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.     

Best proximity point theorems generalizing the contraction principle

The primary objective of this article is to elicit some interesting extensions of the simple but powerful Banach’s contraction principle to the case of non-self-mappings. In fact, due to the fact that best proximity point theorems fit the bill to this end, the proposed extensions are presented as best proximity point theorems for non-self proximal contractions which are more general than the notion of self-contractions.     

Best proximity point theorems for contractive mappings

This article is concerned with some new best proximity point theorems for principal cyclic contractive mappings, proximal cyclic contractive mappings, and proximal contractive mappings. As a consequence, an interesting fixed point theorem, due to Edelstein, for a contractive mapping is obtained from all those best proximity point theorems.     

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.     

Best proximity point theorems for cyclic strongly quasi-contraction mappings

In this paper, we introduce a new class of maps, called cyclic strongly quasi-contractions, which contains the cyclic contractions as a subclass. Then we give some convergence and existence results of best proximity point theorems for cyclic strongly quasi-contraction maps. An example is given to support our main results.     

Best proximity point theorems for single- and set-valued non-self mappings

We study the existence of best proximity points for single-valued non-self mappings. Also, we prove a best proximity point theorem for set-valued non-self mappings in metric spaces with an appropriate geometric property. Examples are given to support the usability of our results.     

Best proximity point theorems for cyclic orbital Meir-Keeler contraction maps

We introduce a notion of cyclic orbital Meir-Keeler contraction and give sufficient conditions for the existence of fixed points and best proximity points of such a map. Our main result is a generalization of a best proximity point result due to Di Bari et al. [C. Di Bari, T. Suzuki, C. Vetro, Best proximity points for cyclic Meir-Keeler contractions, Nonlinear Anal. 69 (2008) 3790-3794].     

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.     

Fixed point theorems for multi-valued contractions in complete metric spaces

In this paper the concept of a contraction for multi-valued mappings in a metric space is introduced and the existence theorems for fixed points of such contractions in a complete metric space are proved. Presented results generalize and improve the recent results of Y. Feng, S. Liu [Y. Feng, S. Liu, Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings, J. Math. Anal. Appl. 317 (2006) 103-112], D. Klim, D. Wardowski [D. Klim, D. Wardowski, Fixed point theorems for set-valued contractions in complete metric spaces, J. Math. Anal. Appl. 334 (2007) 132-139] and several others. The method used in the proofs of our results is new and is simpler than methods used in the corresponding papers. Two examples are given to show that our results are genuine generalization of the results of Feng and Liu and Klim and Wardowski.     

Best proximity point theorems: resolution of an important non-linear programming problem

The purpose of this article is to resolve the non-linear programming problem of globally minimizing the real valued function ${x \longrightarrow d(x, Sx)}$ where S is a non-self-mapping in the setting of a metric space with the distance function ‘d’. An iterative algorithm is also furnished to find a solution of such global optimization problems. As a consequence, one can determine an optimal approximate solution to some equations of the form Sx = x.     

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.     

Fixed point theorems for set-valued contractions in complete metric spaces

The fixed point theory of set-valued contractions which was initiated by Nadler [S.B. Nadler Jr., Multi-valued contraction mappings, Pacific J. Math. 30 (1969) 475-488] was developed in different directions by many authors, in particular, by [S. Reich, Fixed points of contractive functions, Boll. Unione Mat. Ital. 5 (1972) 26-42; N. Mizoguchi, W. Takahashi, Fixed point theorems for multivalued mappings on complete metric spaces, J. Math. Anal. Appl. 141 (1989) 177-188; Y. Feng, S. Liu, Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings, J. Math. Anal. Appl. 317 (2006) 103-112]. In the present paper, the concept of contraction for set-valued maps in metric spaces is introduced and the conditions guaranteeing the existence of a fixed point for such a contraction are established. One of our results essentially generalizes the Nadler and Feng-Liu theorems and is different from the Mizoguchi-Takahashi result. The second result is different from the Reich and Mizoguchi-Takahashi results. The method used in the proofs of our results is inspired by Mizoguchi-Takahashi and Feng-Liu's ideas. Comparisons and examples are given.     

Multidimensional fixed point theorems in partially ordered complete metric spaces

In this paper we propose a notion of coincidence point between mappings in any number of variables and we prove some existence and uniqueness fixed point theorems for nonlinear mappings verifying different kinds of contractive conditions and defined on partially ordered metric spaces. These theorems extend and clarify very recent results that can be found in [T. Gnana-Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65 (7)(2006) 1379–1393], [V. Berinde, M. Borcut, Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces, Nonlinear Anal. 74 (2011) 4889–4897] and [M. Berzig, B. Samet, An extension of coupled fixed point’s concept in higher dimension and applications, Comput. Math. Appl. 63 (8) (2012) 1319–1334].     

Common best proximity points theorems in metric spaces

In this paper, we first give an existence and uniqueness common best proximity point theorem for a pair of non-self mappings, one of which weakly dominates the other proximally. Moreover, an algorithm is exhibited to determine such unique common best proximity point. An example is also given to support our main result. Our main result extends and unifies some well-known results in the literature.     

Best proximity results in regular cone metric spaces

First, we define the notion of distance between two subsets in regular cone metric spaces. Then, we establish some conditions which guarantee the existence of best proximity points for cyclic contraction mappings on regular cone metric spaces.     

Three fixed point theorems for generalized contractions with constants in complete metric spaces

We prove three fixed point theorems for generalized contractions with constants in complete metric spaces, which are generalizations of very recent fixed point theorems due to Suzuki. We also raise one problem concerning the constants.     

Fixed point theorems in metric spaces

In this paper, we establish two general theorems for equivalence between the Meir–Keeler type contractive conditions and the contractive definitions involving gauge functions. One of these theorems is an extension of a recent result of Lim (On characterization of Meir–Keeler contractive maps, Nonlinear Anal. 46 (2001) 113–120).