Hybrid Method In Mathematical Programming Research Articles

APPROXIMATING COMMON FIXED POINTS OF NONEXPANSIVE SEMIGROUPS IN BANACH SPACES BY METRIC PROJECTIONS

In this paper, we prove a strong convergence theorem by the hybrid method for nonexpansive semigroups in Banach spaces. Using this theorem, we obtain some strong convergence theorems in Banach spaces. Let H be a real Hilbert space with inner product �· , ·� and norm �·� and let C be a nonempty closed convex subset of H. Then, a mapping T : C → C is called nonexpansive (5) ifTx − Ty �≤� x − yfor all x, y ∈ C. We denote by F (T ) the set of fixed points of T. We know iteration procedures for finding a fixed point of a nonexpansive mapping; see, for instance, (11, 14). In 2003, Nakajo and Takahashi (13) studied the following iteration procedure of finding a fixed point of a nonexpansive mapping in a Hilbert space by using the hybrid method in mathematical programming:

Improved gradient method for monotone and lipschitz continuous mappings in banach spaces

Let C be a nonempty closed convex subset of a 2-uniformly convex and uniformly smooth Banach space E and {An}nɛN be a family of monotone and Lipschitz continuos mappings of C into E*. In this article, we consider the improved gradient method by the hybrid method in mathematical programming [10] for solving the variational inequality problem for {An} and prove strong convergence theorems. And we get several results which improve the well-known results in a real 2-uniformly convex and uniformly smooth Banach space and a real Hilbert space.

Approximation of common fixed points of left Bregman strongly nonexpansive mappings and solutions of equilibrium problems

Abstract In this paper we propose an iterative algorithm based on the hybrid method in mathematical programming for approximating a common fixed point of an infinite family of left Bregman strongly nonexpansive mappings which also solves a finite system of equilibrium problems in a reflexive real Banach space. We further prove that our iterative sequence converges strongly to a common fixed point of an infinite family of left Bregman strongly nonexpansive mappings which is also a common solution to a finite system of equilibrium problems. Our result extends many recent and important results in the literature.

Strong convergence for gradient projection method and relatively nonexpansive mappings in Banach spaces

Let C be a nonempty closed convex subset of a 2-uniformly convex and uniformly smooth Banach space E, A be a single valued monotone and Lipschitz continuous mapping of C into E* and T be a single valued relatively nonexpansive mapping of C into itself. In this paper, we consider the composition and the convex combination of T and the gradient projection method for A which Goldstein (1964) proposed and proved the strong convergence to a common element of solutions of the variational inequality problem for A and fixed points of T by the hybrid method in mathematical programming (Haugazeau, 1968). And we get several results which improve the well-known results in a 2-uniformly convex and uniformly smooth Banach space and a Hilbert space.

An Iterative Method for Equilibrium, Variational Inequality, and Fixed Point Problems for a Nonexpansive Semigroup in Hilbert Spaces

The purpose of this paper is to present a new iteration method based on the hybrid method in mathematical programming, extragradient method, and Mann’s method for finding a common element of the solution set of equilibrium problems, the solution set of variational inequality problems for a monotone, Lipschitz continuous mapping and the set of fixed points for a nonexpansive semigroup in Hilbert spaces. We obtain a strong convergence theorem for the sequences generated by this process. The results in this paper generalize and extend some well-known strong convergence theorems in the literature.

A STRONGLY CONVERGENT SHRINKING DESCENT-LIKE HALPERN’S METHOD FOR MONOTONE VARIATIONAL INEQUALITY AND FIXED POINT PROBLEMS

In this paper, we introduce a new iteration method based on the hybrid method in mathematical programming, the descent-like method and the Halpern’s iteration method for finding a common element of the solution set for a variational inequality and the set of common fixed points of a countably infinite family of nonexpansive mappings in Hilbert spaces. The result in this paper modifies and improves some well-known results in the literature for a more general problem.

Hybrid Mann–Halpern Iteration Methods for Finding Fixed Points Involving Asymptotically Nonexpansive Mappings and Semigroups

In this paper, we introduce some new iteration methods combining the hybrid method in mathematical programming with Mann’s iterative method and the Halpern method for finding a fixed point of an asymptotically nonexpansive mapping and a common fixed point of an asymptotically nonexpansive semigroup in a Hilbert space. The main results in this paper modify and improve some well-known results in the literature.

Strong Convergence Theorems for Two Total Quasi-ϕ-Asymptotically Nonexpansive Mappings in Banach Spaces

The purpose of this paper is to establish some strong convergence theorems for a common fixed point of two total quasi-ϕ-asymptotically nonexpansive mappings in Banach space by means of the hybrid method in mathematical programming. The results presented in this paper extend and improve on the corresponding ones announced by Martinez-Yanes and Xu (2006), Plubtieng and Ungchittrakool (2007), Qin et al. (2009), and many others.

A New Hybrid Method for Variational Inequality and Fixed Point Problems

In this paper, we introduce a new iteration method based on the hybrid method in mathematical programming, the descent-like method and the Halpern iteration method for finding a common element of the solution set for a variational inequality and the set of common fixed points of a countably infinite family of nonexpansive mappings in Hilbert spaces. The result in this paper modifies and improves some well-known results in the literature.

