On variational inclusion and common fixed point problems in Hilbert spaces with applications

The purpose of this paper is to introduce a general iterative method for finding a common element of the solution set of quasi-variational inclusion problems and of the common fixed point set of an infinite family of nonexpansive mappings in the framework Hilbert spaces. Strong convergence of the sequences generated by the purposed iterative scheme is obtained.     

A note on two fixed point problems

We extend the applicability of the Exterior Ellipsoid Algorithm for approximating n-dimensional fixed points of directionally nonexpanding functions. Such functions model many practical problems that cannot be formulated in the smaller class of globally nonexpanding functions. The upper bound 2n²ln(2/) on the number of function evaluations for finding -residual approximations to the fixed points remains the same for the larger class. We also present a modified version of a hybrid bisection-secant method for efficient approximation of univariate fixed point problems in combustion chemistry.     

Solving variational inequality and fixed point problems by line searches and potential optimization

We introduce a general adaptive line search framework for solving fixed point and variational inequality problems. Our goals are to develop iterative schemes that (i) compute solutions when the underlying map satisfies properties weaker than contractiveness, for example, weaker forms of nonexpansiveness, (ii) are more efficient than the classical methods even when the underlying map is contractive, and (iii) unify and extend several convergence results from the fixed point and variational inequality literatures. To achieve these goals, we introduce and study joint compatibility conditions imposed upon the underlying map and the iterative step sizes at each iteration and consider line searches that optimize certain potential functions. As a special case, we introduce a modified steepest descent method for solving systems of equations that does not require a previous condition from the literature (the square of the Jacobian matrix is positive definite). Since the line searches we propose might be difficult to perform exactly, we also consider inexact line searches.Preparation of this paper was supported, in part, from the National Science Foundation NSF Grant 9634736-DMI, as well as the Singapore-MIT AllianceAcknowledgments.We are grateful to the associate editor and the referees for their insightful comments and suggestions that have helped us improve both the exposition and the content of this paper.     

Monotone CQ iteration processes for nonexpansive semigroups and maximal monotone operators

Nakajo and Takahashi [K. Nakajo, W. Takahashi, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl. 279 (2003) 372–379] proved strong convergence theorems for nonexpansive mappings, nonexpansive semigroups and the proximal point algorithm for zero-point of monotone operators in Hilbert spaces by the CQ iteration method. The purpose of this paper is to modify the CQ iteration method of K. Nakajo and W. Takahashi using the monotone CQ method, and to prove strong convergence theorems. In the proof process of this article, the Cauchy sequence method is used, so we proceed without use of the demiclosedness principle and Opial’s condition, and other weak topological techniques.     

A viscosity of extragradient approximation method for finding equilibrium problems, variational inequalities and fixed point problems for nonexpansive mappings

In this paper, we investigate the problem for finding the set of solutions for equilibrium problems, the set of solutions of the variational inequalities for k-Lipschitz continuous mappings and fixed point problems for nonexpansive mappings in a Hilbert space. We introduce a new viscosity extragradient approximation method which is based on the so-called viscosity approximation method and extragradient method. We show that the sequence converges strongly to a common element of the above three sets under some parameters controlling conditions. Finally, we utilize our results to study some convergence problems for finding the zeros of maximal monotone operators. Our results are generalization and extension of the results of Kumam [P. Kumam, Strong convergence theorems by an extragradient method for solving variational inequalities and equilibrium problems in a Hilbert space, Turk. J. Math. 33 (2009) 85–98], Wangkeeree [R. Wangkeeree, An extragradient approximation method for equilibrium problems and fixed point problems of a countable family of nonexpansive mappings, Fixed Point Theory and Applications, 2008, Article ID 134148, 17 pages, doi:10.1155/2008/134148], Yao et al. [Y. Yao, Y.C. Liou, R. Chen, A general iterative method for an finite family of nonexpansive mappings, Nonlinear Analysis 69 (5–6) (2008) 1644–1654], Qin et al. [X. Qin, M. Shang, Y. Su, A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces, Nonlinear Analysis (69) (2008) 3897–3909], and many others.     

