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Common coupled coincidence and coupled fixed point results in two generalized metric spaces

Fixed Point Theory and Applications20112011:80

https://doi.org/10.1186/1687-1812-2011-80

  • Received: 14 June 2011
  • Accepted: 16 November 2011
  • Published:

Abstract

In this article, we prove the existence of common coupled coincidence and coupled fixed point of generalized contractive type mappings in the context of two generalized metric spaces. These results generalize several comparable results from the current literature. We also provide illustrative examples in support of our new results.

2000 MSC: 47H10.

Keywords

  • coupled coincidence point
  • common coupled fixed point
  • weakly compatible maps
  • generalized metric space

1 Introduction and preliminaries

The study of common fixed points of mappings satisfying certain contractive conditions has been at the center of rigorous research activity [15]. Mustafa and Sims [4] generalized the concept of a metric space and call it a generalized metric space. Based on the notion of generalized metric spaces, Mustafa et al. [59] obtained some fixed point theorems for mappings satisfying different contractive conditions. Abbas and Rhoades [10] initiated the study of common fixed point theory in generalized metric spaces (see also [11]). Saadati et al. [12] proved some fixed point results for contractive mappings in partially ordered G-metric spaces. Abbas et al. [13] obtained some periodic point results in generalized metric spaces. Shatanawi [14] obtained some fixed point results for contractive mappings satisfying Φ-maps in G-metric spaces (see also [15]).

Bhashkar and Lakshmikantham [16] introduced the concept of a coupled fixed point of a mapping F : X × XX (a nonempty set) and established some coupled fixed point theorems in partially ordered complete metric spaces. Later, Lakshmikantham and Ćirić [3] proved coupled coincidence and coupled common fixed point results for nonlinear mappings F : X × XX and g : XX satisfying certain contractive conditions in partially ordered complete metric spaces. Recently, Abbas et al. [17] obtained some coupled common fixed point results in two generalized metric spaces. Choudhury and Maity [18] also proved the existence of coupled fixed points in generalized metric spaces. Recently, Aydi et al. [19] generalized the results of Choudhury and Maity [18]. For other works on G-metric spaces, we refer the reader to [20, 21].

The aim of this article is to prove some common coupled coincidence and coupled fixed points results for mappings defined on a set equipped with two generalized metrics. It is worth mentioning that our results do not rely on continuity of mappings involved therein. Our results extend and unify various comparable results in [17, 22, 23].

Consistent with Mustafa and Sims [4], the following definitions and results will be needed in the sequel.

Definition 1.1. Let X be a nonempty set. Suppose that a mapping G : X × X × XR+ satisfies:
  1. (a)

    G(x, y, z) = 0 if x = y = z;

     
  2. (b)

    0 < G(x, y, z) for all x, y X, with xy;

     
  3. (c)

    G(x, x, y) ≤ G(x, y, z) for all x, y, z X, with yz;

     
  4. (d)

    G(x, y, z) = G(x, z, y) = G(y, z, x) = ... (symmetry in all three variables); and

     
  5. (e)

    G(x, y, z) ≤ G(x, a, a) + G(a, y, z) for all x, y, z, a X.

     

Then, G is called a G-metric on X and (X, G) is called a G-metric space.

Definition 1.2. A sequence {x n } in a G-metric space X is:
  1. (i)

    a G-Cauchy sequence if, for any ε > 0, there is an n0 N (the set of natural numbers) such that for all n, m, ln0, G(x n , x m , x l ) < ε,

     
  2. (ii)

    a G-convergent sequence if, for any ε > 0, there is an x X and an n0 N, such that for all n, mn0, G(x, x n , x m ) < ε.

     

A G-metric space on X is said to be G-complete if every G-Cauchy sequence in X is G-convergent in X. It is known that {x n } G-converges to x X if and only if G(x m , x n , x) → 0 as n, m → ∞ [4].

Proposition 1.3. [4] Let X be a G-metric space. Then, the following are equivalent:
  1. 1.

    {x n } is G-convergent to x.

     
  2. 2.

    G(x n , x n , x) → 0 as n → ∞.

     
  3. 3.

    G(x n , x, x) → 0 as n → ∞.

     
  4. 4.

    G(x n , x m , x) → 0 as n, m → ∞.

     

Definition 1.4. [16] An element (x, y) X × X is called:

(C1) a coupled fixed point of mapping T : X × XX if x = T (x, y) and y = T (y, x);

(C2) a coupled coincidence point of mappings T : X × XX and f : XX if f(x) = T(x,y) and f(y) = T(y,x), and in this case (fx,fy) is called coupled point of coincidence;

(C3) a common coupled fixed point of mappings T : X × XX and f : XX if x = f(x) = T(x, y) and y = f(y) = T(y, x).

Definition 1.5. An element (x, y) X × X is called:

(CC1) a common coupled coincidence point of the mappings T, S : X × XX and f : XX if T(x, y) = S(x, y) = fx and T(y, x) = S(y, x) = fy, and in this case (fx, fy) is called a common coupled point of coincidence;

(CC2) a common coupled fixed point of mappings T, S : X × XX and f :
X X if T ( x , y ) = S ( x , y ) = f ( x ) = x and T ( y , x ) = S ( y , x ) = f ( y ) = y .

Definition 1.6. [22] Mappings T : X × XX and f : XX are called

(W1) w-compatible if f(T(x, y)) = T(fx,fy) whenever f(x) = T(x,y) and f(y) = T(y, x);

(W2) w*-compatible if f(T(x,x)) = T(fx, fx) whenever f(x) = T(x,x).

