Common Fixed Point Theorem for Three Selfmaps of A Complete G-Metric Space

Common Fixed Point Theorem for Three Selfmaps of A Complete G-Metric Space

V. Kiran ¹ , K. Rajani Devi* ² and J. Niranjan Goud ³

1 Department of Mathematics,

Osmania University Hyderabad, 500007, INDIA.

2 Department of Mathematics,

KVR(W) GDC Kurnool, Andhra Pradesh, INDIA.

3 Department of Mathematics,

Dr.B.R.R. GDC Jadcharla, Telangana, INDIA.

email:[email protected] ¹ , [email protected]* ² and [email protected] ³

(Received on: June 2, 2019) ABSTRACT

In this paper we prove a common fixed point theorem for three weakly compatible self maps of a complete G-metric space.

Keywords: G-metric space, fixed point theorem.

1. INTRODUCTION

Weakly commuting maps were introduced by Sessa ⁷ as a generalization of commuting maps. Subsequently G.Jungck ^4,5 initiated compatibility, later Jungck and Rhoades ⁶ initiated the concept of weakly compatible mappings. Gahler ^2,3 introduced the notion of 2-metric spaces, Dhage ¹ initiated the notion of D- metric spaces. Later Shaban Sedghi, Nabi Shobe and Haiyun Zhou ⁸ introduced D∗ metric spaces. In 2006, Zead Mustafa and Brailey Sims ^9,10 initiated G- metric spaces. Among these generalizations, G -metric spaces are noteworthy, as several results established by many researchers on these.

The purpose of this paper is to prove a common fixed point theorem for three weakly compatible self maps of a complete G –metric space.

2. PRELIMINARIES

Definition 2.1[10]: Let X be a non-empty set and G X :

[0, ) be a function satisfying:

(G1) G x y z ( , , ) 0 if x y z

(G2) 0 G x x y ( , , ) for all x y , X with x y

(G3) G x x y ( , , ) G x y z ( , , ) for all x y z , , X with y z

(G4) G x y z ( , , ) G ( ( , , )) x y z for all x y z , , X , where ( , , ) x y z X is a permutation of the set { , , } x y z and

(G5) G x y z ( , , ) G x w w ( , , ) G w y z ( , , ) for all x y z w , , , X

then G is called a G - metric on X and the pair ( X G , ) is called a G - metric Space.

Example 2.2 Let (X, d) be a metric space. Define G

: X

[0, ) by 1

( , , ) ( , ) ( , ) ( , ) 3

d

G

x y z d x y d y z d z x for x y z , , X .Then (X, G

) is a G-metric Space.

Lemma 2.3. If ( X G , ) is a G-metric space then G x y y ( , , ) 2 ( , , ) G y x x for all x y , X Definition 2.4. Let ( X G , ) be a G-metric Space. A sequence { } x _n in X is said to be G- convergent if there is a x

X such that to each 0 there is a natural number N for which

(

,

,

)

G x x x for all ^{n N} .

Definition 2.5. Let ( X G , ) be a G-metric space, then a sequence { } x

in X is said to be G- Cauchy if for each > 0, there exists a natural number N such that G x x (

,

, x

) for all

, , n m l N .

Note that every G-convergent sequence in a G-metric space ( X G , ) is G-Cauchy.

Definition 2.6. A G-metric space ( X G , ) is said to be G-complete if every G -Cauchy sequence in ( X G , ) is G-convergent in ( X G , )

Definition 2.7: Suppose f and g are selfmaps of a G-metric space (X, G) such that

lim

,

,

0

n

G fgx gfx gfx for every sequence { } x

in X with lim _n lim _n ,

n fx n gx t

for some t X , then f and g are said to be compatible mappings.

Definition 2.8. Suppose f and g are self maps of a G-metric space ( X G , ) . The pair f and g is said to be weakly compatible if G fgx gfx gfx ( , , ) 0 whenever G fx gx gx ( , , ) 0 that is the mappings f and g are said to be weakly compatible if they commute at their coincident points.

