Some Common Fixed Point Theorems in S-Metric Space

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ISSN(Online): 2319-8753 ISSN (Print) : 2347-6710

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(An ISO 3297: 2007 Certified Organization)

Vol. 4, Issue 9, September 2015

Some Common Fixed Point Theorems in

S-Metric Space

Tanmoy Mitra

Asst. Teacher, Akabpur High School, Akabpur, Nadanghat, Burdwan, West Bengal, India

ABSTRACT: In 2012 S. Sedghi, N. Shobe and A. Aliouche have introduced the concept of S-metric space. In this

paper some common fixed point results for two mappings has been established in S-metric space.

KEYWORDS: S-metric space, Fixed point.

AMS Subject Classification (2010): 47H10, 54H25.

I. INTRODUCTION AND PRELIMINARIES

Definition 1 [1]: Let be a nonempty set. An S-metric on is a function : →[0,∞) that satisfies the following

conditions for all , , , ∈

(S1) ( , , ) = 0 if and only if = = .

(S2) ( , , )≤ ( , , ) + ( , , ) + ( , , ).

The pair ( , ) is called an S-metric space.

The concept of S-metric space is a generalization of a G-metric space [11] and a D*- metric space [14].

Definition 2 [2]: Let be a nonempty set. A b-metric on is a function : →[0,∞) such that, for a real number

≥1, the following conditions hold for all , , ∈ .

(i) ( , ) = 0 if and only if = .

(ii) ( , ) = ( , ).

(iii) ( , )≤ [ ( , ) + ( , )].

The pair ( , ) is called a b-metric space.

Definition 3 [1]: Let ( , ) be an S-metric space. For > 0 and ∈ , we define the open ball ( , ) and the

closed ball [ , ] with centre and radius as follows ( , ) = { ∈ : ( , , ) < },

[ , ] = { ∈ : ( , , )≤ }.

Definition 4 [1]: Let ( , ) be an S-metric space.

(1) A sequence { } in is said to converge , if ( , , )→0 as →∞. That is for any > 0, there exists ∈ ℕ such that for all ≥ , we have ( , , ) < .

(2) A sequence { } in is a Cauchy sequence if ( , , )→0 as , →∞. That is for every > 0, there exists ∈ ℕ such that for all , ≥ , ( , , ) < .

(3) The S-metric space ( , ) is complete if every Cauchy sequence is a convergent sequence.

Lemma 5 [1]: In an S-metric space, we have ( , , ) = ( , , ), ∀ , ∈ .

Lemma 6 [1]: Let ( , ) be an S-metric space. If → and → then

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II. MAIN RESULTS

Theorem 1: Let , be two self-maps on a complete S-metric space ( , ) and

( , , )≤ ( , , ) + ( , , ) + ( , , ), ∀ , ∈ and , , are non-negative reals with, + + < 1, + 3 < 1, and + 3 < 1.

Then the mappings , have a unique common fixed point in .

Proof: Let ∈ .

Let us define a sequence { } in as

= , = , = 0,1,2, … … … ..

Now ( , , ) = ( , , )

≤ ( , , ) + ( , , ) +

( , , )

= ( , , ) + ( , , ) +

( , , )

≤

( , , ) + [2 ( , , ) +

( , , )] [ By S2]

= ( , , ) + [2 ( , , ) + ( , , )] [By Lemma 5] Thus,

( , , )≤ ( , , ). Let =ℎ . Since 0≤ + 3 < 1, 0≤ ℎ < 1.

Thus

( , , )≤ ℎ ( , , ) .

Again,

( , , ) = ( , , )

= ( , , )

≤ ( , , ) + ( , , ) +

( , , )

= ( , , ) + ( , , ) +

( , , )

≤ ( , , ) + [2 ( , , ) +

( , , )]

= ( , , ) + [2 ( , , ) +

( , , )

Thus,

( , , )≤ ( , , )

Let =ℎ . Since 0≤ + 3 < 1, 0≤ ℎ < 1.

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Take ℎ= max {ℎ ,ℎ }. Now

( , , )≤ ℎ ( , , )

And also

( , , )≤ ℎ ( , , )

Thus

( , , )≤ ℎ ( , , )≤ ℎ ( , , )≤ ⋯… …≤ ℎ ( , , )

Now for > ,

( , , )≤2 ( , , ) + ( , , )

≤2 ( , , ) + 2 ( , , ) + ( , , )

≤ ………. ≤2 ( , , ) + 2 ( , , ) +⋯… … … … . . +

2 ( , , ) + ( , , )

≤2[ ( , , ) + ( , , ) +⋯… … … + ( , , )

≤2[ℎ +ℎ +⋯… … … . +ℎ ] ( , , )

≤ ( , , )

⟶0 as ⟶∞.

Thus { } is a Cauchy sequence.

Now since ( , ) is complete, there exists ∈ , such that → as →∞.

Now,

( , , )≤2 ( , , ) + ( , , )

≤2[ ( , , ) + ( , , ) + ( , , )] + ( , , )

= 2[ ( , , ) + ( , , ) + ( , , )] +

( , , )

Letting →∞, we get, ( , , )≤2 ( , , )

Thus, (1− ) ( , , )≤0.