Strong convergence for asymptotical pseudocontractions with the demiclosedness principle in banach spaces

The aim of this article is to give an answer to an interesting question proposed in Zhou. At the end of his article, he remarked that it was of great interest to extend his results to certain Banach spaces. So in this article, we extend the demiclosedness principle from Hilbert spaces to Banach spaces. A strong convergence theorem for asymptotical pseudo-contractions in Banach spaces is established. The approaches are based on the extended demiclosedness principle, and the generalized projective operator, and the hybrid method in mathematical programming. Our results extend the previous known results from Hilbert spaces to Banach spaces. MSC: 47H10; 47H09; 47H05.

Hybrid Mann–Halpern iteration methods for nonexpansive mappings and semigroups

In this paper, we introduce some new iteration methods based on the hybrid method in mathematical programming, the Mann’s iterative method and the Halpern’s method for finding a fixed point of a nonexpansive mapping and a common fixed point of a nonexpansive semigroup Hilbert spaces.

Approximations for nonlinear mappings by the hybrid method in Hilbert spaces

The hybrid method in mathematical programming was introduced by Haugazeau (1968) [1] and he proved a strong convergence theorem for finding a common element of finite nonempty closed convex subsets of a real Hilbert space. Later, Bauschke and Combettes (2001) [2] proposed some condition for a family of mappings (the so-called coherent condition) and established interesting results by the hybrid method. The authors (Nakajo et al., 2009) [10] extended Bauschke and Combettes’s results. In this paper, we introduce a condition weaker than the coherent condition and prove strong convergence theorems which generalize the results of Nakajo et al. (2009) [10]. And we get strong convergence theorems for a family of asymptotically κ -strict pseudo-contractions, a family of Lipschitz and pseudo-contractive mappings and a one-parameter uniformly Lipschitz semigroup of pseudo-contractive mappings.

Hybrid Ishikawa iterative methods for a nonexpansive semigroup in Hilbert space

In this paper, on the base of the Ishikawa iteration method and the hybrid method in mathematical programming, we give two new strong convergence methods for finding a point in the common fixed point set of a nonexpansive semigroup in Hilbert space.

Strong convergence theorem of an iterative method for variational inequalities and fixed point problems in Hilbert spaces

In this paper, we introduce a new iteration method based on the hybrid method in mathematical programming and the descent-like method for finding a common element of the set of solutions for a variational inequality and the set of fixed points for a nonexpansive mapping in Hilbert spaces. Our method modifies and improves some methods in literature.

Strong convergence theorem for nonexpansive semigroups in Hilbert space

The purpose of this paper is to prove the strong convergence of a method combining the descent method and the hybrid method in mathematical programming for finding a point in the common fixed point set of a semigroup of nonexpansive mappings in Hilbert space.

Iterative algorithm by using the hybrid method in mathematical programming for solving variational inequality problems and equilibrium problems

In this paper, we introduce an iterative scheme by a new hybrid method for finding a common element of the set of fixed points of a countable family of nonexpansive mappings, the set of solutions of an equilibrium problem and the set of solutions of the variational inequality for α -inverse-strongly monotone mappings in a real Hilbert space. We show that the iterative sequence converges strongly to a common element of the above three sets, which solves some fixed point problems, variational inequality problems and equilibrium problems by using the hybrid method in mathematical programming which connected with optimization problems. The results are connected with Kumam’s results [11, 12], Shinzato and Takahashi’s result [19], Tada and Takahashi’s result [23] and Takahashi’s et al. result [26] and many others.

Strong convergence theorems for a countable family of quasi-Lipschitzian mappings and its applications

We use the hybrid method in mathematical programming to obtain strong convergence to common fixed points of a countable family of quasi-Lipschitzian mappings. As a consequence, several convergence theorems for quasi-nonexpansive mappings and asymptotically κ -strict pseudo-contractions are deduced. We also establish strong convergence of iterative sequences for finding a common element of the set of fixed point, the set of solutions of an equilibrium problem, and the set of solutions of a variational inequality. With an appropriate setting, we obtain the corresponding results due to Tada–Takahashi and Nakajo–Shimoji–Takahashi.

On some algorithms in Banach spaces finding fixed points of nonlinear mappings

By using a modification of the hybrid method in mathematical programming, we give some new algorithm to find out a fixed point of mappings which are in some sense non-expansive. Our results hold in reflexive, strictly convex, smooth Banach spaces with the Kadec˘–Klee property.

Strong Convergence Theorems for Countable Lipschitzian Mappings and Its Applications in Equilibrium and Optimization Problems

The purpose of this paper is to propose a modified hybrid method in mathematical programming and to obtain some strong convergence theorems for common fixed points of a countable family of Lipschitzian mappings. Further, we apply our results to solve the equilibrium and optimization problems. The results of this paper improved and extended the results of W. Nilsrakoo and S. Saejung (2008) and some others in some respects.

Strong convergence theorems by the hybrid method for families of mappings in Banach spaces

Let C be a nonempty closed convex subset of a uniformly convex Banach space E whose norm is Gâteaux differentiable and let { T n } be a family of mappings of C into itself such that the set of all common fixed points of { T n } is nonempty. We consider a sequence { x n } generated by the hybrid method in mathematical programming. And we give the conditions of { T n } under which { x n } converges strongly to a common fixed point of { T n } and generalize the results of [K. Nakajo, K. Shimoji, W. Takahashi, Strong convergence theorems by the hybrid method for families of nonexpansive mappings in Hilbert spaces, Taiwanese J. Math. 10 (2006) 339–360; S. Ohsawa, W. Takahashi, Strong convergence theorems for resolvents of maximal monotone operators in Banach spaces, Arch. Math. 81 (2003) 439–445].