2 Common coupled fixed points

We extend some recent results of Abbas et al. [17, 22] and Sabetghadam [23] to the setting of two generalized metric spaces.

Theorem 2.1. Let G1 and G2 be two G-metrics on X such that G2(x,y, z) ≤ G1(x, y, z) for all x, y, z X, S,T : X × XX, and f : XX be mappings satisfying
G 1 S ( x , y ) , T ( u , v ) , T ( s , t ) a 1 G 2 f x , f u , f s + a 2 G 2 S x , y , f x , f x + a 3 G 2 T x , v , f u , f s + a 4 G 2 f y , f v , f t + a 5 G 2 S x , y , f u , f s + a 6 G 2 T u , v , T s , t , f x
(2.1)

for all x, y, u, v, s, t X, where a i ≥ 0, for i = 1, 2,..., 6 and a1 + a4 + a5 + 2(a2 + a3 + a6) < 1. If S(X × X) f(X), T(X × X) f(X), f(X) is G1-complete subset of X, then S, T, and f have a unique common coupled coincidence point. Moreover, if S or T is w* -compatible with f, then f, S, and T have a unique common coupled fixed point.

Proof. As S, T, and f satisfy condition (2.1), so for all x, y, u, v X, we have
G 1 S ( x , y ) , T ( u , v ) , T ( s , v ) a 1 G 2 f x , f u , f s + a 2 G 2 S x , y , f x , f x + a 3 G 2 T x , v , f u , f u + a 4 G 2 f y , f v , f v + a 5 G 2 S x , y , f u , f u + a 6 G 2 T u , v , T u , v , f x .
(2.2)
Let x0,y0 X. We choose x1,y1 X such that fx1 = S(x0, y0) and fy1 = S(y0, x0), this can be done in view of S(X × X) f(X). Similarly, we can choose x2,y2 X such that fx2 = T(x1, y1) and fy2 = T(y1,x1) since T(X × X) f(X). Continuing this process, we construct two sequences {x n } and {y n } in X such that
f x 2 n + 1 = S x 2 n , y 2 n , f x 2 n + 2 = T x 2 n + 1 , y 2 n + 1
(2.3)
and
f y 2 n + 1 = S y 2 n , x 2 n , f y 2 n + 2 = T y 2 n + 1 , x 2 n + 1 .
(2.4)
From (2.2), we have
G 1 f x 2 n + 1 , f x 2 n + 2 , f x 2 n + 2 = G 1 S x 2 n , y 2 n , T x 2 n + 1 , y 2 n + 1 , T x 2 n + 1 , y 2 n + 1 a 1 G 2 f x 2 n , f x 2 n + 1 , f x 2 n + 1 + a 2 G 2 S x 2 n , y 2 n , f x 2 n , f x 2 n + a 3 G 2 T x 2 n + 1 , y 2 n + 1 , f x 2 n + 1 , f x 2 n + 1 + a 4 G 2 f y 2 n , f y 2 n + 1 , f y 2 n + 1 + a 5 G 2 S x 2 n , y 2 n , f x 2 n + 1 , f x 2 n + 1 + a 6 G 2 T x 2 n + 1 , y 2 n + 1 , T x 2 n + 1 , y 2 n + 1 , f x 2 n = a 1 G 2 f x 2 n , f x 2 n + 1 , f x 2 n + 1 + a 2 G 2 f x 2 n + 1 , f x 2 n , f x 2 n + a 3 G 2 f x 2 n + 2 , f x 2 n + 1 , f x 2 n + 1 + a 4 G 2 f y 2 n , f y 2 n + 1 , f y 2 n + 1 + a 5 G 2 f x 2 n + 1 , f x 2 n + 1 , f x 2 n + 1 + a 6 G 2 f x 2 n + 2 , f x 2 n + 2 , f x 2 n a 1 + 2 a 2 + a 6 G 2 f x 2 n , f x 2 n + 1 , f x 2 n + 1 + 2 a 3 + a 6 G 2 f x 2 n + 1 , f x 2 n + 2 , f x 2 n + 2 + a 4 G 2 f y 2 n , f y 2 n + 1 , f y 2 n + 1 ,
which implies that
G 1 ( f x 2 n + 1 , f x 2 n + 2 , f x 2 n + 2 ) 1 1 - 2 a 3 - a 6 [ ( a 1 + 2 a 2 + a 6 ) G 2 ( f x 2 n + 1 , f x 2 n + 1 , f x 2 n + 1 ) + a 4 G 2 ( f y 2 n , f y 2 n + 1 , f y 2 n + 1 ) ] .
(2.5)
Similarly, we obtain
G 1 ( f y 2 n + 1 , f y 2 n + 2 , f y 2 n + 2 ) 1 1 - 2 a 3 - a 6 [ ( a 1 + 2 a 2 + a 6 ) G 2 ( f y 2 n , f y 2 n + 1 , f y 2 n + 1 ) + a 4 G 2 ( f x 2 n , f x 2 n + 1 , f x 2 n + 1 ) ] .
(2.6)
Now, from (2.5) and (2.6), we obtain
G 1 ( f x 2 n + 1 , f x 2 n + 2 , f x 2 n + 2 ) + G 1 ( f y 2 n + 1 , f y 2 n + 2 , f y 2 n + 2 ) λ [ G 2 ( f x 2 n , f x 2 n + 1 , f x 2 n + 1 ) + G 2 ( f y 2 n , f y 2 n + 1 , f y 2 n + 1 ) ] ,

where λ = a 1 + a 4 + 2 a 2 + a 6 1 - 2 a 3 - a 6 . Obviously, 0 ≤ λ < 1.

