THREE-STEP ITERATIONS FOR NONEXPANSIVE MAPPINGS AND INVERSE-STRONGLY MONOTONE MAPPINGS

This paper introduces a three-step iteration for finding a common element of the set of fixedpoints of a nonexpansive mapping and the set of solutions of the variational inequality for an inverse-strongly monotone mapping by viscosity approximation methods in a Hilbert space.The authors showthat the iterative sequence converges strongly to a common element of the two sets,which solves somevariational inequality.Subsequently,the authors consider the problem of finding a common fixed pointof a nonexpansive mapping and a strictly pseudo-contractive mapping and the problem of finding acommon element of the set of fixed points of a nonexpansive mapping and the set of zeros of an inverse-strongly monotone mapping.The results obtained in this paper extend and improve the correspondingresults announced by Nakajo,Takahashi,and Toyoda.

A NEW ITERATIVE METHOD FOR FINDING COMMON SOLUTIONS OF GENERALIZED EQUILIBRIUM PROBLEM,FIXED POINT PROBLEM OF INFINITE k-STRICT PSEUDO-CONTRACTIVE MAPPINGS,AND QUASI-VARIATIONAL INCLUSION PROBLEM

In this article, we introduce a hybrid iterative scheme for finding a common element of the set of solutions for a generalized equilibrium problems, the set of common fixed point for a family of infinite k-strict pseudo-contractive mappings, and the set of solutions of the variational inclusion problem with multi-valued maximal monotone mappings and inverse-strongly monotone mappings in Hilbert space. Under suitable conditions, some strong convergence theorems are proved. Our results extends the recent results in G.L.Acedo and H.K.Xu [2], Zhang, Lee and Chan [8], Wakahashi and Toyoda [9], Takahashi and Takahashi [I0] and S. S. Chang, H. W. Joseph Lee and C. K. Chan [II], S.Takahashi and W.Takahashi [12]. Moreover, the method of proof adopted in this article is different from those of [4] and [12].

SOME RESULTS ON GENERALIZED EQUILIBRIUM PROBLEMS INVOLVING STRICTLY PSEUDOCONTRACTIVE MAPPINGS

In this paper,we consider an iterative sequence for generalized equilibrium problems and strictly pseudocontractive mappings.We show that the iterative sequence converges strongly to a common element of the solution set of generalized equilibrium problems and of the fixed point set of strictly pseudocontractive mappings.

Some Surjectivity Theorems for Maximal Monotone Operators and Its Application

The main purpose of this paper is to study the surjectivity theorems for maximal monotone mapping in reflexive Banach spaces by using fixed point theory. We prove some new surjectivity theorems under some conditions and give its application in differential equation.

Generalized mixed equilibrium problem in Banach spaces

This paper uses a hybrid algorithm to find a common element of the set of solutions to a generalized mixed equilibrium problem, the set of solutions to variational inequality problems, and the set of common fixed points for a finite family of quasi-C- nonexpansive mappings in a uniformly smooth and strictly convex Banach space. As applications, we utilize our results to study the optimization problem. It shows that our results improve and extend the corresponding results announced by many others recently.

APPROXIMATION OF COMMON SOLUTIONS OF VARIATIONAL INEQUALITIES VIA STRICT PSEUDOCONTRACTIONS

In this paper, a convex feasibility problem is considered. We construct an iterative method to approximate a common element of the solution set of classical variational inequalities and of the fixed point set of a strict pseudocontraction. Strong convergence theorems for the common element are established in the framework of Hilbert spaces.

Algorithms of common solutions to quasi variational inclusion and fixed point problems

The purpose of this paper is to present an iterative scheme for finding a common element of the set of solutions to the variational inclusion problem with multivalued maximal monotone mapping and inverse-strongly monotone mappings and the set of fixed points of nonexpansive mappings in Hilbert space.Under suitable conditions, some strong convergence theorems for approximating this common elements are proved. The results presented in the paper not only improve and extend the main results in Korpelevich（Ekonomika i Matematicheskie Metody,1976,12（4）：747-756）,but also extend and replenish the corresponding results obtained by Iiduka and Takahashi（Nonlinear Anal TMA,2005,61（3）：341-350）,Takahashi and Toyoda（J Optim Theory Appl,2003, 118（2）：417-428）,Nadezhkina and Takahashi（J Optim Theory Appl,2006,128（1）：191- 201）,and Zeng and Yao（Taiwan Residents Journal of Mathematics,2006,10（5）：1293-1303）.

Generalized Strongly Nonlinear Quasi-Complementarity Problems

Using the algorithm in this paper, we prove the existence of solutions to the gene-ralized strongly nonlinear quasi-complementarity problems and the convergence of theiterative sequences generated by the algorithm. Our results improve and extend thecorresponding results of Noor and Chang-Huang. Moreover, a more general iterativealgorithm for finding the approximate solution of generalized strongly nonlinear quasi-complementarity problems is also given. It is shown that the approximate solution ob-tained by the iterative scheme converges to the exact solution of this quasi-com-plementarity problem.

