A FIXED POINT THEOREM FOR   MAPINGS ON   COMPLETE FUZZY METRIC SPACES USING IMPLICIT RELATIONS

KIKINA, KRISTAQ; KIKINA (GJONI), LULJETA; SHARA, JOLLANDA

AKTET - Journal of Institute Alb-Shkenca
AKTET, Volume 4, Issue 3, Matematike, Statistike, 2011

Title: A FIXED POINT THEOREM FOR MAPINGS ON COMPLETE FUZZY METRIC SPACES USING IMPLICIT RELATIONS

Titulli shqip: NJE TEOREMË E PIKAVE FIKSE PËR PASQYRIME NË HAPËSIRA TË PLOTA FUZZY METRIKE DUKE PËRDORUR RELACIONE IMPLICITE

Abstract: In this paper, we give some fixed point theorems for two, three and generally, for m mappings in m fuzzy metric spaces, m-2 of mappings must be continuous. These results extend, generalize, unify and fuzzyfy some of well-known fixed point theorems for contractive-type mappings in metric spaces for example the theorem of Nung, Jain et al., Popa, Telci and Fisher. The extending and generalization of these known results for an arbitrary number m of fuzzy metric spaces is obtained using implicit relations introduced as follows: Let Φm be the set of continuous functions with m variables: φ:[01]m→[0,1],m∈N with the following properties: 1.φ is no decreasing on t1,t2,...,tm variables and 2.φ(t,t,...,t)≥t,for t∈[0,1]. After that, we prove our theorem from which a several corollaries follow according as the forms of implicit function φ. A counterexample proves that the continuity of m-2 between m mappings is necessary.

Abstrakti: Ne kete artikull, japim disa teorema te pikave fikse per dy, tre dhe ne pergjithesi per m pasqyrime ne m hapesira fuzzy metrike, m-2 prej pasqyrimeve duhet te jene te vazhdueshem. Keto rezultate shtrijne, pergjithesojne, unifikojne dhe fuzzifikojne disa teorema te njohura te pikave fikse per pasqyrime tipit kontraktiv ne hapesira metrike, p.sh. teorema Nung, Jain, Popa, Telci dhe Fisher. Shtrirja dhe pergjithesimi i ketyre rezultateve per nje numer cfardoshem m hapesirash fuzzy metrike realizohet duke perdorur relacionet implicite si me poshte : Le te jete Φm bashkesia e funksioneve te vazhdueshem me m varibla : φ:[01]m→[0,1],m∈N me vetite e meposhtme : 1.φ eshte jo zbrites ne lidhje me variablat t1,t2,...,tm dhe 2.φ(t,t,...,t)≥t,per t∈[0,1]. Me pas, provojme teoremen tone, nga e cila fitojme nje sere rrjedhimesh sipas trajtes se funksionit implicit φ. Nje kundrshembull provon qe vazhdueshmeria e m-2 nga m pasqyrimet eshte e nevojshme.

Key words: fixed point, fuzzy metric space, implicit relation

Fjalë kyçe: pika fikse, hapesira fuzzy metrike, relacion implicit

Corresponding author: LULJETA KIKINA (GJONI) (gjonileta@yahoo.com)

Authors: KRISTAQ KIKINA, UNIVERSITY OF GJIROKASTRA, Gjirokastër, Albania
LULJETA KIKINA (GJONI), UNIVERSITY OF GJIROKASTRA, Gjirokastër, Albania
JOLLANDA SHARA, UNIVERSITY OF GJIROKASTRA, Gjirokastër, Albania

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	AKTET Journal of Institute Alb-Shkenca Revistë Shkencore e Institutit Alb-Shkenca ( ISSN 2073-2244 )
(english)

Title:	A FIXED POINT THEOREM FOR MAPINGS ON COMPLETE FUZZY METRIC SPACES USING IMPLICIT RELATIONS
Titulli shqip:	NJE TEOREMË E PIKAVE FIKSE PËR PASQYRIME NË HAPËSIRA TË PLOTA FUZZY METRIKE DUKE PËRDORUR RELACIONE IMPLICITE
Abstract:	In this paper, we give some fixed point theorems for two, three and generally, for m mappings in m fuzzy metric spaces, m-2 of mappings must be continuous. These results extend, generalize, unify and fuzzyfy some of well-known fixed point theorems for contractive-type mappings in metric spaces for example the theorem of Nung, Jain et al., Popa, Telci and Fisher. The extending and generalization of these known results for an arbitrary number m of fuzzy metric spaces is obtained using implicit relations introduced as follows: Let Φm be the set of continuous functions with m variables: φ:[01]m→[0,1],m∈N with the following properties: 1.φ is no decreasing on t1,t2,...,tm variables and 2.φ(t,t,...,t)≥t,for t∈[0,1]. After that, we prove our theorem from which a several corollaries follow according as the forms of implicit function φ. A counterexample proves that the continuity of m-2 between m mappings is necessary.
Abstrakti:	Ne kete artikull, japim disa teorema te pikave fikse per dy, tre dhe ne pergjithesi per m pasqyrime ne m hapesira fuzzy metrike, m-2 prej pasqyrimeve duhet te jene te vazhdueshem. Keto rezultate shtrijne, pergjithesojne, unifikojne dhe fuzzifikojne disa teorema te njohura te pikave fikse per pasqyrime tipit kontraktiv ne hapesira metrike, p.sh. teorema Nung, Jain, Popa, Telci dhe Fisher. Shtrirja dhe pergjithesimi i ketyre rezultateve per nje numer cfardoshem m hapesirash fuzzy metrike realizohet duke perdorur relacionet implicite si me poshte : Le te jete Φm bashkesia e funksioneve te vazhdueshem me m varibla : φ:[01]m→[0,1],m∈N me vetite e meposhtme : 1.φ eshte jo zbrites ne lidhje me variablat t1,t2,...,tm dhe 2.φ(t,t,...,t)≥t,per t∈[0,1]. Me pas, provojme teoremen tone, nga e cila fitojme nje sere rrjedhimesh sipas trajtes se funksionit implicit φ. Nje kundrshembull provon qe vazhdueshmeria e m-2 nga m pasqyrimet eshte e nevojshme.
Key words:	fixed point, fuzzy metric space, implicit relation
Fjalë kyçe:	pika fikse, hapesira fuzzy metrike, relacion implicit
Corresponding author:	LULJETA KIKINA (GJONI) (gjonileta@yahoo.com)
Authors:	KRISTAQ KIKINA, UNIVERSITY OF GJIROKASTRA, Gjirokastër, Albania LULJETA KIKINA (GJONI), UNIVERSITY OF GJIROKASTRA, Gjirokastër, Albania JOLLANDA SHARA, UNIVERSITY OF GJIROKASTRA, Gjirokastër, Albania
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