S. Sadiq Basha

Best proximity points: global optimal approximate solutions

Let A and B be non-empty subsets of a metric space. As a non-self mapping

Best Proximity Points: Optimal Solutions

{T:A\longrightarrow B}

Best proximity point theorems for generalized proximal contractions

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.

Transient analysis of a single server queue with catastrophes, failures and repairs

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 Banachs 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.

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

This article elicits a best proximity point theorem for non-self-proximal contractions. As a consequence, it ascertains the existence of an optimal approximate solution to some equations for which it is plausible that there is no solution. Moreover, an algorithm is exhibited to determine such an optimal approximate solution designated as a best proximity point. It is interesting to observe that the preceding best proximity point theorem includes the famous Banach contraction principle.This article elicits a best proximity point theorem for non-self-proximal contractions. As a consequence, it ascertains the existence of an optimal approximate solution to some equations for which it is plausible that there is no solution. Moreover, an algorithm is exhibited to determine such an optimal approximate solution designated as a best proximity point. It is interesting to observe that the preceding best proximity point theorem includes the famous Banach contraction principle.

Full length article: Best proximity point theorems

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.

A single server feedback retrial queue with collisions

Best proximity point theorems unravel the techniques for determining an optimal approximate solution, designated as a best proximity point, to the equation Tx = x which is likely to have no solution when T is a non-self mapping. This article presents best proximity point theorems for new classes of non-self mappings, known as generalized proximal contractions, in the setting of metric spaces. Further, the famous Banachs contraction principle and some of its generalizations and variants are realizable as special cases of the aforesaid best proximity point theorems.Mathematics Subject Classification: 41A65; 46B20; 47H10.

Best proximity point theorems: An exploration of a common solution to approximation and optimization problems

Abstract A transient solution is obtained analytically using continued fractions for the system size in an M/M/1 queueing system with catastrophes, server failures and non-zero repair time. The steady state probability of the system size is present. Some key performance measures, namely, throughput, loss probability and response time for the system under consideration are investigated. Further, reliability and availability of the system are analysed. Finally, numerical illustrations are used to discuss the system performance measures.