Gâteaux differentiable norm, D a nonempty closed convex subset of E, and T:D→K(E) a nonself multimap such that F(T)≠0̸ and PT is nonexpansive, where F(T) is the fixed point set of T, K(E) is the family of nonempty compact subsets of E and PT(x)={ux∈Tx:‖x−ux‖=d(x,Tx)}. Suppose that D is a nonexpansive retract of E and that for each v∈D and t∈(0,1), the contraction St defined by Stx=tPTx+(1−t)v has a fixed point xt∈D. Let {αn},{βn} and {γn} be three real sequences in (0,1) satisfying approximate conditions. Then for fixed u∈D and arbitrary x0∈D, the sequence {xn} generated by converges strongly to a fixed point of T.Nonlinear Analysis: Theory, Methods & Applications. -
Article: The viscosity approximation method for asymptotically nonexpansive mappings in Banach spaces
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ABSTRACT: A recent trend in the iterative methods for constructing fixed points of nonlinear mappings is to use the viscosity approximation technique. The advantage of this technique is that one can find a particular solution to the associated problems, and in most cases this particular solution solves some variational inequality. In this paper, we try to extend this technique to find a particular common fixed point of a finite family of asymptotically nonexpansive mappings in a Banach space which is reflexive and has a weakly continuous duality map. Both implicit and explicit viscosity approximation schemes are proposed and their strong convergence to a solution to a variational inequality is proved.Nonlinear Analysis: Theory, Methods & Applications. -
Article: A relaxed extragradient-like method for a generalized mixed equilibrium problem, a general system of generalized equilibria and a fixed point problem
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ABSTRACT: Very recently, Takahashi and Takahashi [S. Takahashi, W. Takahashi, Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space, Nonlinear Anal. 69 (2008) 1025–1033] suggested and analyzed an iterative method for finding a common solution of a generalized equilibrium problem and a fixed point problem of a nonexpansive mapping in a Hilbert space. In this paper, based on Takahashi–Takahashi’s iterative method and well-known extragradient method we introduce a relaxed extragradient-like method for finding a common solution of a generalized mixed equilibrium problem, a general system of generalized equilibria and a fixed point problem of a strictly pseudocontractive mapping in a Hilbert space and then obtain a strong convergence theorem. Utilizing this theorem, we establish some new strong convergence results in fixed point problems, variational inequalities, mixed equilibrium problems and systems of generalized equilibria.Nonlinear Analysis: Theory, Methods & Applications. -
Article: A hybrid iterative scheme for mixed equilibrium problems and fixed point problems
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ABSTRACT: The purpose of this paper is to investigate the problem of finding a common element of the set of solutions of a mixed equilibrium problem (MEP) and the set of common fixed points of finitely many nonexpansive mappings in a real Hilbert space. First, by using the well-known KKM technique we derive the existence and uniqueness of solutions of the auxiliary problems for the MEP. Second, by virtue of this result we introduce a hybrid iterative scheme for finding a common element of the set of solutions of MEP and the set of common fixed points of finitely many nonexpansive mappings. Furthermore, we prove that the sequences generated by the hybrid iterative scheme converge strongly to a common element of the set of solutions of MEP and the set of common fixed points of finitely many nonexpansive mappings.Journal of Computational and Applied Mathematics. -
Article: On the convergence analysis of inexact hybrid extragradient proximal point algorithms for maximal monotone operators
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ABSTRACT: In this paper we introduce general iterative methods for finding zeros of a maximal monotone operator in a Hilbert space which unify two previously studied iterative methods: relaxed proximal point algorithm [H.K. Xu, Iterative algorithms for nonlinear operators, J. London Math Soc. 66 (2002) 240–256] and inexact hybrid extragradient proximal point algorithm [R.S. Burachik, S. Scheimberg, B.F. Svaiter, Robustness of the hybrid extragradient proximal-point algorithm, J. Optim. Theory Appl. 111 (2001) 117–136]. The paper establishes both weak convergence and strong convergence of the methods under suitable assumptions on the algorithm parameters.Journal of Computational and Applied Mathematics. -
Article: Fixed point solutions of variational inequalities for a finite family of asymptotically nonexpansive mappings without common fixed point assumption
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ABSTRACT: Let E be a real Banach space with a uniformly Gâteaux differentiable norm and which possesses uniform normal structure, K a nonempty bounded closed convex subset of E, a finite family of asymptotically nonexpansive self-mappings on K with common sequence , {tn},{sn} be two sequences in (0, 1) such that sn+tn=1(n≥1) and f be a contraction on K. Under suitable conditions on the sequences {sn},{tn}, we show the existence of a sequence {xn} satisfying the relation where n=lnN+rn for some unique integers ln≥0 and 1≤rn≤N. Further we prove that {xn} converges strongly to a common fixed point of , which solves some variational inequality, provided ‖xn−Tixn‖→0 as n→∞ for i=1,2,…,N. As an application, we prove that the iterative process defined by , converges strongly to the same common fixed point of .Computers & Mathematics with Applications.
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Institutions
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2008–2012
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Shanghai Normal University
Shanghai, Shanghai Shi, China
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2007
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National Sun Yat-sen University
- Department of Applied Mathematics
Kaohsiung, Kaohsiung, Taiwan
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