International Journal of Trend in Scientific Research and Development (IJTSRD)

Volume 4 Issue 3, April 2020 Available Online: www.ijtsrd.com e-ISSN:

ISSN: 2456 – 6470

Some New Fixed Point Theorems on S-Metric

Metric Spaces
J. Gayathri1, Dr. K. Sathya2
1Research
Scholar, 2Head,
1,2PG & Research Department of Mathematics,

1,2Sakthi College off Arts and Science for Women, Dindigul, Tamil Nadu,
adu, India

ABSTRACT How to cite this paper: J. Gayathri | Dr.

In this paper, we present some new type of contractive mappings and prove K. Sathya "Some New Fixed Point
new fixed point theorems on S-metric
metric Spaces. Theorems on S
S-Metric Spaces"
Published in
KEYWORDS: Fixed point, S-Metric
Metric spaces, Contractive mapping International
Journal of Trend in
Scientific Research
and Development
(ijtsrd),
jtsrd), ISSN: 2456-
2456
6470, Volume-4
Volume | IJTSRD30311
Issue-3,
3, April 2020,
pp.182-186,
186, URL:
www.ijtsrd.com/papers/ijtsrd30311.pdf

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20 by author(s) and
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1. INTRODUCTION
Metric spaces are very important in various area of mathematics such as analysis, topology, applied mathematics etc. so
various generalized of metric spaces have been studied and several fixed point results were obtained.

Definition: 1.1
Let X be a nonempty set and S: X3→ [0, ∞) be a function satisfying the following conditions for all x, y, z, a∈ X:
A. s (x, y, z) ≥ 0
B. s (x, y, z) = 0 If and only if x=y=z
C. s (x, y, z) ≤ s (x, x, a) + s (y, y, a) + s (z, z, a)

Then the pair (x, s) is called an s-metric

metric space.

Let (x, s) be an s-metric space and T be a mapping from x into x.

We define,

S(Tx, Tx, Ty) < max { s(x, x, y), s(Tx, Tx, x) s(Ty, Ty, y), s(Ty, Ty, x) s(Tx, Tx, y)} s(25)

for each x,y ∈ X, x≠y

N. Yilmaz orgur and N. Tas presented the notion of a Cs-mapping

Cs mapping and obtained some fixed point theorems using such
mappings under (s25) in 1.

Motivated by the above studies, we modify the notion of s-metric

s metric spaces and define a new type of contractive mappings.

tractive mapping condition (V25), defining the notions of a Cv-mapping

We introduce new contractive Cv mapping on s-metric
s spaces. Also we
give some counter examples and prove some fixed point theorems using the notions of a Cv-mapping.
Cv mapping.

2. A New type of contractive mappings on s-metric

s spaces
In this section, we recall some definitions, lemmas, a remark and corollary which are needed in the sequel.

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30311 | Volume – 4 | Issue – 3 | March-April
April 2020 Page 182
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Definition: 2.1
Let (x, s) be an s-metric space and A⊂x
A. A subset A of X is called s- bounded if there exist r> 0 such that s(x, x, y) < r for all x, y ∈ A.
B. A sequence {xn} in X converges to x if and only if s(xn, xn, x) → 0
as n → ∞ that there exists  ∈N such that for all n ≥ no, S(xn, xn, X)< 
 
for each  > 0.we denote this by →∞  =   →∞ ( ,  , )= 0.
C. A sequences { xn} in X is called cauchy’s sequence if s (xn, xn, xm) →0
as n, m → ∞. That is there exists no∈N such that for all n, m ≥ no.
s (xn, xn, xm)< є for each є > o.
D. The s-metric space (x, s) is called complete if every Cauchy sequence is convergent.

Lemma: 2.2
Let (x,s) be an s-metric space then s (x,x,y) = s (y,y,x) for all x,y єX (2.1)

Lemma: 2.3
Let (x, s) be an s-metric spaces if {xn} and {yn} are sequence in X such that xn→x. yn→y. Then s (xn, xn, yn) → s (x, x, y).

Remark:
Every s-metric space is topologically equivalent to a B-metric space.

Definition: 2.4
Let (x, s) be an metric space and T be a mapping of x, we define
S(Tx, Tx, Ty) < max { s (x, x, y), s (Tx, Tx. x), s(Ty, Ty, y), s(Ty,Ty,x) +s(Tx,Tx,y) } V(25)
2
for any x ,y єX , x≠y.