In a similar way, we obtain
G 1 ( f x 2 n , f x 2 n + 1 , f x 2 n + 1 ) + G 1 ( f y 2 n , f y 2 n + 1 , f y 2 n + 1 ) λ [ G 2 ( f x 2 n - 1 , f x 2 n , f x 2 n ) + G 2 ( f y 2 n - 1 , f y 2 n , f y 2 n ) ] .
Thus, for all n ≥ 0,
G 1 ( f x n , f x n + 1 , f x n + 1 ) + G 1 ( f y n , f y n + 1 , f y n + 1 ) λ [ G 2 ( f x n - 1 , f x n , f x n ) + G 2 ( f y n - 1 , f y n , f y n ) ] .
Repetition of above process n times gives
G 1 ( f x n , f x n + 1 , f x n + 1 ) + G 1 ( f y n , f y n + 1 , f y n + 1 ) λ [ G 2 ( f x n - 1 , f x n , f x n ) + G 2 ( f y n - 1 , f y n ) ] λ 2 [ G 2 ( f x n - 2 , f x n - 1 , f x n - 1 ) + G 2 ( f y n - 2 , f y n - 1 , f y n - 1 ) ] λ n [ G 2 ( f x 0 , f x 1 , f x 1 ) + G 2 ( f y 0 , f y 1 , f y 1 ) ] .
For any m > n ≥ 1, repeated use of property (e) of G-metric gives
G 1 ( f x n , f x m , f x m ) + G 1 ( f y n , f y m , f y m ) G 2 ( f x n , f x n + 1 , f x n + 1 ) + G 2 ( f x n + 1 , x x + 2 , x n + 2 ) + G 2 ( f y n , f y n + 1 , f y n + 1 ) + G 2 ( f x y + 1 , x y + 2 , x y + 2 ) + + G 2 ( f x m - 1 , f x m , f x m ) + G 2 ( f y m - 1 , f y m , f y m ) ( λ n + λ n + 1 + + λ m - 1 ) [ G 2 ( f x 0 , f x 1 , f x 1 ) + G 2 ( f y 0 , f y 1 , f y 1 ) ] λ n 1 - λ [ G 2 ( f x 0 , f x 1 , f x 1 ) + G 2 ( f y 0 , f y 1 , f y 1 ) ] ,

and so G1(fx n ,fx m , fx m ) + G1(fy n , fy m , fy m ) → 0 as n, m → ∞. Hence, {fx n } and {fy n } are G1-Cauchy sequences in f(X). By G1-completeness of f(X), there exists fx, fy f(X) such that {fx n } and {fy n } converge to fx and fy, respectively.

Now, we prove that S(x,y) = fx and T(y,x) = fy. Using (2.2), we have
G 1 ( f x , T ( x , y ) , T ( x , y ) ) G 1 ( f x 2 n + 1 , T ( x , y ) , T ( x , y ) ) + G 1 ( f x , f x 2 n + 1 , f x 2 n + 1 ) = G 1 ( S ( s 2 n , y 2 n ) , T ( x , y ) , T ( x , y ) ) + G 1 ( f x 2 n + 1 , f x 2 n + 1 , f x ) a 1 G 2 ( f x 2 n , f x , f x ) + a 2 G 2 ( S ( x 2 n , y 2 n ) , f x 2 n , f x 2 n ) + a 3 G 2 ( T ( x , y ) , f x , f x ) + a 4 G 2 ( f y 2 n , f y , f y ) + a 5 G 2 ( S ( x 2 n , y 2 n ) , f x , f x ) + a 6 G 2 ( T ( x , y ) , T ( x , y ) , f x 2 n ) + G 1 ( f x 2 n + 1 , f x 2 n + 1 , f x ) a 1 G 2 ( f x 2 n , f x , f x ) + a 2 G 1 ( f x 2 n + 1 , f x 2 n , f x 2 n ) + 2 a 3 G 3 ( T ( x , y ) , T ( x , y ) , f x ) + a 4 G 2 ( f y 2 n , f y , f y ) + a 5 G 2 ( f x 2 n + 1 , f x , f x ) + a 6 G 2 ( T ( x , y ) , T ( x , y ) , f x 2 n ) + G 1 ( f x 2 n + 1 , f x 2 n + 1 , f x ) ,
which further implies that
G 1 ( f x , T ( x , y ) , T ( x , y ) ) 1 1 2 a 3 [ a 1 G 2 ( f x 2 n , f x , f x ) + a 2 G 2 ( f x 2 n , f x 2 n ) + a 4 G 2 ( f y 2 n , f y , f y ) + a 5 G 2 ( f x 2 n + 1 , f x , f x ) + a 6 G 2 ( T ( x , y ) , T ( x , y ) , f x 2 n ) + G 1 ( f x 2 n + 1 , f x 2 n + 1 , f x ) ] .
Taking limit as n → ∞, we have
G 1 ( f x , T ( x , y ) , T ( x , y ) ) a 6 1 - 2 a 3 G 1 ( T ( x , y ) , T ( x , y ) , f x ) .

As a 6 1 - 2 a 3 < 1 , so we have G1(fx, T(x, y), T (x, y)) = 0, and T (x, y) = fx.

