Fixed Point Theorems for Occasionally Weakly Compatible Maps in G-Metric Space

In 1992, Dhage[1] introduced the concept of D - metric space. Recently, Mustafa and Sims[2,3] shown that most of the results concerning Dhage’s D - metric spaces are invalid. Therefore, they introduced a improved version of the generalized metric space structure and called it as G - metric space. For more details on G - metric spaces, one can refer to the papers [4]-[11].

Now we give basic definitions and some basic results ([2]-[11]) which are helpful for proving our main result.

In 2006, Mustafa and Sims[3] introduced the concept of G- metric spaces as follows:

Definition 1.1.[3] Let X be a nonempty set, and let G: X × X × X → R⁺ be a function satisfying the following axioms:

(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 e X with z + У,

(G4) G(x, y, z) = G(x, z, y) = G(y, z, x) = … (symmetry in all three variables) and

(G5) G(x, y, z) < G(x, a, a) + G(a, y, z) for all x, y, z, a ∈ X, (rectangle inequality)

then the function G is called a generalized metric, or, more specifically a G – metric on X and the pair (X, G) is called a G – metric space.

If condition (G6) also satisfied then (X, G) is called Symmetric G-metric space.

(G6) G(x, y, y) = G(x, x, y) for all x, y in X.

Definition 1.2.[3] Let (X,G) be a G -metric space then for x 0 ∈ X, r > 0 , the G -ball with centre x 0 and radius r is

B G (x 0 , r) = {y ∈ X : G(x 0 , y, y ) < r }.

Proposition 1.1.[3] Let (X,G) be a G- metric space then for any x 0 ∈ X, r > 0, we have,

• (1)    if G(x 0 , x, y ) < r

then x, y ∈ B G (x 0 , r),

• (2)    if y ∈ BG(x0, r) then there exists a δ > 0 such that BG(y, δ ) ⊆ BG(x0, r).

It follows from (2) of the above proposition that the family of all G -balls,

B = {B_G(x, r): x e X, r > 0} is the base of a topology T (G) on X , the G -metric topology.

Proposition 1.2.[3] Let (X,G) be a G- metric space then for all x ₀ e X and r > 0, we have,

BG f x0,1 r )- Bd= ^ x0, r ^ - BG ( x0 r )

where d G (x,y) = G(x,y,y) + G(x,x,y), for all x, y in X.

Consequently, the G -metric topology T (G) coincides with the metric topology arising from d G . Thus, while ‘isometrically’ distinct, every G -metric space is topologically equivalent to a metric space. This allows us to readily transport many results from metric spaces into G -metric spaces settings.

Definition 1.3.[3 ] Let (X, G) be a G– metric space, and let { x n } a sequence of points in X , a point ‘ x’ in X is said to be the limit of the sequence { x n } if lim G(x, x n , m , n ^^

xm) = 0, and one says that sequence {xn } is G– convergent to x.

Thus, that if xn ^ x or lim xn = x in a g-metric space (X, G) then for each n ^^

£ >  0, there exists a positive integer N such that

G (x, x n , x_m) <  £ for all m, n > N.

Proposition 1.3.[3] Let (X, G) be a G – metric space. Then the following are equivalent:

• (1)  { x n } is G -convergent to x,

• (2)    G (x n , x n , x) ^ 0 as n ^ да,

• (3)    G (x_n, x, x) ^ 0 as n ^ да,

• (4)    G (x_m, x n , x) ^ 0 as m, n ^ да.

Definition 1.4.[3] Let ( X, G ) be a G – metric space. A sequence { x n } is called G - Cauchy if, for each £ > 0 there exists a positive integer N such that

G (x n , x_m, x l ) <  £ for all n, m, l > N ;

• i .e. if G (x n , x m , x l ) →0 as n, m, l→∞

Proposition 1.4.[3] If ( X, G ) is a G – metric space then the following are equivalent:

• (1)    The sequence {x n } is G – Cauchy,

• (2)    for each ^£ > 0 , there exist a positive integer N such that

G(x n , x m , x m ) <  £ for all n, m > N.

Proposition 1.5.[3] Let ( X, G ) be a G – metric space. Then the function G(x, y, z) is jointly continuous in all three of its variables.

Definition 1.5.[3] 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.

Proposition 1.6.[3] A G – metric space ( X, G ) is G – complete if and only if ( X, d G ) is a complete metric space.