Fixed point iteration processes for asymptotic pointwise nonexpansive mappings in Banach spaces

Let X be a uniformly convex Banach space with the Opial property. Let T:C→C be an asymptotic pointwise nonexpansive mapping, where C is bounded, closed and convex subset of X. In this paper, we prove that the generalized Mann and Ishikawa processes converge weakly to a fixed point of T. In addition, we prove that for compact asymptotic pointwise nonexpansive mappings acting in uniformly convex Banach spaces, both processes converge strongly to a fixed point.     

Average optimality for continuous-time Markov decision processes with a policy iteration approach

This paper deals with the average expected reward criterion for continuous-time Markov decision processes in general state and action spaces. The transition rates of underlying continuous-time jump Markov processes are allowed to be unbounded, and the reward rates may have neither upper nor lower bounds. We give conditions on the system's primitive data and under which we prove the existence of the average reward optimality equation and an average optimal stationary policy. Also, under our conditions we ensure the existence of ?-average optimal stationary policies. Moreover, we study some properties of average optimal stationary policies. We not only establish another average optimality equation on an average optimal stationary policy, but also present an interesting “martingale characterization” of such a policy. The approach provided in this paper is based on the policy iteration algorithm. It should be noted that our way is rather different from both the usually “vanishing discounting factor approach” and the “optimality inequality approach” widely used in the previous literature.     

An iterative method for finding common solutions of equilibrium and fixed point problems

We introduce an iterative method for finding a common element of the set of solutions of an equilibrium problem and of the set of fixed points of a finite family of nonexpansive mappings in a Hilbert space. We prove the strong convergence of the proposed iterative algorithm to the unique solution of a variational inequality, which is the optimality condition for a minimization problem.     

Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces

In this paper, we introduce 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. Then, we prove a strong convergence theorem which is connected with Combettes and Hirstoaga's result [P.L. Combettes, S.A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal. 6 (2005) 117-136] and Wittmann's result [R. Wittmann, Approximation of fixed points of nonexpansive mappings, Arch. Math. 58 (1992) 486-491]. Using this result, we obtain two corollaries which improve and extend their results.     

A random fixed point iteration for three random operators on uniformly convex Banach spaces

In the present paper we introduce a random iteration scheme for three random operators defined on a closed and convex subset of a uniformly convex Banach space and prove its convergence to a common fixed point of three random operators. The result is also an extersion of a known theorem in the corresponding non-random case.     

A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces

In this paper, we introduce two iterative schemes by the general iterative method for finding a common element of the set of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space. Then, we prove two strong convergence theorems for nonexpansive mappings to solve a unique solution of the variational inequality which is the optimality condition for the minimization problem. These results extended and improved the corresponding results of Marino and Xu [G. Marino, H.K. Xu, A general iterative method for nonexpansive mapping in Hilbert spaces, J. Math. Anal. Appl. 318 (2006) 43-52], S. Takahashi and W. Takahashi [S. Takahashi, W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (1) (2007) 506-515], and many others.     

Algorithms of common solutions for variational inclusions, mixed equilibrium problems and fixed point problems

In this paper, we present an iterative algorithm for finding a common element of the set of solutions of a mixed equilibrium problem and the set of fixed points of an infinite family of nonexpansive mappings and the set of a variational inclusion in a real Hilbert space. Furthermore, we prove that the proposed iterative algorithm has strong convergence under some mild conditions imposed on algorithm parameters.     