Again from (2.2), we have
G 1 ( S ( x , y ) , f x , f x ) = G 1 ( S ( x , y ) , T ( x , y ) , T ( x , y ) ) a 1 G 2 ( f x , f x , f x ) + a 2 G 2 ( S ( x , y ) , f x , f x ) + a 3 G 2 ( T ( x , y ) , f x , f x ) + a 4 G 2 ( f y , f y , f y ) + a 5 G 2 ( S ( x , y ) , f x , f x ) + a 6 G 2 ( T ( x , y ) , T ( x , y ) , f x ) = ( a 2 + a 5 ) G 2 ( S ( x , y ) , f x , f x ) ( a 2 + a 5 ) G 1 ( S ( x , y ) , f x , f x ) .

That is G1(S(x,y), fx, fx) = 0, and S(x,y) = fx. Thus, T(x,y) = S(x,y) = fx. Similarly, it can be shown that T(y, x) = S(y, x) = fy. Thus, (fx, fy) is a coupled point of coincidence of mappings f, S, and T.

To show that fx = fy, we proceed as follows: Note that
G 1 ( f x 2 n + 1 , f y 2 n + 2 , f y 2 n + 2 ) = G 1 ( S ( x 2 n , y 2 n ) , T ( y 2 n + 1 , x 2 n + 1 ) , T ( y 2 n + 1 , x 2 n + 1 ) a 1 G 2 ( f x 2 n , f y 2 n + 1 , f y 2 n + 1 ) + a 2 G 2 ( S ( x 2 n , y 2 n ) , f x 2 n , f x 2 n ) + a 3 G 2 ( T ( y 2 n + 1 , x 2 n + 1 ) , f y 2 n + 1 , f y 2 n + 1 ) + a 4 G 2 ( f y 2 n , f x 2 n + 1 , f x 2 n + 1 ) + a 5 G 2 ( S ( x 2 n , y 2 n ) , f y 2 n + 1 , f y 2 n + 1 ) + a 6 G 2 ( T ( y 2 n + 1 , x 2 n + 1 ) , T ( y 2 n + 1 , x 2 n + 1 ) , f x 2 n ) = a 1 G 2 ( f x 2 n , f y 2 n + 1 , f y 2 n + 1 ) + a 2 G 2 ( f x 2 n + 1 , f x 2 n , f x 2 n ) + a 3 G 2 ( f y 2 n + 2 , f y 2 n + 1 , f y 2 n + 1 ) + a 4 G 2 ( f y 2 n , f x 2 n + 1 , f x 2 n + 1 ) + a 5 G 2 ( f x 2 n + 1 , f y 2 n + 1 , f y 2 n + 1 ) + a 6 G 2 ( f y 2 n + 2 , f y 2 n + 2 , f x 2 n ) .
Taking limit as n → ∞, we obtain
G 1 ( f x , f y , f y ) ( a 1 + a 5 + a 6 ) G 2 ( f x , f y , f y ) + a 4 G 2 ( f x , f x , f y ) .
This implies that
G 1 ( f x , f y , f y ) a 4 1 - ( a 1 + a 5 + a 6 ) G 1 ( f x , f x , f y ) .
(2.7)
In the similar way, we can show that
G 1 ( f y , f x , f x ) a 4 1 - ( a 1 + a 5 + a 6 ) G 1 ( f y , f y , f x ) .
(2.8)
Since a 4 1 - ( a 1 + a 5 + a 6 ) < 1 , from (2.7) and (2.8), we must have G1(fx, fy, fy) = 0. So that fx = fy. Thus, (fx, fx) is a coupled point of coincidence of mappings f, S and T. Now, if there is another x* X such that (fx*,fx*) is a coupled point of coincidence of mappings f, S, and T, then
G 1 ( f x , f x * , f x * ) = G 1 ( S ( x , x ) , T ( x * , x * ) , T ( x * , x * ) ) a 1 G 2 ( f x , f x * , f x * ) + a 2 G 2 ( S ( x , x ) , f x , f x ) + a 3 G 2 ( T ( x * , x * ) , f x * , f x * ) + a 4 G 2 ( f x , f x * , f x * ) + a 5 G 2 ( S ( x , x ) , f x * , f x * ) + a 6 G 2 ( T ( x * , x * ) , T ( x * , x * ) , f x ) = a 1 G 2 ( f x , f x * , f x * ) + a 2 G 2 ( f x , f x , f x ) + a 3 G 2 ( f x * , f x * , f x * ) + a 4 G 2 ( f x , f x * , f x * ) + a 5 G 2 ( f x , f x * , f x * ) + a 6 G 2 ( f x * , f x * , f x ) ( a 1 + a 4 + a 5 + a 6 ) G 2 ( f x , f x * , f x * )

implies that G1(fx,fx*,fx*) = 0 and so fx* = fx. Hence, (fx, fx) is a unique coupled point of coincidence of mappings f, S, and T.

Now, we show that f, S, and T have common coupled fixed point.

For this, let f(x) = u. Then, we have u = fx = T(x, x). By w*-compatibility of f and T, we have
f ( u ) = f ( f x ) = f ( T ( x , x ) ) = T ( f x , f x ) = T ( u , u ) .

Then, (fu, fu) is a coupled point of coincidence of f, S, and T. By the uniqueness of coupled point of coincidence, we have fu = fx. Therefore, (u, u) is the common coupled fixed point of f, S, and T.

