IJPAM: Volume 109, No. 4 (2016)

ON SOME MISCONCEPTIONS AND
CHATTERJEE-TYPE $G$-CONTRACTION

T. Phaneendra$^1$, S. Saravanan$^2$
$^{1,2}$Department of Mathematics
School of Advanced Sciences,
VIT University
Vellore-632014, Tamil Nadu, INDIA

Abstract. Let $(X,d)$ be a $G$-metric space, $f$, a self-map on $X$ and $x_0\in X$. Some misconceptions are brought about in findings of Mustafa et al [2], and a fixed point theorem for a Chatterjee-type $G$-contraction on a complete $G$-metric space is proved. More over, the unique fixed point p will be its contractive fixed point, in the sense that for each each $x_0\in X$, the $f$-iterates $x_0, fx_0, ..., f^nx_0, ...$ converge to $p$.

Received: Jule 2, 2016

Revised: September 13, 2016

Published: October 9, 2016

AMS Subject Classification: 54H25

Key Words and Phrases: $G$-metric space, Chatterjee-type $G$-contraction, fixed point, $G$-contractive fixed point
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Bibliography

1
S.K. Chatterjea, Fixed-point theorems, C.R. Acad. Bulgare Sci., 25 (1972), 727-730, MR 48#2845.

2
Z. Mustafa, H. Obiedat, F. Awawdeh, Some fixed point theorems for mapping on complete G-Metric Spaces, Fixed Point Theory and Applications (2008), Article ID 189870, 1-12, doi: 10.1155/2008/189870.

3
Z. Mustafa, B. Sims, A new approach to generalized metric spaces, Jour. Nonlinear and Convex Anal., 7, No. 2 (2006), 289-297.

4
T. Phaneendra, K. Kumara Swamy, Unique fixed point in G-metric space through greatest lower bound properties, Novi Sad J. Math., 43 (2013), 107-115.

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DOI: 10.12732/ijpam.v109i4.3 How to cite this paper?

Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2016
Volume: 109
Issue: 4
Pages: 789 - 798


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