Since 0≤ + 3 < 1, ( , , ) = 0.

Thus = .

Also, ( , , ) = ( , , )

≤ ( , , ) + ( , , ) + ( , , )

= ( , , )

Thus, (1− ) ( , , )≤0.

Since 0≤ + 3 < 1, ( , , ) = 0.

Thus = .

Therefore, = = .

i.e. is a common fixed point of and .

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Now, ( , , ∗_{) = (} _, _, ∗₎

≤ ( , , ∗_{) +} ₍ _, _, ∗_{) +} ₍ ∗_, ∗_{, )}

= ( , , ∗_{) +} _{( , ,} ∗_{) +} ₍ ∗_, ∗_{, )}

= ( , , ∗_{) +} _{( , ,} ∗_{) +} _{( , ,} ∗₎

Thus, (1− − − ) ( , , ∗₎_≤_0.

Since 0≤ + + < 1, ( , , ∗_{) = 0.}

Thus = ∗_.

Thus is the unique common fixed point of and .

Corollary 1.1: Let be a self-maps on a complete S-metric space ( , ) and

( , , )≤ ( , , ) + ( , , ) + ( , , ), ∀ , ∈ ,

, , are non-negative reals with + + < 1, + 3 < 1, + 3 < 1.

Then the mapping has a unique fixed point in .

Proof: Setting = = in the above theorem, we get the result.

Theorem 2: Let ( , ) be a complete S-metric space. Let , be two self-maps on , satisfying

( , , )≤ ( , , ) + ( , , ) (_{( , , )}, , ) , where 0≤ + < 1.

Then , have a unique common fixed point in .

Proof: Let ∈ .

Let us define a sequence { } in as

= , = , = 0,1,2, … … … ..

Now ( , , ) = ( , , )

≤ ( , , ) + ( , , ) ( , , )

( , , )

= ( , , ) + ( , , ) ( , , )

( , , )

= ( , , )

Again,

( , , ) = ( , , )

= ( , , )

≤ ( , , ) + ( , , ) ( , , )

( , , )

= ( , , ) + ( , , ) ( , , )

( , , )

= ( , , )

Now,

( , , )≤ ( , , )≤ ( , , )≤ ⋯… … .≤ ( , , ).

Now,

( , , )≤ ( , , )≤ ( , , )≤ ⋯… … .≤ ( , , ).

Now, for > ,

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≤2[ ( , , ) + ( , , ) +⋯… … . + ( , , )]

≤2[ + +⋯… … … + ] ( , , )

≤ ( , , )

→0, as → ∞ [Since 0≤ + < 1. ]

Thus { } is a Cauchy sequence.

Since ( , ) is complete, there exists ∈ , such that → as → ∞.

Now,

( , , )≤2 ( , , ) + ( , , )

= 2 ( , , ) + ( , , )

≤2 ( , , ) + ( , , _{( , ,}) ( ₎, , ) + ( , , )

= 2 ( , , ) + ( , , _{( , ,}) ( ₎, , ) + ( , , )

Letting → ∞ and using the lemma 5 and lemma 6, we get,

( , , )≤0

Hence = .

Again,

( , , ) = ( , , )

≤ ( , , ) + ( , , ) ( , , )

( , , )

= 0

Thus, ( , , ) = 0 and hence = .

Therefore, = = .

i.e. is a common fixed point of and .

For Uniqueness, let ∗_∈ _{, such that} ∗₌ ∗₌ ∗_.

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( , , ∗_{) = (} _, _, ∗₎

≤ ( , , ∗_{) +} ( , , ∗) ( ∗, ∗, ) ( , , ∗)

= ( , , ∗_{) +} ( , ,∗) (∗, ∗, ) ( , , ∗₎

≤ ( , , ∗_{) +} _{( , ,} ∗₎

Thus, (1− − ) ( , , ∗₎_≤_0.

Since 0≤ + < 1, ( , , ∗_{) = 0.}

Thus = ∗_.

Hence is the unique common fixed point of and .

III. CONCLUSON

Many other common fixed point results for two mappings, satisfying different types of rational inequalities in S-metric space can also be proved similarly.

REFFERENCES

[1] S. Sedghi, N. Shobe, A. Aliouche, A generalization of fixed point theorem in S-metric spaces, Mat. Vesnik 64 (2012), 258 – 266. [2] I. A. Bakhtin, The contraction principle in quasimetric spaces, Func. An., Ulianowsk, Gos. Ped. Ins. 30 (1989), 26 – 37. [3] S. Sedghi, N. V. Dzung, Fixed Point Theorems on S-metric Spaces, Mat. Vesnik (01.11.2012).

[4] R. Kannan, Some results on fixed points II, Amer. Math. Monthly 76 (1969), 405 – 408.

[5] S. Sedghi, N. Shobe, H. Zhou, A Common fixed point theorem in ∗ metric spaces, Fixed Point Theory Appl. 2007 (2007), 1 – 13.