International Research Journal of Engineering and Technology (IRJET)
e-ISSN: 2395-0056
Volume: 10 Issue: 08 | Aug 2023
p-ISSN: 2395-0072
www.irjet.net
ON SOME FIXED POINT RESULTS IN GENERALIZED METRIC SPACE WITH SELF MAPPINGS UNDER THE BOUNDS Rohit Kumar Verma Associate Professor, Department of Mathematics, Bharti Vishwavidyalaya, Durg, C.G., India. ---------------------------------------------------------------------------***--------------------------------------------------------------------------Abstract Generalized metric spaces are important in many fields and are regarded as mathematical tools. The idea of generalized metric space is introduced in this study, and various sequence convergence qualities are demonstrated. We also go over the continuous and self mappings fixed point extended result.
Keyword: Generalized metric spaces, Continuous mappings, Self mappings, Fixed point theory. 1. INTRODUCTION The study of fixed point theory has been at the center of vigorous activity, although they arise in many other areas of mathematics. In 1992, Dhage [1] developed the concept of generalized metric space, often known as D-metric space, and demonstrated the existence of a single fixed point for a self-map that satisfies a contractive condition. Rhoades [4] discovered certain fixed point theorems and generalized Dhage's contractive condition. The Rhoades contractive condition was also extended by Dhage's [3] to two maps in D-metric space. Dhage [2] discovered a singular common fixed point [6] on a D-metric space by applying the idea of weak compatibility of self-mappings. () ( ) A generalized metric on set X is a function such that for any and ( ) ) )), where ) if and only if , ( ) ( ( ( is a permutation, and ( ) ( ( ) ( ) ( ). The pair ( ) is referred to be generalized metric space after that. A triangle with the ). vertices and has a peridiameter defined by a generalized metric ( Definition 1.1 If ( )( )
(
)
)( )) ⟺
Definition 1.2 If ( )( )
(
( )
)( )) ⟺
(
Definition 1.3 A sequence * ) , ( . Definition 1.4
(
) then (
)
(
(
) then (
)
(
)
)
)( (
) )
* +
)( )
(
(
) for all
,
- and
)
( (
)( (
)
(
) for all
,*
*
+
Then
* (
)
)
(
has a unique fixed point
in
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,
.
( - and
. , there exists
such that for all
( )
+.
) be a complete bounded D-metric space and Theorem 1.5 Let ( , ) such that for all exists a if (
)( )
+ in a D-metric space is said to be Cauchy if for any given
is said to be orbitally continuous if for each
( )
( (
) and
(
)
(
)
be a self map of (
satisfying the condition that if there
)+.
is continuous at .
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