Proposition 1.7.[3] Let ( X, G ) be a G – metric space. Then, for any x, y, z, a in X it follows that:

• (i)    If G(x, y, z) = 0, then x = y = z,

• (ii)    G(x, y, z) ≤ G(x, x, y) + G(x, x, z),

• (iii)    G(x, y, y) ≤ 2G(y, x, x),

• (iv)    G(x, y, z) ≤ G(x, a, z) + G(a, y, z),

• (v)    G(x, y, z) ≤ ²/ 3 (G(x, y, a) +

G(x, a, z) + G(a, y, z)),

• (vi)    G(x, y, z) ≤ (G(x, a, a) +

G(y, a, a) + G(z, a, a)).

In 1996, Jungck [12] introduced the notion of weakly compatible maps as follows:

Definition 1.6.[12] A pair of self mappings     ( f , g )

of a metric space is said to be weakly compatible if they commute at the coincidence points i.e. fu = gu for some u in X , then fgu = gfu.

Definition 1.7[13,14]: Two self mappings f and g of a Symmetric G -metric space ( X, G ) are said to be occasionally weakly compatible ( owc ) iff there is a point x in X which is coincidence point of f and g at which f and g commute.

Lemma 1.1[13,14]: Let ( X, G ) be a Symmetric G -metric space. f and g be self maps on X and let f and g have a unique point of coincidence, w = fx = gx, then w is the unique common fixed point of f and g.

Proof: Since f and g are owc , there exists a point x in X such that fx = gx = w and fgx = gfx . Thus, ffx = fgx = gfx, which says that ffx is also a point of coincidence of f and g . Since the point of coincidence w = fx is unique by hypothesis, gfx = ffx = fx, and w = fx is a common fixed point of f and g .

Moreover, if z is any common fixed point of f and g , then z = fz =gz = w by the uniqueness of the point of coincidence.

In this paper, we prove new common fixed point theorem in Symmetric G -metric space. While proving our result, we utilize the idea of occasionally weakly compatible maps due to Al-Thagafi and N. Shahzad. Our result substantially generalize and improve a multitude of relevant common fixed point theorems of the existing literature in G - metric space.

• II.    Main Result

Following to Matkowski[15], let Ф be the set of all functions ф such that ф : [0, +^ ) ^ [0, +w ) be a non-decreasing function with lim ф ( t ) = 0 for all n ^+^

t G [0, +w ) . If ф G Ф , then ф is called Ф - map. If ф is Ф - map, then it is an easy matter to show that

• (1)    ф ( t ) <  t for all t G [0, +w ) ;

• (2)    ф (0) = 0.

From now unless otherwise stated, we mean by ф the Ф - map. Now, we introduce and prove our result.

Theorem 2.1: Let ( X, G ) be a Symmetric G-metric space. If f and g are owc self maps on X and

G ( fx , fy , fy ) <  ф

max <

^{G (} gx , fy , fy ), G ( gy , fx , fx ), . ^{G (} gy , fy , fy )

for all x, y in X . Then f and g have a unique common fixed point in X .

Proof: Since f and g are owc , there exist a point u in X such that fu = gu and fgu = gfu. We claim that fu is the unique common fixed point of f and g . We first assert that fu is a fixed point of f.

For, if ffu ≠ fu, then from equation (1), we get

' G ( gu , gfu , gfu ),'

G ( fu , ffu , ffu ) <  ф

max <

G ( gu , ffu , ffu ), G ( gfu , fu , fu ),

I G ( gfu , ffu , ffu ) J

[ G ( fu , ffu , ffu ), 1

= ф

max <

G ( fu , ffu , ffu ), G ( ffu , fu , fu ),

IG (ffu, ffu, ffu) J max <

= ^ф

= ^ф

max <

[ ^{G ( fu} , ffu , ^ffu ), ] G ( fu , ffu , ffu ), G (, ffu , fu , fu ), 0

' G ( fu , ffu , ffu ),' G ( fu , ffu , ffu ), G ( fu , ffu , ffu ), "

= ф [ G ( fu , ffu , ffu ) ] <  G ( fu , ffu , ffu )

a contradiction. So ffu = fu and ffu = fgu = gfu = fu . Hence fu is a common fixed point of f and g.