Definition 2.9: Let ( , ) X G be a G-metric space and f g , and h be self maps G -metric space ( , ) X G such that f X ( ) h X ( ) and g X ( ) h X ( ) For any x ₀ X , we can find a sequence { } x _n in X such that fx _{2 n} hx _{2 n 1} , gx _{2 n 1} hx _{2 n 2}

. for n 0 . Then { } x _n is

called an associated sequence of x ₀ relative to the self maps f g , and h

3. MAIN RESULT

Theorem 3.1. suppose f g , and h are three selfmaps of a complete G -metric space ( X G , ) satisfying the following conditions

(i) f X ( ) h X ( ) and g X ( ) h X ( ) (ii) h X ( ) is closed sub subset of X

(iii) G fx gy gy ( , , ) c max{ ¹ ₂ G hx hy hy G hx fx fx G hy gy gy ( , , ), ( , , ), ( , , ) ¹ ₂ G hx gy gy ( , , ), ¹ ₂ G hy fx fx ( , , )}

for all x y , X and 0 c ¹ ₂

(iv) The pairs ( , ) f h and ( , ) g h are weakly compatible Then f g , and h have a unique common fixed point in X .

Proof. Let x ₀ X be an arbitrary point. Then we can construct an associated sequence { } x _n in X such that

2 _n 2 _n x 2 _n 1 , 2 _n 1 2 _n 1 2 _n 2 fo 0,1, 2.. . ( ) 1

y fx h y gx hx r n (1)

From the condition (iii) of the Theorem 3.1 we have

2 2 1 2 1 2 2 1 2 1

1

2 2 1 2 1 2 2 2 2 1 2 1 2 1

2

1 1

2 2 1 2 1 2 1 2 2

2 2

1

2 1 2 2 2 1 2 2

2

( , , ) ( , , )

max{ ( , , ), ( , , ), ( , , ),

( , , ), ( , , )}

max{ ( , , ), ( , , ),

n n n n n n

n n n n n n n n n

n n n n n n

n n n n n n

G y y y G fx gx gx

c G hx hx hx G hx fx fx G hx gx gx

G hx gx gx G hx fx fx

c G y y y G y y y _{2 2 1 2 1}

1 1

2 1 2 1 2 1 2 2 2

2 2

1

2 1 2 2 2 1 2 2 2 2 1 2 1

2 1

2 1 2 2 2 2 1 2 1

2

( , , ),

( , , ), ( , , )}

max{ ( , , ), ( , , ), ( , , ),

[ ( , , ) ( , , )], 0} (2)

n n n

n n n n n n

n n n n n n n n n

n n n n n n

G y y y

G y y y G y y y

c G y y y G y y y G y y y

G y y y G y y y

(2)

If G y ( _{2 n} , y _{2 n 1} , y _{2 n 1} ) G y ( _{2 n 1} , y _{2 n} , y _{2 n} ) then from the inequality (2)

we get G y ( _{2 n} , y _{2 n 1} , y _{2 n 1} ) cG y ( _{2 n} , y _{2 n 1} , y _{2 n 1} ), which is contradiction since

1

0 c 2

Hence G y ( _{2 n} , y _{2 n 1} , y _{2 n 1} ) cG y ( _{2 n 1} , y _{2 n} , y _{2 n} ) (3) (3) where 0 c ¹ ₂

Similarly, we can show that

2 1 2 2 2 2 2 2 1 2 1

( _n , _n , _n ) ( _n , _n , _n ) (4)

G y y y cG y y y (4)

Therefore from equations (3) and (4) we have

1 1 1 0 1 1

(

,

,

) (

,

,

) ....

( , , ) (5)

G y y y cG y y y c G y y y (5)

We now show that { } y _n is a Cauchy sequence

For every m n , with m n we have

1 1 1 2 2 1

1 1

0 1 1 0 1 1 0 1 1

2 1

0 1 1

0 1 1

( , , ) ( , , ) ( , , ) ... ( , , )

( , , ) ( , , ) ... ( , , )

(1 ... ) ( , , )

1 ( , , ) 0

1

n m m n n n n n n m m m

n n m

n m n

n

G y y y G y y y G y y y G y y y

c G y y y c G y y y c G y y y

c c c c G y y y

c G y y y as n

c

proving that { } y _n is a Cauchy sequence.

