Best proximity point theorems in the frameworks of fairly and proximally complete spaces

Let us contemplate the problem of solving the linear or non-linear equations of the form T x = g x in the framework of metric space. When T is a non-self mapping and g is a self-mapping, it may cause the non-existence of a solution to the preceding equation. At this juncture, one is of course intere...

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Published in:Fixed point theory and algorithms for sciences and engineering Vol. 19; no. 3; pp. 1939 - 1951
Main Author: Basha, Sadiq
Format: Journal Article
Language:English
Published: Cham Springer International Publishing 01.09.2017
Springer Nature B.V
Subjects:
Analysis
Approximation
Euclidean space
Linear equations
Mapping
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Metric space
Nonlinear equations
Theorems
Global minimization
47H10
Fairly complete space
Proximal contraction
Best proximity point
Proximally complete space
Fixed point
ISSN:1661-7738, 1661-7746, 2730-5422
Online Access:Get full text
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Summary:Let us contemplate the problem of solving the linear or non-linear equations of the form T x = g x in the framework of metric space. When T is a non-self mapping and g is a self-mapping, it may cause the non-existence of a solution to the preceding equation. At this juncture, one is of course interested in computing an approximate solution x ∗ in the space such that T x ∗ is as close to g x ∗ as possible. To be precise, if T is from A to B and g is from A to A , where A and B are subsets of a metric space, one is concerned with the computation of a global minimizer of the mapping x ⟶ d ( g x , T x ) which serves as a measure of closeness between Tx and gx . This paper is concerned with the resolution of the aforesaid global minimization problem if T is a proximal contraction and g is an isometry in the frameworks of fairly and proximally complete spaces.
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ISSN:1661-7738
1661-7746
2730-5422
DOI:10.1007/s11784-016-0324-x