Now we prove uniqueness. Suppose that u, v in X such that fu = gu = u and fv = gv = v and u ≠ v. Then from equation (1),

G ⁽ и , v , v ) = G ( ft , fv , fv )

' G(gu, gv, gv),

^{G ( fx} , gy , gy ),

G (fx,fx,fx),

< Ф

max -

G ( gu , fv , fv ), G ( gv , fu , fu ),

= max 1

G ( gy , gy , gy ),

G ( fx , gy , gy ),

. G ⁽ gv , fv , ^fv )

_G (gy,fx,fx)

G ( и , v , v ),

'^{G ( fx} , gy , gy ),

= ф

max 1

G ( u , v , v ),

G ( v , u , u ),

G ( fx , fx , fx ),

= max 1

^{G (} gy , gy , gy ), *

_ G ( v , v , v )

G ( и , v , v ),

^{G ( fx} , gy , gy ), G ( gy , fx , fx )

= ф

	G ( и , v , v ),
max -	* G ( u , v , v ),

'^{G ( fx} , gy , gy ),

	0,0,
= max -	* G ( fx , gy , gy ),

= ф [ G ( и , v , v ) ] < G ( и , v , v )

a contradiction. So u = v. Therefore, the common fixed point of f and g is unique.

Theorem 2.2: Let (X, G) be a Symmetric G-metric space. Suppose that f, g, S, T are self maps on X and that the pairs {f , S} and {g , T} are each owc. If f G (Sx, Ty, Ty),

G ( Sx , fx , fx ),

G ( fx , gy , gy ) <  max 1 G(Ty , gy , gy ), *

G ( Sx , gy , gy ), _ G ( Ty , fx , fx )

for all x, y in X . Then f, g, S and T have a unique common fixed point in X .

Proof: By hypothesis, there exists points x, y in X such that fx = Sx and gy = Ty. We claim that fx = gy. For, otherwise, by (2)

f G ( Sx , Ty , Ty ),

. ^{G ( fx} , gy , gy )

= ^{G ( fx} , gy , gy )

a contradiction. This implies that fx = gy . So fx = Sx = gy = Ty. Moreover, if there is another point z such that fz = Sz, then, using (2) it follows that fz = Sz = gy = Ty or fx = fz and w = fx = Sx is the unique point of coincidence of f and S . Then by lemma 1.1, it follows that w is the unique common fixed point of f and S. By symmetry, there is a unique common fixed point z in X such that z = gz = Tz.

Now, we claim that w = z. Suppose that w ≠ z. Using (2),

G ( w, z , z ) = G ( fw , gz , gz )

	'G ( Sw , Tz , Tz ), ' G ( Sw , fw , fw ),
< max -	G (Tz , gz , gz ), G ( Sw , gz , gz ), _ G(Tz , fw , fw )

G ( Sx , fx , fx ),

G ( fx , gy , gy ) <  max -

G ( Ty , gy , gy ),

G ( Sx , gy , gy ), _ G ( Ty , fx , fx )

= max <

= max <

G ( w, z, z ), ' G ( w, w, w ), G ( z, z, z ), G ( w, z, z ), G ( z, w, w ) G ( w, z, z ), 0,0, G ( w, z, z ),

^ G ( w, z, z )

this is a contradiction. Therefore w = z and w is a unique point of coincidence of f, g, S, T. By lemma 1.1, w is the unique common fixed point of f, g, S, T.

Corollary 2.1: Let ( X, G ) be a Symmetric G -metric space. Suppose that f, g, S, T are self maps on X and that the pairs { f , S } and { g , T } are each owc.

If G(fx,gy,gy) ≤ h m(x,y,y)

where

’ G ( Sx, Ty, Ty ), G ( Sx, fx, fx), m (x, y, y) = max • G(Ty, gy, gy), (3) 1 " G (Sx, gy, gy) +^ 2 v G (Ty, fx, fx)  v for all x, y in X and 0 ≤ h < 1, then f, g, S and T have a unique common fixed point in X.

Proof: Since (3) is a special case of (2), the result follows immediately from Theorem 2.2.

• III.    Conclusions

In this paper, as an application of occasionally weakly compatible mappings, we prove common fixed point theorems under contractive conditions that extend the scope of the study of common fixed point theorems from the class of weakly compatible mappings to a wider class of mappings. Our result substantially generalize and improve a multitude of relevant common fixed point theorems of the existing literature in G -metric spaces.