Since X is complete G-metric space, then there exists a point z X such that lim _n

n y z

Which implies

2 2 1 2 1 2 2

lim _n lim _n lim _n lim _n (6)

n fx n hx n gx n hx z (6)

Suppose that h X ( ) is a closed subset of X . Then there exists a point u X such that hu z We now show that fu z . If fu z then G fu z z ( , , ) 0

From the condition (iii) of the Theorem 3.1 we have

1

2 1 2 1 2 2 1 2 1 2 1 2 1 2 1

1 1

2 1 2 1 2 1

2 2

( , , ) max{ ( , , ), ( , , ), ( , , ),

( , , ), ( , , )}

n n n n n n n

n n n

G fu gx gx c G hu hx hx G hu fu fu G hx gx gx G hu gx gx G hx fu fu

On letting n we get

1 1 1

2 2 2

1 2

( , , ) max{ ( , , ), ( , , ), ( , , ) ( , , ), ( , , )}

max{0, ( , , ), 0, 0, ( , , )}

max{0, 2 ( , , ), 0, 0, ( , , )}

2 ( , , )

G fu z z c G z z z G z fu fu G z z z G z z z G z fu fu

c G z fu fu G z fu fu

c G fu fu z G fu z z

cG fu z z

which is a contradiction, since 0 c ¹ ₂ Hence G fu z z ( , , ) 0 gives fu z

We now show that gu z . If gu z then G z gu gu ( , , ) 0 From the condition (iii) of the Theorem 3.1 we have

1 1 1

2 2 2

1 1 1

2 2 2

1 2

( , , ) max{ ( , , ), ( , , ), ( , , ) ( , , ), ( , , )}

( , , ) max{ ( , , ), ( , , ), ( , , ) ( , , ), ( , , )}

max{0, 0, ( , , ), ( , , ), 0}

( , , )

G fu gu gu c G hu hu hu G hu fu fu G hu gu gu G hu gu gu G hu fu fu G z gu gu c G z z z G z z z G z gu gu G z gu gu G z z z

c G z gu gu G z gu gu cG z gu gu

which leads to a contradiction since 0 c ¹ ₂ Hence G z gu gu ( , , ) 0 which gives gu z Therefore we have fu gu hu z .

Now, since the pair (f,h) is weakly compatible then fhu hfu implies fz hz

We now prove that fz z .If fz z then G fz z z ( , , ) 0

From the condition (iii) of the Theorem 3.1 we have

1 1 1

2 2 2

1 1 1

2 2 2

1 1

2 2

( , , ) max{ ( , , ), ( , , ), ( , , ) ( , , ), ( , , )}

( , , ) max{ ( , , ), ( , , ), ( , , ) ( , , ), ( , , )}

max{ ( , , ), 0, 0, ( , , ), ( , , )}

G fz gu gu c G hz hu hu G hz fz fz G hu gu gu G hz gu gu G hu fz fz G fz z z c G fz z z G fz fz fz G z z z G fz z z G z fz fz

c G fz z z G fz z z G z fz fz

1

max{ 2 ( , , ), ( , , )}

( , , )

c G fz z z G fz z z cG fz z z

which leads to a contradiction since 0 c ¹ ₂ Hence G fz z z ( , , ) 0 which gives fz z Showing that fz hz z .

Again since the pair (g,h) is weakly compatible then ghu hgu implies gz hz We now prove that gz z .If gz z then G z gz gz ( , , ) 0

From the condition (iii) of the Theorem 3.1 we have

1 1 1

2 2 2

1 1 1

2 2 2

1 2

( , , ) max{ ( , , ), ( , , ), ( , , ), ( , , ), ( , , )}

( , , ) max{ ( , , ), ( , , ), ( , , ) ( , , ), ( , , )}

max{0, ( , , ), 0, 0, ( , , )}

max{

G fz gz gz c G hz hz hz G hz fz fz G hz gz gz G hz gz gz G hz fz fz G z gz gz c G gz gz gz G gz z z G gz gz gz G gz gz gz G gz z z

c G gz z z G gz z z c 2 ( , , ), ( , , )}

2 ( , , )

G z gz gz G z gz gz cG z gz gz

which leads to a contradiction since 0 c ¹ ₂ Hence G z gz gz ( , , ) 0 which gives gz z proving that gz hz z .