To prove the uniqueness, let v X with uv such that (v, v) is the common coupled fixed point of f, S, and T. Then, using (2.2),
G 1 u , v , v = G 1 s u , u , T v , v , T v , v a 1 G 2 f u , f v , f v + a 2 G 2 S u , u , f u , f u + a 3 G 2 T v , v , f v , f v + a 4 G 2 f u , f v , f v + a 5 G 2 S u , u , f v , f v + a 6 G 2 T v , v , T v , v , f u = a 1 + a 4 + a 5 + a 6 G 2 f u , f v , f v = a 1 + a 4 + a 5 + a 6 G 2 u , v , v a 1 + a 4 + a 5 + a 6 G 1 u , v , v .

Since a1 + a4 + a5 + a6 < 1, so that G1(u, v, v) = 0 and u = u*. Thus, f, S, and T have a unique common coupled fixed point.

In Theorem 2.1, take S = T, to obtain Theorem 2.1 of Abbas et al. [22] as the following corollary.

Corollary 2.2. Let G1 and G2 be two G-metrics on X such that G2(x, y, z) ≤ G1(x, y, z), for all x, y, z X, T : X × XX, and f : XX be mappings satisfying
G 1 T x , y , T u , v , T s , t a 1 G 2 f x , f u , f s + a 2 G 2 T x , y , f x , f x + a 3 G 2 T u , v , f u , f s + a 4 G 2 f y , f v , f t + a 5 G 2 T x , y , f u , f s + a 6 G 2 T u , v , T s , t , f x
(2.9)

for all x, y, u, v, s, t X, where a i ≥ 0, for i = 1, 2,..., 6 and a1 + a4 + a5 + 2(a2+a3 + a6) < 1. If T(X × X) f(X), f(X) is G1-complete subset of X, then T and f have a unique common coupled coincidence point. Moreover, if T is w*-compatible with f, then T and f have a unique common coupled fixed point.

In Theorem 2.1, take s = u and t = v, to obtain the following corollary which extends and generalizes the corresponding results of [17, 22, 23].

Corollary 2.3 Let G1 and G2 be two G-metrics on X such that G2(x, y, z) ≤ G1(x, y, z), for all x, y, z X, S, T :X × XX, and f : XX be mappings satisfying
G 1 S x , y , T u , v , T u , v a 1 G 2 f x , f u , f u + a 2 G 2 S x , y , f x , f x + a 3 G 2 T u , v , f u , f u + a 4 G 2 f y , f v , f v + a 5 G 2 S x , y , f u , f u + a 6 G 2 T u , v , T s , t , f x
(2.10)

for all x, y, u, v X, where a i ≥ 0, for i = 1, 2,..., 6 and a1 + a4 + a5 + 2(a2 + a3 + a6) < 1. If S(X × X) f(X), T(X × X) f(X), f(X) is G1-complete subset of X, then S, T, and f have a unique common coupled coincidence point. Moreover, if S or T is w*-compatible with f, then f, S, and T have a unique common coupled fixed point.

Example 2.4. Let X = 0,1, G-metrics G1 and G2 on X be given as (in [22]):
G 1 a , b , c = a - b + b - c + c - a G 2 a , b , c = 1 2 a - b + b - c + c - a .
Define S, T : X × XX and f : XX as
S ( x , y ) = x 2 8 , T x , y = 0 and f ( x ) = x 2 for all x , y X .
For x, y, u, v X, we have
G 1 S x , y , T u , v , T u , v = G 1 x 2 8 , 0 , 0 = x 2 4 = 1 4 1 2 2 x 2 = 1 4 G 2 0 , 0 , x 2 = 1 4 G 2 T u , v , T u , v , f x .

Thus, (2.10) is satisfied with a1 = a2 = a3 = a4 = a5 = 0 and a 6 = 1 4 , where a1 + a2 + a3 + a4 + a5 + a6 < 1. It is obvious to note that S is w*-compatible with f. Hence, all the conditions of Corollary 2.4 are satisfied. Moreover, (0, 0) is the unique common coupled fixed point of S, T, and f.

If we take α = a1, β = a4, γ = a5, and a2 = a3 = a6 = 0 in Theorem 2.1, then the following corollary is obtained which extends and generalizes the comparable results of [17, 22, 23].

Corollary 2.5. Let G1 and G2 be two G-metrics on X such that G2(x, y, z) ≤ G1(x, y, z), for all x, y, z X, and S, T : X × XX, f : XX be mappings satisfying
G 1 S x , y , T u , v , T s , t α G 2 f x , f u , f s + β G 2 f y , f v , f t + γ G 2 S x , y , f u , f s
(2.11)

for all x, y, u, v, s, t X, where α, β, γ ≥ 0, and α + β + γ < 1. If S(X × X) f(X), T(X × X) f(X), f(X) is G1-complete subset of X, then S, T, and f have a unique common coupled coincidence point. Moreover, if S or T is w*-compatible with f, then f, S, and T have a unique common coupled fixed point.

Corollary 2.6. Let G1 and G2 be two G-metrics on X such that G2(x, y, z) ≤ G1(x, y, z), for all x, y, z X, T : X × XX, and f : XX be mappings satisfying
G 1 T x , y , T u , v , T s , t α G 2 f x , f u , f s + β G 2 f y , f v , f t + γ G 2 S x , y , f u , f s

for all x, y, u, v, s, t X, where α, β, γ ≥ 0, and α + β + γ < 1. If T(X × X) f(X), f(X) is G1-complete subset of X, then T and f have a unique common coupled coincidence point. Moreover, if T is w*-compatible with f, then f and T have a unique common coupled fixed point.

