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Remark on “Tripled Coincidence Point Theorem for Compatible Maps inFuzzy Metric Spaces”

Deepmala1*, Manish Jain2 and Vandana3

1Mathematics Discipline, PDPM Indian Institute of Technology, Design and Manufacturing (IIITDM) Jabalpur-482005, India

2Department of Mathematics, Ahir College, Rewari 123401, India

3School of Studies in Mathematics, Pt. Ravishankar Shukla University, Raipur, Raipur, Chhattisgarh, India

*Corresponding Author:

Deepmala
Mathematics Discipline
PDPM Indian, Institute of Technology
Design and Manufacturing (IIITDM)
Jabalpur-482005, India
Tel: 09674189903
E-mail: [email protected]

Received Date: November 10, 2016 Accepted Date: November 10, 2016 Published Date: December 18, 2016

Citation: Deepmala, Jain M, Vandana (2016) Remark on “Tripled Coincidence Point Theorem for Compatible Maps in Fuzzy Metric Spaces”. Fluid Mech Open Acc 3: 140.

Copyright: © 2016 Deepmala, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

In this note, we point out and rectify an error in a recently published paper “PP Murthy, Rashmi, VN Mishra, Tripled Coincidence Point Theorem For Compatible Maps In Fuzzy Metric Spaces, Electronic Journal of Mathematical Analysis and Applications, Vol. 4(2) July 2016, pp. 96-106”.

Keywords

Fuzzy metric space; Hadžic type t-norm; Compatible mappings; Tripled coincidence points.

MSC (2010): Primary: 47H10; Secondary: 54H25.

In ref. [1], the authors showed the existence of tripled coincidence points for the pair of compatible mappings in the setup of complete fuzzy metric spaces with Hadžić type t-norm. Authors accompanied the main result with the help of a suitable example. The reader should consult [1] for terms not specifically defined in this note.

Remark 1

The authors in [1] claimed that Example 20 supports Theorem 17. In Theorem 17, the t-norm considered is the Hadžić type t-norm but in Example 20, the t-norm * is defined by a*b=ab for all a, b ∈ [0, 1], which is actually not a Hadžić type t-norm. Hence, we conclude that the Example 20 does not support Theorem 17 in [1].

We now rectify Example 17 as follows:

Example 2: Let X=[-1, 1] and a*b=,in{a.b}for all a,b∈[0,1].Let for all t>0 and EquationThen (X, M,*) is a complete fuzzy metric space such that M(x, y, t)→ 1 as t→∞, for all x, y∈X and * being the Hadžić type t-norm.

Let us define the mappings g:X→X and F: X×X→X respectively by g(x)=x2 and Equation

Then, F(X×X) ⊂ g(X), the pair (F, g) of the mappings is compatible and g is continuous.

Equation

Next, we verify inequality (3.2) of Theorem 17 in [1], that is

Equation

Equation(1)

for all x, y, z, u, v, w ∈ X, t > 0.

If the inequality (1) does not holds, then for some t > 0 and x, y, z, u, v, w∈X, we have

Equation

Equation

that is,

Equation

that is,

Equation

that is,

Equation

Equation

 

and Equation

that is,

Equation

Equation

Equation

and, Combining the above three last inequalities, we get |(x2-u2)+(y2-v2)+(z2-w2)| > |x2-u2| + |y2-v2| +|z2-w2|, which is impossible for x2,<u2, y2>v2 and z2<w2. Hence, (1) holds.

Thus all the conditions of Theorem 17 in [1] are satisfied. Hence on applying Theorem 17 [1], we obtain that (0,0,0) is the coupled coincidence point of the mappings F and g.

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