An iterative method for finding coincidence points of two mappings

The problem of finding coincidence points of two mappings of which one is a covering, while the other satisfies the Lipschitz condition, is examined. An iterative method for finding an approximate solution to this problem is discussed. It is based on the a priori estimates derived in the paper.     

On w-compatible mappings and common coupled coincidence point in cone metric spaces

Recently, Abbas et al. [M. Abbas, M.A. Khan, S. Radenovi?, Common coupled fixed point theorems in cone metric spaces for $w$-compatible mapping, Applied Mathematics and Computation (2010) doi:10.1016/j.amc.2010.05.042] introduced the concept of $w$-compatible mappings and obtained results on coupled coincidence point for nonlinear contractive mappings in a cone metric space. In the present paper, we introduce the concept of a common coupled coincidence point of the mappings $F,G:X\times X\to X$ and $f:X\to X$ and we prove some theorems for nonlinear contractive mappings in a cone metric space with a cone having nonempty interior. Our results generalize several well known comparable results in the literature.     

A coupled coincidence point result in partially ordered metric spaces for compatible mappings

In this paper we introduce the notion of compatibility of mappings in a partially ordered metric space and use this notion to establish a coupled coincidence point result. Our work extends the work of Bhaskar and Lakshmikantham [T. Gnana Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. TMA 65 (2006) 1379-1393]. An example is also given.     

On coincidence points for vector mappings

For mappings acting in the product of metric spaces we propose a concept of vector covering. This concept is a natural extension of the notion of covering formappings inmetric spaces. The statements on the solvability of systems of operator equations are proved for the case when the left-hand side of an equation is a value of a vector covering mapping and the right-hand side is Lipschitzian vector mapping. In the scalar case the obtained statements are equivalent to the coincidence point theorems by A. V. Arutyunov. As an application, we prove a statement on the existence of n-fold coincidence points and obtain estimates of the points. The sufficient conditions for n-fold fixed points existence, including the well-known theorems on double fixed point, follow from the obtained results.     

Strong convergence theorems for a common fixed point of two hemi-relatively nonexpansive mappings

The purpose of this article is to prove strong convergence theorems for common fixed points of two closed hemi-relatively nonexpansive mappings in Banach spaces. In order to get the strong convergence theorems, the monotone hybrid algorithms are presented and are used to approximate the common fixed points. Finally, a new simplified hybrid algorithm has been proposed and relative convergence theorem has been proved by using the new method for proofs. The results of this article modify and improve the results of Matsushita, Takahashi [S. Matsushita, W. Takahashi, A strong convergence theorem for relatively nonexpansive mappings in a Banach space, J. Approx. Theory 134 (2005) 257–266] and the results of Plubtieng, Ungchittrakool [S. Plubtieng, K. Ungchittrakool, Strong convergence theorems for a common fixed point of two relatively nonexpansive mappings in a Banach space, J. Approx. Theory 149 (2007) 103–115], and many others.     

Common fixed point theorems for two pairs of non-self mappings

Some common fixed point theorems for two pairs of non-self mappings in complete metrically convex metric spaces are proved which generalize earlier results due to Khan et al. [11], Assad [2], Chatterjea [3] and several others. Some related results and illustrative examples are also discussed.     

Equilibrium structures of two supply chains with price and displayed-quantity competition

This paper studies the equilibrium structure of two competing supply chains, each of which consists of one manufacturer and one retailer who faces the demand influenced by price and displayed quantity. Each chain has two structure options: integration or decentralization. Under linear demand, we present the optimal pricing/displayed quantity of all members in the two chains under possible structures: two integrated chains (II), two decentralized chains (DD), and one integrated chain and one decentralized chain (ID or DI). We then analyse the impact of the intensities of price and displayed-quantity competition on the equilibrium structure of two supply chains. The results show that both price and displayed-quantity competition intensities influence significantly the equilibrium structure. Moreover, under certain specific conditions, both price and displayed-quantity competition can have the two chains fall into the prisoner’s dilemma and play a game of chicken as well.     

Stability of coincidence points and properties of covering mappings

Properties of closed set-valued covering mappings acting from one metric space into another are studied. Under quite general assumptions, it is proved that, if a given α-covering mapping and a mapping satisfying the Lipschitz condition with constant β < α have a coincidence point, then this point is stable under small perturbations (with respect to the Hausdorff metric) of these mappings. This assertion is meaningful for single-valued mappings as well. The structure of the set of coincidence points of an α-covering and a Lipschitzian mapping is studied. Conditions are obtained under which the limit of a sequence of α-covering set-valued mappings is an (α–?)-covering for an arbitrary ? > 0.     

Fixed point theorems for the sum of two weakly sequentially continuous mappings

In the present paper, we prove some Krasnosel’skii-Leray-Schauder type fixed point theorems for weak topology. Some fixed point theorems for the sum of two weakly sequentially continuous mappings are also presented. Our results extend and improve on ones from several earlier works.     

Strong convergence theorems for a common fixed point of two relatively nonexpansive mappings in a Banach space

In this paper, we establish strong convergence theorems for a common fixed point of two relatively nonexpansive mappings in a Banach space by using the hybrid method in mathematical programming. Our results extend and improve the recent ones announced by Matsushita and Takahashi [A strong convergence theorem for relatively nonexpansive mappings in a Banach space, J. Approx. Theory 134 (2005) 257-266], Matinez-yanes and Xu [Strong convergence of the CQ method for fixed point iteration processes, Nonlinear Anal. 64 (2006) 2400-2411], and many others.     

Inertial methods for fixed point problems and zero point problems of the sum of two monotone mappings

ABSTRACT

The purpose of this paper is to investigate the problem of finding a common element of the set of zero points of the sum of two operators and the fixed point set of a quasi-nonexpansive mapping. We introduce modified forward-backward splitting methods based on the so-called inertial forward-backward splitting algorithm, Mann algorithm and viscosity method. We establish weak and strong convergence theorems for iterative sequences generated by these methods. Our results extend and improve some related results in the literature.     

On coincidence points of multivalued vector mappings of metric spaces

The notion of metric regularity can be extended to multivalued mappings acting in the products of metric spaces. A vector analog of Arutyunov’s coincidence-point theorem for two multivalued mappings is proved. Statements on the existence and estimates of solutions of systems of inclusions of special form occurring in the multiple fixed-point problem are obtained. In particular, these results imply some well-known double-point theorems.     

Equations in free groups and coincidence of mappings on surfaces

Let be a continuous map and a constant map between closed orientable surfaces of genus h,g, respectively. By definition the pair (f,c) has the Wecken property if f can be deformed into a map such that every coincidence classes of (f',c) is essential and consists of exactly one point. The main result is that (f,c) has the Wecken property if and only if where . Certain quadratic equations in free groups closely related to the coincidence problem are solved. Received January 26, 1999; in final form December 10, 1999 / Published online March 12, 2001