Best proximity point theorems on partially ordered sets

The main purpose of this article is to address a problem that amalgamates approximation and optimization in the setting of a partially ordered set that is endowed with a metric. Indeed, if A and B are non-void subsets of a partially ordered set that is equipped with a metric, and S is a non-self mapping from A to B, this paper scrutinizes the existence of an optimal approximate solution, called a best proximity point of the mapping S, to the operator equation Sx = x where S is a continuous, proximally monotone, ordered proximal contraction. Further, this paper manifests an iterative algorithm for discovering such an optimal approximate solution. As a special case of the result obtained in this article, an interesting fixed point theorem on partially ordered sets is deduced.     

Extensions of Banach's Contraction Principle

Let A and B be nonempty subsets of a metric space. As a non-self mapping T: A → B does not necessarily have a fixed point, it is of considerable interest to find an element x that is as close to Tx as possible. In other words, if the fixed point equation Tx = x has no exact solution, then it is contemplated to find an approximate solution x such that the error d(x, Tx) is minimum. Indeed, best proximity point theorems investigate the existence of such optimal approximate solutions, called best proximity points, of the fixed point equation Tx = x when there is no exact solution. As d(x, Tx) is at least d(A, B), a best proximity point theorem achieves an absolute minimum of the error d(x, Tx) by stipulating an approximate solution x of the fixed point equation Tx = x to satisfy the condition that d(x, Tx) = d(A, B). This article furnishes extensions of Banach's contraction principle to the case of non-self mappings. On account of the preceding argument, the proposed generalizations are formulated as best proximity point theorems for non-self contractions.     

Best proximity point theorems

Let us assume that A and B are non-empty subsets of a metric space. In view of the fact that a non-self mapping T:A?B does not necessarily have a fixed point, it is of considerable significance to explore the existence of an element x that is as close to Tx as possible. In other words, when the fixed point equation Tx=x has no solution, then it is attempted to determine an approximate solution x such that the error d(x,Tx) is minimum. Indeed, best proximity point theorems investigate the existence of such optimal approximate solutions, known as best proximity points, of the fixed point equation Tx=x when there is no solution. Because d(x,Tx) is at least d(A,B), a best proximity point theorem ascertains an absolute minimum of the error d(x,Tx) by stipulating an approximate solution x of the fixed point equation Tx=x to satisfy the condition that d(x,Tx)=d(A,B). This article establishes best proximity point theorems for proximal contractions, thereby extending Banach’s contraction principle to the case of non-self mappings.     

Discrete optimization in partially ordered sets

The primary aim of this article is to resolve a global optimization problem in the setting of a partially ordered set that is equipped with a metric. Indeed, given non-empty subsets A and B of a partially ordered set that is endowed with a metric, and a non-self mapping ${S : A \longrightarrow B}$ , this paper discusses the existence of an optimal approximate solution, designated as a best proximity point of the mapping S, to the equation Sx?=?x, where S is a proximally increasing, ordered proximal contraction. An algorithm for determining such an optimal approximate solution is furnished. Further, the result established in this paper realizes an interesting fixed point theorem in the setting of partially ordered set as a special case.     

Global optimal approximate solutions

Given non-void subsets A and B of a metric space and a non-self mapping T:A? B{T:A\longrightarrow B}, the equation T x = x does not necessarily possess a solution. Eventually, it is speculated to find an optimal approximate solution. In other words, if T x = x has no solution, one seeks an element x at which d(x, T x), a gauge for the error involved for an approximate solution, attains its minimum. Indeed, a best proximity point theorem is concerned with the determination of an element x, called a best proximity point of the mapping T, for which d(x, T x) assumes the least possible value d(A, B). By virtue of the fact that d(x, T x) ≥ d(A, B) for all x in A, a best proximity point minimizes the real valued function x? d(x, T x){x\longrightarrow d(x, T\,x)} globally and absolutely, and therefore a best proximity in essence serves as an ideal optimal approximate solution of the equation T x = x. The aim of this article is to establish a best proximity point theorem for generalized contractions, thereby producing optimal approximate solutions of certain fixed point equations. In addition to exploring the existence of a best proximity point for generalized contractions, an iterative algorithm is also presented to determine such an optimal approximate solution. Further, the best proximity point theorem obtained in this paper generalizes the well-known Banach’s contraction principle.     

Best Proximity Pair Results for Relatively Nonexpansive Mappings in Geodesic Spaces

Given A and B two nonempty subsets in a metric space, a mapping T: A ∪ B → A ∪ B is relatively nonexpansive if d(Tx, Ty) ≤ d(x, y) for every x ∈ A, y ∈ B. A best proximity point for such a mapping is a point x ∈ A ∪ B such that d(x, Tx) = dist(A, B). In this work, we extend the results given in Eldred et al. (2005) [A.A. Eldred, W.A. Kirk, P. Veeramani, Proximal normal structure and relatively nonexpansive mappings, Studia Math. 171, 283–293] for relatively nonexpansive mappings in Banach spaces to more general metric spaces. Namely, we give existence results of best proximity points for cyclic and noncyclic relatively nonexpansive mappings in the context of Busemann convex reflexive metric spaces. Moreover, particular results are proved in the setting of CAT(0) and uniformly convex geodesic spaces. Finally, we show that proximal normal structure is a sufficient but not necessary condition for the existence in A × B of a pair of best proximity points.     