Therefore z is a common fixed point of f g , and h

We now prove that z is a unique common fixed point of f g , and h Uniqueness: Suppose w is another common fixed point of f g , and h . If w z then G z w w ( , , ) 0

From the condition (iii) of the Theorem 3.1 we have

1 1 1

2 2 2

1 1 1

2 2 2

1 2

( , , ) max{ ( , , ), ( , , ), ( , , ), ( , , ), ( , , )}

( , , ) max{ ( , , ), ( , , ), ( , , ), ( , , ), ( , , )}

max{ ( , , ), 0, 0, ( , , ), ( , , , )}

( , , )

G fz gw gw c G hz hw hw G hz fz fz G hw gw gw G hz gw gw G hw fz fz G z w w c G z w w G z z z G w w w G z w w G w z z

c G z w w G z w w G z w w cG z w w

which leads to a contradiction since 0 c ¹ ₂ Hence G z w w ( , , ) 0 which gives z w

proving that z is a unique common fixed point of f g , and h .

we have the following example

Example 3.2: Let X [0,1] with G x y z ( , , ) | x y | | y z | | z x | for x y z , , X . Then G is a G-metric on X . Define f : X X , g X : X , h X : X by

11 1

20 2

1 1

1 2 2 1 1

11 1

2 2 2

3 1

20 2

20 2

[0, )

, [0, ]

, ( ,1]

( ,1]

if x

fx x X gx if x hx if x

if x if x

3

1 1 11 1 11

2 2 20 20 2 20

( ) { }, ( ) { , }, ( ) { , , }

f X g X h X Clearly f X ( ) h X ( ), ( g X ) h X ( )

Also h X ( ) is closed sub subset of X . The pairs ^{( , ) f h} and ( , ) g h commute at their coincident point ¹ ₂ and hence they are weakly compatible

Hence the selfmaps f g , and h have a unique common fixed point in X . Clearly ¹ ₂ is unique common fixed point of f g , and h .

Corollary 3.3: Let f and h be selfmaps of a complete G -metric space ( X G , ) satisfying the following conditions

(i) f X ( ) h X ( )

(ii) h X ( ) is closed sub subset of X

(iii) G fx fy fy ( , , ) c max{ ¹ ₂ G hx hy hy G hx fx fx G hy fy fy ( , , ), ( , , ), ( , , ) ¹ ₂ G hx fy fy ( , , ), ¹ ₂ G hy fx fx ( , , )}

for all x y , X and 0 c ¹ ₂

(iv) The pair ( , ) f h is weakly compatible

Then f and h have a unique common fixed point in X .

Proof. On taking g f _, the corollary follows from the Theorem 3.1

REFERENCES

1. B.C.Dhage, Generalized metric space and mapping with fixed point, Bulletin of the Calcutta Mathematical Society. 84, 329-336 (1992).

2. Gahler,S, 2-metriche Raume and Ihre Topologische Strukture, Mathematische Nachriten.

26, 115-148 (1963).

3. Gahler.S, Zur Geometric 2-metriche raume, Revue Roumaine de mathematiques pure et Appliquees. 11, 664-669 (1966).

4. Jungck.G, Compatible mappings and Common fixed points, International Journal of Mathematics and Mathematical Sciences. 4, 771-779 (1986).

5. Jungck.G, Common fixed points for non-continuous non selfmaps on non-metric spaces, Far East J. math Sci. 4 .No 2, 199-225 (1996).

6. Jungck.G, Rhoades,B.E. Fixed point for set valued Functions without continuity. Indian

Journal of Pure Applied Mathematics. 29, 227-238 (1998).

7. Sessa,S On a weak commutativity condition of mappings in a fixed point considerations, Publications De L’Institut Mathematique. 62, 149-153 (1982).

8. Sedgi.S, Shobe,N., and Zhou,H.Y., A common fixed point Theorem in D ^* -metric space, Journal of Fixed Point Theory and Applications,1-13 (2007).

9. Mustafa. Z, Sims.B, Some remarks concerning D-metric spaces, Proceedings of International Conference on Fixed Point Theory and Applications. Yokohama Publishers, Valencia, 19, 189-198 (2004).

10. Mustafa.Z. Sims.B, A New Approach to a Generalized Metric spaces, Journal of Nonlinear

and Convex Analysis. 2, 289-297 (2006).