A SOLUTION OF A GENERAL EQUILIBRIUM PROBLEM

Under the framework of a real Hilbert space, we introduce a new iterative method for finding a common element of the set of solution of a general equilibrium problem and the set of fixed points of a nonexpansive semigroup. Moreover, a numerical example is presented. This example grantee the main result of the paper.

ON A CLASS OF GENERALIZED NONLINEAR IMPLICIT QUASIVARIATIONAL INCLUSIONS

In this paper, a new class of generalized nonlinear implicit quasivariational inclusions involving a set-valued maximal monotone wrapping are studied. A existence theorem of solutions for this class of generalized nonlinear implicit quasivariational inclusions is Proved without compactness assumptions. A new iterative algorithm for finding approximate solutions of the generalized nonlinear implicit quasivariational inclusions is suggested and analysed and the convergence of iterative sequence generated by the new algorithm is also given, As special cases, some known results in this field are also discussed.

Quadratic minimization for equilibrium problem variational inclusion and fixed point problem

The purpose of this paper is to find the solutions to the quadratic mini- mization problem by using the resolvent approach. Under suitable conditions, some new strong convergence theorems are proved for approximating a solution of the above min- imization problem. The results presented in the paper extend and improve some recent results.

Hybrid projection method for generalized mixed equilibrium problems,variational inequality problems,and fixed point problems in Banach spaces

A new hybrid projection iterative scheme is introduced to approximate a common element of the solution set of a generalized mixed equilibrium problem, the solution set of a variational inequality problem, and the set of fixed points of a relatively weak nonexpansive mapping in the Banach spaces. The obtained results generalize and improve the recent results announced by many other authors.

ON THE PARTIALLY ORDERED METHODS IN THE STUDY OF IMPLICIT VARIATIONAL INEQUALITIES

By using the partially ordered methods, the existence problem of solutions formore general implicit varational inequalities of monotone type in Hausdorfftopological linear spaces are considered. As application we utilize the results presentedin this paper to study the existence of solutions for Nash equilibrium problem and thesemi-linear elliptic differential equations.

Strong Convergence by the Shrinking Projection Method for a Generalized Equilibrium Problems and Hemi-Relatively Nonexpansive Mappings

Motivated by the recent result obtained by Takahashi and Zembayashi in 2008,we prove a strong convergence theorem for finding a common element of the set of solutions of a generalized equilibrium problem and the set of fixed points of a hemi-relatively nonexpansive mapping in a Banach space by using the shrinking projection method.The main results obtained in this paper extend some recent results.

Chaotic Dynamics of Monotone Twist Maps

For a monotone twist map,under certain non-degenerate condition,we showed the existence of infinitely many homoclinic and heteroclinic orbits between two periodic neighboring minimal orbits with the same rotation number,which indicates chaotic dynamics.Our results also apply to geodesics of smooth Riemannian metrics on the two-dimension torus.

A Single Species Model with Impulsive Diffusion

In most models of population dynamics, diffusion between patches is assumedto be continuous or discrete, but in practice many species diffuse only during a single period. Inthis paper we propose a single species model with impulsive diffusion between two patches, whichprovides a more natural description of population dynamics. By using the discrete dynamical systemgenerated by a monotone, concave map for the population, we prove that the map always has a globallystable positive fixed point. This means that a single species system with impulsive diffusionalways has a globally stable positive periodic solution. This result is further substantiated bynumerical simulation. Under impulsive diffusion the single species survives in the two patches.

The Common Solution for the Question of Fixed Points and the Question of Variational Inclusions

In this paper, we introduce an iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inclusion for an inverse-strongly monotone mapping and a maximal monotone mapping in a real Hilbert space. Then we show that the sequence converges strongly to a common element of two sets. Using the result, we consider the problem of finding a common fixed point of a nonexpansive mapping and a strictly pseudocontractive mapping in a real Hilbert space.

A New Iterative Method for Finding Common Solutions of Generalized Equilibrium Problem and Fixed Point Problem in Hilbert Spaces

In this paper,we introduce a new iterative scheme for finding a common element of the set of solutions for a generalized equilibrium problems and the set of fixed points for nonexpansive mappings in Hilbert space.Under suitable conditions,some strong convergence theorems are proved.Our results extend and improve some recent results.

Simultaneous Solutions to Fixed Point and Minimax-Type Inequality Problems(Ⅱ)

As important applications of minimax-type inequalities for a family of functions in [3], in the present paper there are given a number of existence theorems on simultaneous solutions to fixed point and minimax-type inequality problems and to fixed point and variational-type inequality problems. not only abolishing the paracompactness hypothesis on the underlying space and weak- ening the others in the results in[1], but also making them still nicer in form with still more concise and straightforward proofs.

Minimax-Type Inequalities for a Family of Functions with Applications

In the present paper there are given a number of minimax-type inequalities for a family of functions,involving two multivalued mappings,one being strongly decomposable and the other monotone.Hence some applications to the variational-type inequality theory and to the fixed point theory.