A RELATED FIXED POINT THEOREM FOR 4 MAPINGS ON 4 COMPLETE METRIC SPACES

KIKINA (GJONI), LULJETA; KIKINA, KRISTAQ

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

Title: A RELATED FIXED POINT THEOREM FOR 4 MAPINGS ON 4 COMPLETE METRIC SPACES

Titulli shqip: NJE TEOREME E PIKAVE FIKSE PER 4 PASQYRIME NE 4 HAPESIRA TE PLOTA METRIKE

Abstract: We prove a related fixed point theorem for 4 mappings in 4 metric spaces using an implicit relation. This result extends, generalizes and unifies several of well-known fixed point theorems for contractive-type mappings on metric spaces. Two examples are obtained to illustrate the theorem. One of the conditions of the theorems is the continuity of three from four mappings. A counterexample shows that this condition is necessary. The extending of the well-known results for one, two and three metric spaces to four metric spaces and their generalization, has been obtained by using the implicit relation. Corollaries of our theorem are considered. From each form of implicit relation is taken a respective corollary. In our theorem, the continuity of all mappings except one, is necessary. The investigation of these results in case of an arbitrary number of m mappings to m complete metric spaces is an open problem for future consideration.

Abstrakti: Ne provojme nje teoreme te pikave fikse per 4 pasqyrime n 4 hapesira mtrike duke perdorur relacione implicite. Ky rezultat shtrin, pergjitheson dhe unifikon teorema te njohura te pikave fikse per pasqyrime tipit kontraktiv ne hapesirat metrike. Jane ndertuar dy shembuj ilustrues per teoremen. Nje nga kushtet e teoremes eshte vazhdueshmeria e tre nga kater pasqyrimet. Nje kundershembull tregon se ky kusht eshte i nevojshem. Shtrirja e rezultateve te njohura per nje, dy dhe tre hapesira metrike ne 4 hapesira metrike dhe pergjithesimi i tyre, eshte fituar nga perdorimi i relacionit implicit. Jane shqyrtuar dhe rrjedhime te toremes tone. Per cdo forme te relacionit implicit eshte marre nje rrjedhim perkates. Ne teoremen tone, vazhdueshmeria e gjithe pasqyrimve pervec njerit, eshte e nevojshme. Shqyrtimi i ketyre rezultateve ne rastin e nje numri te cfardoshem m pasqyrimesh te m hapesirave metrike eshte nje problem i hapur per shqyrtim ne te ardhmen.

Key words: Complete metric spaces, fixed points, implicit relation

Fjalë kyçe: hapesire plote metrike, pika fikse, relacion implicit

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

Authors: LULJETA KIKINA (GJONI), UNIVERSITY OF GJIROKASTRA, Gjirokastër, Albania
KRISTAQ KIKINA, 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 RELATED FIXED POINT THEOREM FOR 4 MAPINGS ON 4 COMPLETE METRIC SPACES
Titulli shqip:	NJE TEOREME E PIKAVE FIKSE PER 4 PASQYRIME NE 4 HAPESIRA TE PLOTA METRIKE
Abstract:	We prove a related fixed point theorem for 4 mappings in 4 metric spaces using an implicit relation. This result extends, generalizes and unifies several of well-known fixed point theorems for contractive-type mappings on metric spaces. Two examples are obtained to illustrate the theorem. One of the conditions of the theorems is the continuity of three from four mappings. A counterexample shows that this condition is necessary. The extending of the well-known results for one, two and three metric spaces to four metric spaces and their generalization, has been obtained by using the implicit relation. Corollaries of our theorem are considered. From each form of implicit relation is taken a respective corollary. In our theorem, the continuity of all mappings except one, is necessary. The investigation of these results in case of an arbitrary number of m mappings to m complete metric spaces is an open problem for future consideration.
Abstrakti:	Ne provojme nje teoreme te pikave fikse per 4 pasqyrime n 4 hapesira mtrike duke perdorur relacione implicite. Ky rezultat shtrin, pergjitheson dhe unifikon teorema te njohura te pikave fikse per pasqyrime tipit kontraktiv ne hapesirat metrike. Jane ndertuar dy shembuj ilustrues per teoremen. Nje nga kushtet e teoremes eshte vazhdueshmeria e tre nga kater pasqyrimet. Nje kundershembull tregon se ky kusht eshte i nevojshem. Shtrirja e rezultateve te njohura per nje, dy dhe tre hapesira metrike ne 4 hapesira metrike dhe pergjithesimi i tyre, eshte fituar nga perdorimi i relacionit implicit. Jane shqyrtuar dhe rrjedhime te toremes tone. Per cdo forme te relacionit implicit eshte marre nje rrjedhim perkates. Ne teoremen tone, vazhdueshmeria e gjithe pasqyrimve pervec njerit, eshte e nevojshme. Shqyrtimi i ketyre rezultateve ne rastin e nje numri te cfardoshem m pasqyrimesh te m hapesirave metrike eshte nje problem i hapur per shqyrtim ne te ardhmen.
Key words:	Complete metric spaces, fixed points, implicit relation
Fjalë kyçe:	hapesire plote metrike, pika fikse, relacion implicit
Corresponding author:	LULJETA KIKINA (GJONI) (gjonileta@yahoo.com)
Authors:	LULJETA KIKINA (GJONI), UNIVERSITY OF GJIROKASTRA, Gjirokastër, Albania KRISTAQ KIKINA, UNIVERSITY OF GJIROKASTRA, Gjirokastër, Albania
Full text:	Download Issue 3 [PDF]