# CONTRACTIONS OVER GENERALIZED METRIC SPACES

Volume 2, Issue 3, pp 180-182
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### Authors

I. R. SARMA - FED-II, K.L. University, Vaddeswaram-522502, Guntur district, Andhra Pradesh, India. J. M. RAO2 - Principal, Vijaya Engineering College, Wrya Road, Khammam-507305, Andhra Pradesh, India. S. S. RAO - Department of Basics Sciences and Humanities, Joginpally B.R. Engineering College, Yenkapally(V), Moinabad(M), Ranga Reddy Distric, Andhra Pradesh- 500075, India..

### Abstract

A generalized metric space (g.m.s) has been defined as a metric space in which the triangle inequality is replaced by the ‘Quadrilateral inequality’, $d(x, y) \leq d(x, a) + d(a, b) + d(b, y)$ for all pairwise distinct points $x, y, a$ and $b$ of $X. (X, d)$ becomes a topological space when we define a subset $A$ of $X$ to be open if to each a in $A$ there corresponds a positive number $r_a$ such that $b \in A$ whenever $d(a, b) < r_a$. Cauchyness and convergence of sequences are defined exactly as in metric spaces and a g.m.s $(X, d)$ is called complete if every Cauchy sequence in $(X, d)$ converges to a point of $X$. A.Branciari [1] has published a paper purporting to generalize Banach’s Contraction principle in metric spaces to g.m.s. In this paper we present a correct version and proof of the generalization.

### Share and Cite

##### ISRP Style

I. R. SARMA, J. M. RAO2, S. S. RAO, CONTRACTIONS OVER GENERALIZED METRIC SPACES, Journal of Nonlinear Sciences and Applications, 2 (2009), no. 3, 180-182

##### AMA Style

SARMA I. R., RAO2 J. M., RAO S. S., CONTRACTIONS OVER GENERALIZED METRIC SPACES. J. Nonlinear Sci. Appl. (2009); 2(3):180-182

##### Chicago/Turabian Style

SARMA, I. R., RAO2, J. M., RAO, S. S.. " CONTRACTIONS OVER GENERALIZED METRIC SPACES." Journal of Nonlinear Sciences and Applications, 2, no. 3 (2009): 180-182

### Keywords

• Fixed point
• Contraction mapping
• Generalized metric spaces

•  47H10
•  54H25

### References

• [1] A. Branciari, A fixed point theorem of Banach-Caccippoli type on a class of generalized metric spaces, Publ. Math. Debrecen, 57 (2000), 31–37

• [2] A. Azam, M. Arshad, Kannan fixed point theorem on generalized metric spaces, J. Nonlinear Sci. Appl., 1 (2008), 45–48

• [3] M. Akram, Akhlaq A. Siddiqui, A fixed point theorem for A-Contractions on a class of generalized metric spaces, Korean J. Math. Sciences., 10 (2003), 1-5

• [4] B. K. Lahiri, P. Das, tFixed point of a Ljubomir Ciric’s quasi-contraction mapping in a generalized metric spaces, Publ. Math. Debrecen, 61 (2002), 584-594

• [5] Pratulananda Das, L. K. Dey, A fixed point theorem in a Generalized metric space, Soochow Journal of Mathematics, 33 (2007), 33-39

• [6] P. A. Das, A fixed point theorem on a class of Generalized metric spaces, Korean J.Math.Scienes, 9 (2002), 29-33