Definition: 2.5
Let (x, s) be an s-metric space T be a mapping from x into x. T is called a cv- mapping on x if for any xєX and any positive
integer n≥2 satisfying

Tix ≠ Tjx, 0≤i≤j≤n-1 (2.2)

We have,
 "#$ % ,$ % , &
S (Tnx, Tnx, Tix)<  {(  !,   !, !), } (2.3)
'

Theorem: 2.6
Let (x, s) be an S-metric space and T be a mapping from X into X. If T satisfies the condition (V25). Then T is a CV-mapping.

Proof:
Let x∈X and the condition (V25) be satisfied by T.By using mathematical induction (2.2) This condition is true. In fact for
n=2 by (V25)

we have,
s (T2x, T2x,Tx) <max {s(Tx, Tx,x), s(T2x, T2x,Tx) s(Tx, Tx,x), s (Tx, Tx, Tx)+s(T2x, T2x, x)}
2
And so, s(T2x, T2x,Tx) <max { s(Tx,Tx,x),s(T2x,T2x,x)}
2
Hence that the condition (2.3) is proved.

Suppose (2.3) is true for n=k-1, k≥3 and denote


) = *+ {(  !,   !, !)}

By the induction hypothesis and the condition (v25)

we find,
s(Tkx, Tkx, Tk-1x) < max { s(Tk-1x, Tk-1x, Tk-2x) , s(Tk-1x, Tk-1x, Tk-2x), s(Tk-1x, Tk-1x, Tk-1x)+ s(Tkx, Tkx, Tk-2x) }
2
< max {),s(Tkx, Tkx, Tk-2x) }
2

Also by induction it can be shown that,

s(Tkx, Tkx, Tk-ix) < max {), s(Tkx, Tkx, Tki-1x) }
< max {s(Tix, Tix,x), s(Tkx, Tkx, Tki-1-1x) }
For i=k-1

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s(Tkx, Tkx, Tx) < max { s(Tk-1x,Tk-1x, x), s(Tkx, Tkx, x) }
2

And hence,
 "#$ % ,$ % , &
s(Tkx, Tkx, Tix) < * {(  !,   !, !), }
'

Hence the condition (2.3) is satisfied. The proof is completed. The converse part of theorem is not true for always. Now we
see in the following example.

Examples: 2.7
Let R be the real line. Let us consider the usual s-metric on R is defined in
s (x, y, z)= |x-z| + |y-z| for all x, y, z∈R

Let,
Tx= { x if x∈[0,1]
x-2 if x=4
1 if x=2
Then T is a self mapping on the s-metric space [0,1]U{2,4}.

Solution:
For x=1/3, y=1/4 ∈[0,1] we have,
s (Tx, Tx, Ty)= s(1/3, 1/3, ¼)= 1/6, s (x, x, y) = s(1/3, 1/3, ¼)= 1/6
s (Tx, Tx, x) = s(1/3, 1/3, 1/3)=0, s (Ty, Ty, y) = s(1/4, 1/4, ¼)=0
s (Ty, Ty, x) = s(1/4, 1/4,1/3)=1/6, s (Tx, Tx, y) = s(1/3, 1/3,1/4)= 1/6

And so,
s(Tx, Tx, Ty) = 1/6 < max { 1/6, 0,0,1/6}=1/6 Therefore T does not satisfy the condition (2.2) now we show that T is a CV-
mapping we have the following case for x∈{2,4}.

Case: 1
For x=2, n=2
s (T2, T2 ,T2) < max { s(T2, T2, 2), s(T22 , T2,2) }
2
s(1,1,1) < max { s(1,1,2), s(1,1,2) }
2
0 < max {2, 1} =2

For n>2 using similar arquments we have to see that condition (lemma 2.1.2) holds.

Case: 2
For x=4 and n∈{2,3}
s (T24 , T24,T4) < max s(T4, T4, 4), s(T24 , T24,4) }
2
s (1,1,2) < max { s(2,2,4) , s(1,1,4) }
2
2 < max {4, 3}=4

s (T34 , T34, T24) < max { s(T24 , T24,T4), s(T34 , T34,T4) }

2
s (1,1,1) < max { s(1,1,2), s(1,1,2) }
2
0 < max {2,1} =2
For n>3 using similar arguments we have to see the condition (2.2) holds. Hence T is a CV-Mapping.

Theorem: 2.8
Let T be a CV-mapping from on S-metric space (x,s) into itself. Then T has a fixed point in X if and only if there exist
integers l and m, l>m≥0 and x∈X satisfying ,
Tix=Tmx (2.4)

If this condition is satisfied, then Tmx is a fixed point of T.

