On the Existence and Uniqueness of Fixed Points in 2-Metric Spaces Using Continuity and Cluster Sequences
DOI:
https://doi.org/10.61952/jlabw.v2i1.478Keywords:
Cluster point, Fixed points, 2-metric spacesAbstract
In this paper, we have established a new and significant fixed point result within the framework of 2-metric spaces. The main theorem is formulated under appropriate and carefully structured assumptions that are naturally adapted to the properties structure of 2-metric spaces. In particular, the proof relies essentially on the continuity property of the 2-metric and on the behavior of cluster sequences and their convergence. This continuity of a 2-metric space plays a crucial role in passing to the limit within the maximum - type inequality considered in our result. Furthermore, the concept of cluster sequences and their convergence is employed to guarantee the existence of limit points that satisfy the fixed point condition. The theorem presented here not only ensures the existence of a fixed point but also provides conditions under which the uniqueness of the fixed point can be obtained.
References
[ 1 ] M. Abdn , M . Abu-Donia & K, AbdRabou, New types of common fixed point theorems 2-metric spaces, Chaos, Soliton and Fractals, 41 (2009),1435-1441.
[ 2 ] A. Aliouche, C. Simpson , Fixed points and lines in 2-metric spaces, Adv. Math , 229 (2012), 668-690.
[ 3 ] B. Fisher , D. Turkoglu, Related fixed points for set valued mappings on 2 - metric
spaces, Int. J. Math. Sci. 23 (2000), 205 – 210.
[ 4 ] S. Gahler , 2 - metric spaces and topological structure, Math. Nachr. 26 (1963),115–148.
[ 5 ] K, Iseki , Fixed point theorems in 2-metric Space , Math . Seminar Notes XIX, 1975.
[ 6 ] M. Khan, On Fixed Point Theorems in 2-Metric Space, Pub De Math insttute, 27 ( 4 ) (1980 ),107-112
[ 7 ] S, Naidu , R Prasad , Fixed point theorems in 2-metric spaces, Indian J. Pure . Appl Math,17(8) (1986), 974-993.
[ 8 ] P, Tiwari , Fixed Point Theorems of Self Mapping in a Complete 2-Metric Spaces , Int. J. Contemp. Math. Sciences, Vol.7 (2012), no. 31,1501 - 1507.