Example 2.7. Let X = [0,1], and two G-metrics G1, G2 on X be given as (in [22]):
G 1 a , b , c = a - b + b - c + c - a and G 2 a , b , c = 1 2 a - b + b - c + c - a .
Define T : X × XX and f : XX as
T ( x , y ) = x + y 16 and f ( x ) = x 2 for all x , y X .
Now, for x, y X,
G 1 T x , y , T u , v , T s , t = 1 16 x + y - ( u + v ) + u + v - ( s + t ) + s + t - ( x + y ) 1 16 x - u + y - v + u - s + v - t + s - x + t - y 1 16 x - u + y - v + u - s + v - t + s - x + t - y + x + y 9 - u + u - s + s - x + y 8 = 1 16 x - u + u - s + s - x + y - v + v - t + t - y + x + y 8 - u + u - s + s - x + y 8 = 1 4 1 2 1 2 x - u + 1 2 u - s + 1 2 s - x + 1 4 1 2 1 2 y - v + 1 2 v - t + 1 2 t - y + 1 4 1 2 1 2 x + y 8 - u + 1 2 u - s + 1 2 s - x + y 8 = α G 2 x 2 , u 2 , s 2 + β G 2 y 2 , v 2 , t 2 + γ G 2 x + y 16 , u 2 , s 2 = α G 2 f x , f u , f s + β G 2 f y , f v , f t + γ G 2 T x , y , f u , f s .

Thus, (2.11) is satisfied with α = β = γ = 1 4 where α + β + γ < 1. It is obvious to note that T is w*-compatible with f. Hence, all the conditions of Corollary 2.5 are satisfied. Moreover, (0,0) is the unique common coupled fixed point of T and f.

Corollary 2.8. Let G1 and G2 be two G-metrics on X with G2(x, y, z) ≤ G1(x, y, z), for all x, y, z X and S,T : X × XX, f : XX be two mappings such that
G 1 S x , y , T u , v , T u , v α G 2 f x , f u , f s + β G 2 f y , f v , f u + γ G 2 S x , y , f u , f u
(2.12)

for all x, y, u, v X, where α, β, γ ≥ 0 and α + β + γ < 1. If S(X × X) f(X), T(X × X) f(X), f(X) is G1-complete subset of X, then S, T, and f have a unique common coupled coincidence point. Moreover, if S or T is w*-compatible with f, then f, S, and T have a unique common coupled fixed point.

Theorem 2.9. Let G1 and G2 be two G-metrics on X such that G2(x, y, z) ≤ G1(x, y, z), for all x, y, z X, and S, T : X × XX, f : XX be mappings satisfying
G 1 S x , y , T u , v , T s , t k max G 2 f x , f u , f s + G 2 f y , f v , f t + G 2 S x , y , f u , f s
(2.13)

for all x, y, u, v, s, t X, where 0 k < 1 2 . If S(X × X) f (X), T(X × X) f(X), f(X) is G1-complete subset of X, then S, T, and f have a unique common coupled coincidence point. Moreover, if S or T is w*-compatible with f, then f, S, and T have a unique common coupled fixed point.

