scholarly journals The Existence Theorems of an Optimal Approximate Solution for Generalized Proximal Contraction Mappings

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Wutiphol Sintunavarat ◽  
Poom Kumam
Keyword(s):  
Approximate Solution ◽  
Best Proximity Point ◽  
Existence Theorems ◽  
Contraction Principle ◽  
Optimal Approximate Solution ◽  
Proximal Contraction ◽  
Contraction Mappings ◽  
Generalized Proximal Contraction ◽  
Banach’S Contraction Principle

Recently, Basha (2011) established the best proximity point theorems for proximal contractions of the first and second kinds which are extension of Banach's contraction principle in the case of non-self-mappings. The aim of this paper is to extend and generalize the notions of proximal contractions of the first and second kinds which are more general than the notion of self-contractions, establish the existence of an optimal approximate solution theorems for these non-self-mappings, and also give examples to validate our main results.

scholarly journals Best Proximity Points for Some Classes of Proximal Contractions

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Maryam A. Alghamdi ◽  
Naseer Shahzad ◽  
Francesca Vetro
Keyword(s):  
Approximate Solution ◽  
Nonlinear Programming ◽  
Programming Problem ◽  
Iterative Algorithm ◽  
Best Proximity Point ◽  
Sufficient Conditions ◽  
Nonlinear Programming Problem ◽  
Optimal Approximate Solution ◽  
Unique Point ◽  
Banach’S Contraction Principle

Given a self-mapping and a non-self-mapping , the aim of this work is to provide sufficient conditions for the existence of a unique point , calledg-best proximity point, which satisfies . In so doing, we provide a useful answer for the resolution of the nonlinear programming problem of globally minimizing the real valued function , thereby getting an optimal approximate solution to the equation . An iterative algorithm is also presented to compute a solution of such problems. Our results generalize a result due to Rhoades (2001) and hence such results provide an extension of Banach's contraction principle to the case of non-self-mappings.

scholarly journals On Best Proximity Point Theorems without Ordering

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
A. P. Farajzadeh ◽  
S. Plubtieng ◽  
K. Ungchittrakool
Keyword(s):  
Approximate Solution ◽  
Operator Equation ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Best Proximity Point ◽  
Optimal Approximate Solution ◽  
Proximal Contraction ◽  
Partially Ordered

Recently, Basha (2013) addressed a problem that amalgamates approximation and optimization in the setting of a partially ordered set that is endowed with a metric. He assumed that ifAandBare nonvoid subsets of a partially ordered set that is equipped with a metric andSis a non-self-mapping fromAtoB, then the mappingShas an optimal approximate solution, called a best proximity point of the mappingS, to the operator equationSx=x, whenSis a continuous, proximally monotone, ordered proximal contraction. In this note, we are going to obtain his results by omitting ordering, proximal monotonicity, and ordered proximal contraction onS.

scholarly journals Best Proximity Point for Generalized Proximal Weak Contractions in Complete Metric Space

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Erdal Karapınar ◽  
V. Pragadeeswarar ◽  
M. Marudai
Keyword(s):  
Approximate Solution ◽  
Metric Space ◽  
Existence And Uniqueness ◽  
Metric Spaces ◽  
Best Proximity Point ◽  
Complete Metric Space ◽  
Optimal Approximate Solution ◽  
Weak Contraction ◽  
Complete Metric Spaces ◽  
New Class

We introduce a new class of nonself-mappings, generalized proximal weak contraction mappings, and prove the existence and uniqueness of best proximity point for such mappings in the context of complete metric spaces. Moreover, we state an algorithm to determine such an optimal approximate solution designed as a best proximity point. We establish also an example to illustrate our main results. Our result provides an extension of the related results in the literature.

scholarly journals Best proximity point results for Suzuki-Edelstein proximal contractions via auxiliary functions

Filomat ◽  
2019 ◽  
Vol 33 (2) ◽  
pp. 435-447 ◽  
Author(s):  
Azhar Hussain ◽  
Muhammad Iqbal ◽  
Nawab Hussain
Keyword(s):  
Fixed Point ◽  
Metric Spaces ◽  
Best Proximity Point ◽  
Ordered Metric Spaces ◽  
Proximal Contraction ◽  
Contraction Mappings ◽  
Partially Ordered Metric Spaces ◽  
Complete Metric Spaces ◽  
Partially Ordered ◽  
Auxiliary Functions

In this paper we study the notion of modified Suzuki-Edelstein proximal contraction under some auxiliary functions for non-self mappings and obtain best proximity point theorems in the setting of complete metric spaces. As applications, we derive best proximity point and fixed point results for such contraction mappings in partially ordered metric spaces. Some examples are given to show the validity of our results. Our results extend and unify many existing results in the literature.

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