Proof:
Necessary condition; Let x∈X be a fixed point of T.
that is, Tx1=x1
Then (2.4) is true with l=1, m=0

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Sufficient condition: suppose there exists a point x∈X and an integers l and m, l>m≥0 such that ,
Tix=Tmx

Without loss of generality,

Assume that l is the minimal integer satisfying Tkx=Tmx, k>m. Putting y=Tmx and n=l-m we have,
Tnx = TnTmx = Tl-mTmx
= Tl-m+mx
=Tlx= Tmx
=y

And n is the minimal such integer satisfying ,

Tny=y, n≥1

Now, we show that y is fixed point of T. Suppose not, that is y is not a fixed point of T. Then n≥2, and
Tiy≠Tiy for 0≤i<j≤n-1

Since T is a CV-mapping we have,

s (Tiy, Tiy,y) = s(Tiy, Tiy, Tny)
= s (Tny, Tny, Tiy)
< max { s(Tiy, Tiy, y), s(Tiy, Tiy, y) }
1≤i<j≤n2
< max { s(Tiy, Tiy,y), s(Tiy, Tiy,y) }
1≤i<j≤n2

Then we obtain,
Max s(Tiy,Tiy, y) < max {s(Tiy, Tiy, y), s(Tjy, Tjy, y) }
1 ≤ i < j ≤n-1 1 ≤ i < j ≤n-1 2

That is a contradiction.
Consequently Tm x = y is a fixed point of T.

3. Some Fixed point theorems on s-metric spaces And m is the minimal integer satisfying Tmy = y, m≥1 Now,
In this section, we present some fixed point theorems we show that y is fixed point of T. Suppose not, that is y is
using the notions of a CS-mapping and LS-mapping not fixed point of T. Then m≥2 and Tiy ≠ Tjy, o≤i<j≤m-1
compactness and diameter on S-metric space.
Since T is a CV-mapping we have,
Theorem: 3.1 d (y,Tiy) = d (Tmy, Tiy)
Let T be a CS-mapping on X. Then T has a fixed point in X if <max{ d(Tiy,y) , d(Tjy,y) }
and only if there exists integers p and q. p>q≥ 0 and a 1 ≤ i < j ≤m 2
point x∈X such that
<max { d(Tiy,y), d(Tjy,y) }
Tpx = Tqx (3.1) 1 ≤ i < j ≤m-1 2

If the condition (3.1) is satisfied, then Tqx is a fixed point of Then we obtain,
T. Max d(y,Tiy) <max { d(Tiy,y) , d(Tjy,y) }
1 ≤ i ≤m-1 i < j ≤m-1 2
Proof:
Let  ∈X be a fixed point of T. Which is contradiction.
Therefore y = Tqx is a fixed point of T.
ie.  =-. For P=1, q=0 the condition (3.1) is satisfied.
Corollary: 3.1
Conversely, suppose there exists a point x∈X and two Let (x,d) be an s-metric space and T be a self mapping of X
integers p,q, p>q≥0 such that, Tpx = Tqx satisfying the condition (2.2). Then T has a fixed point in X
if and only if there exist integers p and q, p>q ≥0 and x∈X
Without loss of generality, we assume that P is the Satisfying (3.1) If the condition (3.1) is satisfied . Then Tqx
minimal such integer satisfying Tkx = Tqx, k>q. Putting y = is a fixed point of T.
Tqx and m= p-q,
Theorem: 3.2
We have, Let T be an LS-mapping from an s-metric space (x,s) into
Tmy = Tm.Tqx itself then T has a fixed point in X if and only if there exists
= Tp-q+qx integers p and q.
= Tpx
= Tqx = y p>q ≥0 and x∈X satisfying (3.1) the condition (3.1) is
satisfied. Then Tqx is a fixed point of T.

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It is obvious from theorem (2.8) and theorem (3.1). J. London Math. SOC 41,101-106 (1996)
[4] S. sedghi and N. V. Dung fixed point theorems on s-
Reference:
metric spaces Mat. Veshik 66 (1) (2014)113-124.
[1] s. s chang size on Rhoades open questions and some
fixed point theorems for a class of mappings proc. [5] J. Harjani. B. Lopez K. sandarngani a fixed point
Amer. Math. SOC (97) (2) (1986) 343-346 theorem for mapping satisfying a contractive
condition on a metric space Abst. Appl. Anal (2010).
[2] M. Edelstien on fixed and contractive mappings J.
Lond Math. Soc 37 (1962) 74-79

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