Proof. Let x0, y0 X. We choose x1, y1 X such that fx1 = S(x0, y0) and fy1 = S(y0,x0), this can be done in view of S(X × X) f(X). Similarly, we can choose x2, y2 X such that fx2 = T(x1, y1) and f y2 = T(y1,x1) since T(X × X) f(X). Continuing this process, we construct two sequences {x n } and {y n } in X such that
f x 2 n + 1 = S x 2 n , y 2 n , f x 2 n + 2 = T x 2 n + 1 , y 2 n + 1
and
f y 2 n + 1 = S y 2 n , x 2 n , f y 2 n + 2 = T y 2 n + 1 , x 2 n + 1 .
Now,
G 1 f x 2 n + 1 , f x 2 n + 2 , f x 2 n + 2 = G 1 S x 2 n , y 2 n , T x 2 n + 1 , y 2 n + 1 , T x 2 n + 1 , y 2 n + 1 k max G 2 f x 2 n , f x 2 n + 1 , f x 2 n + 1 , G 2 f y 2 n , f y 2 n + 1 , f y 2 n + 1 , G 2 S x 2 n , y 2 n , f x 2 n + 1 , f x 2 n + 1 = k max G 2 f x 2 n , f x 2 n + 1 , f x 2 n + 1 , G 2 f y 2 n , f y 2 n + 1 , f y 2 n + 1 , G 2 f x 2 n + 1 , f x 2 n + 1 , f x 2 n + 1 ,
which implies that
G 1 f x 2 n + 1 , f x 2 n + 2 , f x 2 n + 2 k max G 2 f x 2 n , f x 2 n + 1 , f x 2 n + 1 , G 2 f y 2 n , f y 2 n + 1 , f y 2 n + 1 .
(2.14)
Similarly, we can show that
G 1 f y 2 n + 1 , f y 2 n + 2 , f y 2 n + 2 k max G 2 f y 2 n , f y 2 n + 1 , f y 2 n + 1 , G 2 f x 2 n , f x 2 n + 1 , f x 2 n + 1 .
(2.15)
Now, from (2.14) and (2.15), we obtain
G 1 f x 2 n + 1 , f x 2 n + 2 , f x 2 n + 2 + G 1 f y 2 n + 1 , f y 2 n + 2 , f y 2 n + 2 k max G 2 f x 2 n , f x 2 n + 1 , f x 2 n + 1 , G 2 f y 2 n , f y 2 n + 1 , f y 2 n + 1 + max G 2 f y 2 n , f y 2 n + 1 , f y 2 n + 1 , G 2 f x 2 n , f x 2 n + 1 , f x 2 n + 1 2 k G 2 f x 2 n , f x 2 n + 1 , f x 2 n + 1 + G 2 f y 2 n , f y 2 n + 1 , f y 2 n + 1 .
In a similar way, we can obtain
G 1 f x 2 n , f x 2 n + 1 , f x 2 n + 1 + G 1 f y 2 n , f y 2 n + 1 , f y 2 n + 1 2 k G 2 f x 2 n - 1 , f x 2 n , f x 2 n + G 2 f y 2 n - 1 , f y 2 n , f y 2 n .
Thus, for all n ≥ 0,
G 1 f x n , f x n + 1 , f x n + 1 + G 1 f y n , f y n + 1 , f y n + 1 2 k G 2 f x n - 1 , f x n , f x n + G 2 f y n - 1 , f y n , f y n .
Since 0 ≤ 2κ < 1. Therefore, repetition of above process n times gives
G 1 f x n , f x n + 1 , f x n + 1 + G 1 f y n , f y n + 1 , f y n + 1 2 k G 2 f x n - 1 , f x n , f x n + G 2 f y n - 1 , f y n , f y n ( 2 k ) 2 G 2 f x n - 2 , f x n - 1 , f x n - 1 + G 2 f y n - 2 , f y n - 1 , f y n - 1 . . . ( 2 k ) n G 2 f x 0 , f x 1 , f x 1 + G 2 f y 0 , f y 1 , f y 1 .
For any m > n ≥ 1, repeated use of property (e) of G-metric gives
G 1 f x n f x m , f x m + G 1 f y n , f y m , f y m G 2 f x n , f x n + 1 , f x n + 1 + G 2 f x n + 1 , x x + 2 , x n + 2 + G 2 f y n + 1 , f y n + 1 + G 2 f x y + 1 , x y + 2 , x y + 2 + . . . + G 2 f x m - 1 , f x m , f x m + G 2 f y m - 1 , f y m , f y m ( 2 k ) n + ( 2 k ) n + 1 + . . . + ( 2 k ) m - 1 G 2 f x 0 , f x 1 , f x 1 + G 2 f y 0 , f y 1 , f y 1 ( 2 k ) n 1 - 2 k G 2 f x 0 , f x 1 , f x 1 + G 2 f y 0 , f y 1 , f y 1

and so G1(fx n , fx m , fx m ) + G1(fy n ,fy m ,fy m ) → 0 as n, m → ∞. Hence, {fx n } and {fy n } are G1-Cauchy sequences in f(X). By G1-completeness of f(X), there exists fx, fy f(X) such that {fx n } and {fy n } converges to fx and fy, respectively.

Now, we prove that S(x,y) = fx and T(y,x) = fy. Using (2.13), we have
G 1 f x , T ( x , y ) , T ( x , y ) G 1 f x 2 n + 1 , T ( x , y ) , T ( x , y ) + G 1 f x , f x 2 n + 1 , f x 2 n + 1 = G 1 S x 2 n , y 2 n , T ( x , y ) , T ( x , y ) + G 1 f x 2 n + 1 , f x 2 n + 1 , f x k max G 2 f x 2 n , f x , f x , G 2 f y 2 n , f y , f y , G 2 S x 2 n , y 2 n , f x , f x + G 1 f x 2 n + 1 , f x 2 n + 1 , f x = k max G 2 f x 2 n , f x , f x , G 2 f y 2 n , f y , f y , G 2 f x 2 n + 1 , f x n , f x + G 1 f x 2 n + 1 , f x 2 n + 1 , f x .

Taking limit as n → ∞, implies that G1(fx, T(x, y), T(x, y)) = 0, and T(x, y) = fx.

Also, further from (2.13), we have
G 1 S ( x , y ) , f x , f x = G 1 S ( x , y ) , T ( x , y ) , T ( x , y ) k max G 2 f x , f x , f x , G 2 f y , f y , f y , G 2 S ( x , y ) , f x , f x = k G 2 S ( x , y ) , f x , f x k G 1 S ( x , y ) , f x , f x ,

that is G1(S(x, y), fx, fx) = 0, and S(x, y) = fx. Thus, T(x, y) = S(x, y) = fx. Similarly, it can be shown that T(y, x) = S(y, x) = fy. Thus, (fx, fy) is coupled point of coincidence of mappings f, S, and T.

