ON SOME FIXED POINT RESULTS IN GENERALIZED METRIC SPACE WITH SELF MAPPINGS UNDER THE BOUNDS

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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.

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Abstract

Generalizedmetricspacesareimportantinmanyfieldsandareregardedasmathematicaltools.Theideaofgeneralizedmetric space is introduced in this study, and various sequence convergence qualities are demonstrated. We also go over the continuousandselfmappingsfixedpointextendedresult.

Keyword: Generalizedmetricspaces,Continuousmappings,Selfmappings,Fixedpointtheory.

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 demonstratedtheexistenceofasinglefixedpointforaself-mapthatsatisfiesacontractivecondition.Rhoades[4]discovered certain fixed point theorems and generalized Dhage's contractive condition. The Rhoades contractive condition was also extendedbyDhage's[3]totwomapsinD-metricspace.Dhage[2]discoveredasingularcommonfixedpoint[6]onaD-metric spacebyapplyingtheideaofweakcompatibilityofself-mappings.

A generalized metric on set X is a function such that for any ( ) ( ) and ( ) if and only if , ( ) ( ) ( ( )), where is a permutation, and ( ) ( ) ( ) ( ) ( ). The pair ( ) is referred to be generalized metric space after that. A triangle with the vertices and hasaperidiameterdefinedbyageneralizedmetric ( )

Definition 1.1 If ( ) ( )then ( )( ) ( ( )( ) (

) ( )( ))⟺ ( ) ( ) ( ) ( )forall , -and

Definition 1.2 If ( ) ( )then ( )( ) ( ( )( ) ( )( ) ( )( ))⟺ ( ) ( ) ( ) ( )forall , - and

Definition 1.3 Asequence* +inaD-metricspaceissaidtobeCauchyifforanygiven ,thereexists suchthatforall , ( ) .

Definition 1.4 issaidtobeorbitallycontinuousifforeach ,* + ( )

( ) * + * +.

Theorem 1.5 Let( )beacompleteboundedD-metricspaceand beaselfmapof satisfyingtheconditionthatifthere existsa , )suchthatforall if

( ) * ( ) ( ) ( ) ( ) ( )+

Then hasauniquefixedpoint in and iscontinuousat .

Theorem 1.6 Let ( ) be a compact D-metric space and be a continuous self map of satisfying for all with ( ) ,

( ) * ( ) ( ) ( ) ( ) ( )+

Then hasauniquefixedpoint

Definition 1.7 Let represent a multi-valued map [7] on the ( ) D-metric space. Let . If ( ), then , ,thenasequence* +in issaidtorepresentanorbitof at denotedby ( ).Ifanorbit'sdiameter isfinite,itissaidtobebounded.IfeveryCauchysequenceinitconvergestoapointon ,thenitissaidtobecomplete.

( ) , ( ) ( ) ( )-isanexample[5]ofD-metric.,where isametricon ,and( ) ( ) ( ) ( ) ( ).

Whenasequence* +, inaD-metricspace( )convergestoapoint ,itissaidtobeD-convergent.Ifthereis an suchthat ( )

If ,( ) - , ( )- and

In this case, is a non-decreasing uppersemi-continuousfunctionwith ( ) and ( ) for .Then,on , isuppersemi-continuous,and isnon-decreasingon .Additionally, ,( ) - ( ).Therefore, .

2. OUR RESULTS

Theorem 2.1 If( )isacompletegeneralizedmetricspacewiththreeself-mappingsof thatsatisfythecriterion and suchthat ( ) , ( ) ( ) ( )-

( ) ( ) ( )- ( )

Forall and satisfyingthecondition .Then hasfixedpoint.

Proof: Let and define a sequence * + of points of Now,applyingthegivenconditionweachievetheresultasfollows ( ) ( )

Butwehavetogiventhat so

because . Hence there existssomepoint suchthat because

Nowweshallshowthat isthecommonfixedpoint of and

Now, when we achieve ( ) and this shows that by symmetry we also achieve .Thus, .

Theorem 2.2 If meets the criteria,let bean orbitally continuous mapping of a boundedcomplete D-metricspace into itselfsuchthat

then

each , the sequence * + converges to a fixed point

Research Journal of Engineering and Technology (IRJET) e-ISSN:2395-0056

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Proof: Let be an arbitrary point of . Now we define a sequence * + by

. If for some , , then is a fixed point of . Now assume that From the hypothesis we have satisfiesthecondition

Fromtheaboveconditionweachievetheresult

andalso

) similarly we achieve ( )

. This concludes the result that

)

) ( ) ( ) approaches to as . Which leads us

andhence* +isaCauchysequencebecause .Onthe otherhand iscomplete.So* +convergestoapoint .Thisimpliesthat ( ) .But isorbitally continuous ( )

) but .Also ( ) forany .Hence .Hene isafixedpointof

Corollary 2.3 If be a complete D-metric space and and be self-mapping on satisfy the criterion . ( )/

) ( ) thepair( )isD-compatibleand iscontinuous,andforsome , ) ,

International Research Journal of Engineering and Technology (IRJET) e-ISSN:2395-0056

Volume: 10 Issue: 08 | Aug 2023 www.irjet.net p-ISSN:2395-0072

Consequently, and share a unique commonfixedpoint.

REFERENCES

1. Dhage, B. C. (1992): Generalizedmetricspacesandmappingwithfixedpoints,Bull.Cal.Math.Soc.,84,329-336.

2. Dhage, B. C. (1999): AcommonfixedpointprincipleinD-metricspace,Bull.Cal.Math.Soc.,91,475-480.

3. Dhage, B. C. (1999): SomeresultsoncommonfixedpointI,IndianJ.PureAppl.Math.,30(1999),827-837.

4. Rhoades, B. E. (1996): A fixed point theorem for generalized metric space, International J. Math. and Math. Sci., 19, 457460.

5. Singh, B. et al. (2005): Semi-compatibilityandfixedpointtheoremsinanunboundedD-metricspace,InternationalJ.Math. andMath.Sci.,5,789-801.

6. Ume, J. S., Kim, J. K. (2000): CommonFixedPointTheoremsinD-MetricSpaceswithLocalBoundedness.IndianJournalof PureAppl.Math.31,865-871.

7. Veerapandi, T., Chandrasekhara Rao, K. (1995): Fixed Point Theorems of Some Multivalued Mappings in a D-Metric Space.Bull.Cal.Math.Soc.87,549-556.

Biography of author

Dr.RohitKumarVerma,AssociateProfessor&HOD,DepartmentofMathematics,BhartiVishwavidyalaya,Durg,C.G.,India.

Heisawellknownauthorinthefieldofthejournalscope.HeobtainedhishighestdegreefromRSU,Raipur(C.G.)andworked inengineeringinstitutionforoveradecade.CurrentlyheisworkinginacapacityofAssociateProfessorandHOD,Department of Mathematics, Bharti Vishwavidyalaya, Durg (C.G.). In addition to 30 original research publications in the best journals, he also published two research book by LAP in 2013 and 2023 in the areas of information theory and channel capacity that fascinatehim.HeistheChairperson,theBoardofStudiesDepartmentofMathematicsatBhartiVishwavidyalayainDurg,C.G., India.Hehaspublishedtwopatentsinavarietyoffieldsofresearch.Inadditiontonumerousotherinternationaljournals, he reviews for the American Journal of Applied Mathematics (AJAM). In addition to the Indian Mathematical Society, he is a memberoftheIndianSocietyforTechnicalEducation(ISTE)(IMS)