A recursive algorithm for the infinity-norm fixed point problem

We present the PFix algorithm for the fixed point problem f(x)=x on a nonempty domain [a,b], where d1, , and f is a Lipschitz continuous function with respect to the infinity norm, with constant q1. The computed approximation satisfies the residual criterion , where >0. In general, the algorithm requires no more than ∑_i=1^dsⁱ function component evaluations, where s≡max(1,log₂(||b−a||_∞/))+1. This upper bound has order as →0. For the domain [0,1]^d with <0.5 we prove a stronger result, i.e., an upper bound on the number of function component evaluations is , where r≡log₂(1/). This bound approaches as r→∞ (→0) and as d→∞. We show that when q<1 the algorithm can also compute an approximation satisfying the absolute criterion , where x^* is the unique fixed point of f. The complexity in this case resembles the complexity of the residual criterion problem, but with tolerance (1−q) instead of . We show that when q>1 the absolute criterion problem has infinite worst-case complexity when information consists of function evaluations. Finally, we report several numerical tests in which the actual number of evaluations is usually much smaller than the upper complexity bound.     

Strong convergence of a general iterative method for variational inequality problems and fixed point problems in Hilbert spaces

In this paper, we introduce a new iterative scheme to investigate the problem of finding a common element of the set of common fixed points of a finite family of nonexpansive mappings and the set of solutions of a variational inequality problem for a relaxed cocoercive mapping by viscosity approximate methods. Our results improve and extend the recent ones announced by Chen et al. [J.M. Chen, L.J. Zhang, T.G. Fan, Viscosity approximation methods for nonexpansive mappings and monotone mappings, doi:10.1016/j.jmaa.2006.12.088], Iiduka and Tahakshi [H. Iiduka, W. Takahashi, Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings, Nonlinear Anal. 61 (2005) 341–350], Yao and Yao [Y.H. Yao, J.C. Yao, On modified iterative method for nonexpansive mappings and monotone mappings, Appl. Math. Comput, doi:10.1016/j.amc.2006.08.062] and Many others.     

Strong convergence of a new composite iterative method for equilibrium problems and fixed point problems

In this paper, we propose a new composite iterative method for finding a common point of the set of solutions of an equilibrium problem and the set of fixed points of a countable family of nonexpansive mappings in a Hilbert space. It is proved that the sequence generated by the iterative scheme converges strongly to a common point of the set of solutions of an equilibrium problem and the set of fixed points of a countable family of nonexpansive mappings. Our results improve and extend the corresponding ones announced by many others.     

Strong convergence of composite iterative methods for equilibrium problems and fixed point problems

We introduce a new composite iterative scheme by viscosity approximation method for finding a common point of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space. It is proved that the sequence generated by the iterative scheme converges strongly to a common point of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping. Our results substantially improve the corresponding results of Takahashi and Takahashi [A. Takahashi, W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2007) 506-515]. Essentially a new approach for finding solutions of equilibrium problems and the fixed points of nonexpansive mappings is provided.     

Existence of fixed points for the sum of two operators

The purpose of this paper is to study the existence of fixed points for the sum of two nonlinear operators in the framework of real Banach spaces. Later on, we give some examples of applications of this type of results (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimal dividend and capital injection problem in the dual model with proportional and fixed transaction costs

In this paper we consider the dividend payments and capital injections control problem in a dual risk model. Such a model might be appropriate for a company that specializes in inventions and discoveries, which pays costs continuously and has occasional profits. The objective is to maximize the expected present value of the dividends minus the discounted costs of capital injections. This paper can be considered as an extension of Yao et al. (2010), we include fixed transaction costs incurred by capital injections in this paper. This leads to an impulse control problem. Using the techniques of quasi-variational inequalities (QVI), this optimal control problem is solved. Numerical solutions are provided to illustrate the idea and methodologies, and some interesting economic insights are included.     

Computing the smallest fixed point of order-preserving nonexpansive mappings arising in positive stochastic games and static analysis of programs

The problem of computing the smallest fixed point of an order-preserving map arises in the study of zero-sum positive stochastic games. It also arises in static analysis of programs by abstract interpretation. In this context, the discount rate may be negative. We characterize the minimality of a fixed point in terms of the nonlinear spectral radius of a certain semidifferential. We apply this characterization to design a policy iteration algorithm, which applies to the case of finite state and action spaces. The algorithm returns a locally minimal fixed point, which turns out to be globally minimal when the discount rate is nonnegative.