Now, we shall show that fx = fy. So that
G 1 f x 2 n + 1 , f y 2 n + 2 , f y 2 n + 2 = G 1 S x 2 n , y 2 n , T y 2 n + 1 , x 2 n + 1 , T y 2 n + 1 , x 2 n + 1 k max G 2 f x 2 n , f x 2 n + 1 , f x 2 n + 1 , G 2 f y 2 n , f y 2 n + 1 , f y 2 n + 1 , G 2 S x 2 n , y 2 n , f y 2 n + 1 , f y 2 n + 1 k max G 2 f y 2 n , f y 2 n + 1 , f y 2 n + 1 , G 2 f x 2 n , f x 2 n + 1 , f x 2 n + 1 , G 2 f x 2 n + 1 , f y 2 n + 1 , f y 2 n + 1 .
On taking the limit as n → ∞, we obtain that
G 1 f x , f y , f y k max G 2 f x , f y , f y , G 2 f x , f x , f y = k G 2 f x , f x , f y k G 1 f x , f x , f y .
(2.16)
In the similar way, we can show that
G 1 f y , f x , f x k G 1 f y , f y , f x .
(2.17)
From (2.16) and (2.17), we must have G1(fx, fy, fy) = 0 which implies that fx = fy. Thus, (fx, fx) is a coupled point of coincidence of mappings f, S, and T. Now, if there is another x* X such that (fx*,fx*) is a coupled point of coincidence of mappings f, S, and T, then
G 1 ( f x , f x * , f x * ) = G 1 ( S ( x , x ) , T ( x * , x * ) , T ( x * , x * ) ) k max G 2 f x , f x * , f x * , G 2 f x , f x * , f x * , G 2 S ( x , x ) , f x * , f x * = k G 2 f x , f x * , f x *

implies that G1(fx, fx*, fx*) = 0 and so fx* = fx. Hence, (fx, fx) is a unique coupled point of coincidence of mappings f, S, and T.

Now, we show that f, S, and T have common coupled fixed point.

For this, let f(x) = u. Then, we have u = fx = T(x, x). By w*-compatibility of f and T, we have
f ( u ) = f ( f x ) = f T ( x , x ) = T ( f x , f x ) = T ( u , u ) .
(2.18)

That is, (fu, fu) is a coupled point of coincidence of f, S, and T. By the uniqueness of coupled point of coincidence, we have fu = fx. Therefore, (u, u) is the common coupled fixed point of f, S, and T.

To prove the uniqueness, we proceed as follows: let v X with uv such that (v, v) is the common coupled fixed point of f, S and T. Using (2.13), we have
G 1 ( u , v , v ) = G 1 S ( u , u ) , T ( v , v ) , T ( u , v ) k max G 2 ( f u , f v , f v ) , G 2 ( f u , f v , f v ) , G 2 S ( u , u ) , f v , f v = k G 2 ( f u , f v , f v ) = k G 2 ( u , v , v ) k G 1 ( u , v , v ) ,

so that G1(u, v, v) = 0 and u = u*. Thus, f, S, and T have a unique common coupled fixed point.

In Theorem 2.9, take S = T, to obtain the following Theorem 2.6 of [22].

Corollary 2.10. Let G1 and G2 be two G-metrics on X such that G2(x, y, z) ≤ G1(x, y, z), for all x, y, z X, T : X × XX, and f : XX be mappings satisfying
G 1 T ( x , y ) , T ( u , v ) , T ( s , t ) k max G 2 ( f x , f u , f s ) , G 2 ( f y , f v , f t ) , G 2 ( T ( x , y ) , f u , f s )
(2.19)

for all x, y, u, v, s, t X, where 0 k < 1 2 . If T(X × X) f(X), f(X) is G1-complete subset of X, then T and f have a unique common coupled coincidence point. Moreover, if T is w*-compatible with f, then T and f have a unique common coupled fixed point.

In Theorem 2.9, take s = u and t = v, to obtain the following corollary which extends and generalizes the corresponding results of [17, 22, 23].

Corollary 2.11 Let G1 and G2 be two G-metrics on X such that G2(x, y, z) ≤ G1(x, y, z), for all x, y, z X, S, T : X × XX, and f : XX be mappings satisfying
G 1 S x , y , T u , v , T s , v k max G 2 f x , f u , f u + G 2 f y , f v , f v + G 2 S x , y , f u , f v
(2.20)

for all x, y, u, v X, where 0 k < 1 2 . If S(X × X) f(X), T(X × X) f(X), f(X) is G1-complete subset of X, then S, T, and f have a unique common coupled coincidence point. Moreover, if S or T is w*-compatible with f, then f, S, and T have a unique common coupled fixed point.

Corollary 2.12. Let G1 and G2 be two G-metrics on X such that G2(x, y, z) ≤ G1(x, y, z), for all x, y, z X, S, T : X × XX, and f : XX be mappings satisfying
G 1 S ( x , y ) , T ( u , v ) , T ( s , t ) h G 2 ( f x , f u , f s )
(2.21)

for all x, y, u, v, s, t X, where 0 ≤ h < 1. If S(X × X) f(X), T(X × X) f(X), f(X) is G1-complete subset of X, then S, T, and f have a unique common coupled coincidence point. Moreover, if S or T is w*-compatible with f, then f, S, and T have a unique common coupled fixed point.

Remark 2.13. By the equivalence of some metrics and cone metric fixed point results in [24]:
  1. (a)

    Theorem 2.1 can be viewed as an extension and generalization of (i) Theorem 2.2, Corollary 2.3, Theorem 2.6, Corollary 2.7 and Corollary 2.8 in [23], (ii) Theorem 2.1, Corollary 2.2, Corollary 2.5 and Corollary 2.5 in [22], (iii) Theorem 2.4 and Corollary 2.5 in [17].

     
  2. (b)

    Theorem 2.9 is a generalization and improvement of (i) Theorem 2.2 and Corollary 2.3 in [23], Theorem 2.6, Corollary 2.7 and Corollary 2.8 in [22].

     

Declarations

Acknowledgements

The authors thank the referees for their appreciation and suggestions regarding this study.

Authors’ Affiliations

(1)
Department of Mathematics, The Hashemite University, Zarqa, 13115, Jordan
(2)
Department of Mathematics, Lahore University of Management Sciences, 54792 Lahore, Pakistan

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