Common Best Proximity Points: Global Optimal Solutions

Let S:A→B and T:A→B be given non-self mappings, where A and B are non-empty subsets of a metric space. As S and T are non-self mappings, the equations Sx=x and Tx=x do not necessarily have a common solution, called a common fixed point of the mappings S and T. Therefore, in such cases of non-existence of a common solution, it is attempted to find an element x that is closest to both Sx and Tx in some sense. Indeed, common best proximity point theorems explore the existence of such optimal solutions, known as common best proximity points, to the equations Sx=x and Tx=x when there is no common solution. It is remarked that the functions x→d(x,Sx) and x→d(x,Tx) gauge the error involved for an approximate solution of the equations Sx=x and Tx=x. In view of the fact that, for any element x in A, the distance between x and Sx, and the distance between x and Tx are at least the distance between the sets A and B, a common best proximity point theorem achieves global minimum of both functions x→d(x,Sx) and x→d(x,Tx) by stipulating a common approximate solution of the equations Sx=x and Tx=x to fulfill the condition that d(x,Sx)=d(x,Tx)=d(A,B). The purpose of this article is to elicit common best proximity point theorems for pairs of contractive non-self mappings and for pairs of contraction non-self mappings, yielding common optimal approximate solutions of certain fixed point equations. Besides establishing the existence of common best proximity points, iterative algorithms are also furnished to determine such optimal approximate solutions.     

Existence and convergence of best proximity points

Consider a self map T defined on the union of two subsets A and B of a metric space and satisfying T(A)⊆B and T(B)⊆A. We give some contraction type existence results for a best proximity point, that is, a point x such that d(x,Tx)=dist(A,B). We also give an algorithm to find a best proximity point for the map T in the setting of a uniformly convex Banach space.     

Best proximity points: global optimal approximate solutions

Let A and B be non-empty subsets of a metric space. As a non-self mapping \({T:A\longrightarrow B}\) does not necessarily have a fixed point, it is of considerable interest to find an element x in A that is as close to Tx in B as possible. In other words, if the fixed point equation Tx = x has no exact solution, then it is contemplated to find an approximate solution x in A such that the error d(x, Tx) is minimum, where d is the distance function. Indeed, best proximity point theorems investigate the existence of such optimal approximate solutions, called best proximity points, to the fixed point equation Tx = x when there is no exact solution. As the distance between any element x in A and its image Tx in B is at least the distance between the sets A and B, a best proximity pair theorem achieves global minimum of d(x, Tx) by stipulating an approximate solution x of the fixed point equation Tx = x to satisfy the condition that d(x, Tx) = d(A, B). The purpose of this article is to establish best proximity point theorems for contractive non-self mappings, yielding global optimal approximate solutions of certain fixed point equations. Besides establishing the existence of best proximity points, iterative algorithms are also furnished to determine such optimal approximate solutions.     

Common best proximity points: global minimal solutions

Let us suppose that A and B are nonempty subsets of a metric space. Let S:A?B and T:A?B be nonself-mappings. Considering the fact S and T are nonself-mappings, it is feasible that the equations Sx=x and Tx=x have no common solution, designated as a common fixed point of the mappings S and T. Eventually, when the equations have no common solution, one contemplates to figure out an element x that is in close proximity to Sx and Tx in the sense that d(x,Sx) and d(x,Tx) are minimum. In fact, common best proximity point theorems scrutinize the existence of such optimal approximate solutions, known as common best proximity points, to the equations Sx=x and Tx=x in the event that the equations have no common solution. Further, one can perceive that the real-valued functions x?d(x,Sx) and x?d(x,Tx) estimate the magnitude of the error involved for any common approximate solution of the equations Sx=x and Tx=x. In light of the fact that the distance between x and Sx, and the distance between x and Tx are at least the distance between A and B for all x in A, a common best proximity point theorem ascertains global minimum of both functions x?d(x,Sx) and x?d(x,Tx) by limiting a common approximate solution of the equations Sx=x and Tx=x to fulfil the requirement that d(x,Sx)=d(A,B) and d(x,Tx)=d(A,B). This article discusses a common best proximity point theorem for a pair of nonself-mappings, one of which dominates the other proximally, thereby yielding common optimal approximate solutions of some fixed point equations when there is no common solution.     

Best proximity point theorems: exposition of a significant non-linear programming problem

The primary goal of this work is to address the non-linear programming problem of globally minimizing the real valued function x → d(x, Tx) where T is presumed to be a non-self mapping that is a generalized proximal contraction in the setting of a metric space. Indeed, an iterative algorithm is presented to determine a solution of the preceding non-linear programming problem that focuses on global optimization. As a sequel, one can compute optimal approximate solutions to some fixed point equations and optimal solutions to some unconstrained non-linear programming problems.     

Best proximity point theorems: unriddling a special nonlinear programming problem

This paper addresses the non-linear programming problem of globally minimizing the real valued function x?d(x,Sx) where S is a generalized proximal contraction in the setting of a metric space. Eventually, one can obtain optimal approximate solutions to some fixed-point equations in the event that they have no solution.     

Existence of Efficient Points in Vector Optimization and Generalized Bishop–Phelps Theorem

In a set without linear structure equipped with a preorder, we give a general existence result for efficient points. In a topological vector space equipped with a partial order induced by a closed convex cone with a bounded base, we prove another kind of existence result for efficient points; this result does not depend on the Zorn lemma. As applications, we study a solution problem in vector optimization and generalize the Bishop–Phelps theorem to a topological vector space setting by showing that the B-support points of any sequentially complete closed subset A of a topological vector space E is dense in A, where B is any bounded convex subset of E.     

Existence and uniqueness of best proximity points in geodesic metric spaces

A mapping T:A∪B→A∪B such that T(A)⊆B and T(B)⊆A is called a cyclic mapping. A best proximity point x for such a mapping T is a point such that d(x,Tx)= dist(A,B). In this work we provide different existence and uniqueness results of best proximity points in both Banach and geodesic metric spaces. We improve and extend some results on this recent theory and give an affirmative partial answer to a recently posed question by Eldred and Veeramani in [A.A. Eldred, P. Veeramani Existence and convergence of best proximity points, J. Math. Anal. Appl. 323 (2) (2006) 1001-1006].     

A Problem in Linear Matrix Approximation

Let A be a normal operator in ??(H), H a complex Hilbert space, and let ? _A = ? {AX - XA:X ∈ ??(H)} be the commutator subspace of ??(H) associated with A. If B in ??(H) commutes with A, then B is orthogonal to ?_A with respect to the spectral norm; i.e., the null operator is an element of best approximation of B in ? _A. This was proved by J. Anderson in 1973 and extended by P. J. Maher with respect to the Schatten p-norm recently. We take a look at their result from a more approximation theoretical point of view in the finite dimensional setting; in particular, we characterize all elements of best approximation of B in R_A and prove that the metric projection of H onto ?_A is continuous.     

Extending partial isometries

We show that a finite metric spaceA admits an extension to a finite metric spaceB so that each partial isometry ofA extends to an isometry ofB. We also prove a more precise result on extending a single partial isometry of a finite metric space. Both these results have consequences for the structure of the isometry groups of the rational Urysohn metric space and the Urysohn metric space. Research supported by NSF grant DMS-0400931. I would like to thank Ward Henson and Alekos Kechris for conversations and emails regarding the paper. Part of this work was done when I visited Caltech in May 2004. I thank the mathematics department there for support     

Common best proximity points: global minimization of multi-objective functions

Given non-empty subsets A and B of a metric space, let ${S{:}A{\longrightarrow} B}$ and ${T {:}A{\longrightarrow} B}$ be non-self mappings. Due to the fact that S and T are non-self mappings, the equations Sx = x and Tx = x are likely to have no common solution, known as a common fixed point of the mappings S and T. Consequently, when there is no common solution, it is speculated to determine an element x that is in close proximity to Sx and Tx in the sense that d(x, Sx) and d(x, Tx) are minimum. As a matter of fact, common best proximity point theorems inspect the existence of such optimal approximate solutions, called common best proximity points, to the equations Sx = x and Tx = x in the case that there is no common solution. It is highlighted that the real valued functions ${x{\longrightarrow}d(x, Sx)}$ and ${x{\longrightarrow}d(x, Tx)}$ assess the degree of the error involved for any common approximate solution of the equations Sx = x and Tx = x. Considering the fact that, given any element x in A, the distance between x and Sx, and the distance between x and Tx are at least d(A, B), a common best proximity point theorem affirms global minimum of both functions ${x{\longrightarrow}d(x, Sx)}$ and ${x{\longrightarrow}d(x, Tx)}$ by imposing a common approximate solution of the equations Sx = x and Tx = x to satisfy the constraint that d(x, Sx) = d(x, Tx) = d(A, B). The purpose of this article is to derive a common best proximity point theorem for proximally commuting non-self mappings, thereby producing common optimal approximate solutions of certain simultaneous fixed point equations in the event there is no common solution.     

Common fixed point results for four mappings satisfying almost generalized (S, T)-contractive condition in partially ordered metric spaces

In this paper, we define the concept of almost generalized (S, T)-contractive condition, and prove some common fixed point results for four mappings satisfying almost generalized (S, T)-contractive condition in partially ordered metric space. An example is given to support the usability of our results.     

Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces

We introduce the concept of a mixed g-monotone mapping and prove coupled coincidence and coupled common fixed point theorems for such nonlinear contractive mappings in partially ordered complete metric spaces. Presented theorems are generalizations of the recent fixed point theorems due to Bhaskar and Lakshmikantham [T.G. Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. TMA 65 (2006) 1379–1393] and include several recent developments.