Abstract

In this paper, we prove a common fixed point theorem for two pairs of weak compatible mappings of type (A) in G_b-metric space. Further our result is verified with the help of example.

1. Introduction and Preliminaries

Metric fixed point theory is one of the most important and fundamental area of analysis. Due to this a flood of research work have been generated from this area. As a part of this study generalisation of metric space becomes one of the most interesting topic in which many researchers have devoted and continued working. Since the introduction of metric space by Frachet, there is a lot of generalisation of this space. Some of them which can be mentioned are 2-metric space, D-metric space, cone metric space, fuzzy metric space, Menger space, probabilistic metric space, partial metric space, quasi metric space, b-metric space, multiplicative metric space, modular metric space, cyclic metric space, S-metric space, b-cone metric space etc.

In a recent paper, Aghajani et.al. ¹ introduced a new generalisation of metric space. They used the concepts of both G-metric ² and b-metric ^{3, 4, 5} and generated a new definition and named it as G_b-metric space. They also pointed out that the class of G_b-metric space is effectively larger than that of G-metric space and G-metric space becomes a particular case of G_b-metric space. They claimed that every G_b-metric space is topologically equivalent to a b-metric space. For more results on G_b-metric space one can study the research papers in ^{6, 7, 8, 9, 10} and references there in.

Definition 1.1 ²

Let X be a nonempty set and G: X³ → R⁺ be a function satisfying the following properties:

1. G(x, y, z) = 0 if and only if x = y = z;

2. 0 < G(x, x, y) for all x, y ∈ X with x ≠ y;

3. G(x, x, y) ≤ G(x, y, z) for all x, y, z ∈ X with z ≠ y;

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

5. 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 G-metric on X and the pair (X, G) is called a G-metric space.

Following definition was given by I. A. Bakhtin ³

Definition 1.2 ³

Let X be a (nonempty) set and b ≥ 1a given real number. A function d: X × X → R⁺ (nonnegative real numbers) is called a b-metric provided that, for all x, y, z ∈ X, the following conditions are satisfied:

1. d(x, y) = 0 if and only if x = y;

2. d(x, y) = d(y, x);

3. d(x, z) ≤ b[d(x, y) + d(y, z)]

The pair (X, d) is called a b-metric space with parameter b.

Definition 1.3 ¹

Let X be a nonempty set and b ≥ 1 be a given real number. Suppose that a mapping G: X × X × X → R⁺ satisfies:

(G_b1) G(x, y, z) = 0 if x = y = z,

(G_b2) 0 <G(x, x, y) for all x, y ∈ X with x, y,

(G_b3) G(x, x, y) ≤ G(x, y, z) for all x, y, z ∈ X with y ≠ z.

(G_b4) G(x, y, z) = G(p[x, y, z]), where p is a permutation of x, y, z (symmetry),

(G_b5) G(x, y, z) ≤ b(G(x, a, a) + G(a, y, z)) for all x, y, z, a ∈ X (rectangle inequality).

Then G is called a generalized b-metric and pair (X, G) is called a generalized b-metric space or G_b-metric space.

It should be noted that, the class of G_b-metric spaces is effectively larger than that of G-metric spaces. Following example given by Aghajani ¹ shows that a G_b-metric on X need not be a G-metric on X.

Example 1.4 ¹

Let (X, G) be a G-metric space, and G_∗(x, y, z) = G(x, y, z)^p, where p > 1 is a real number. Note that G_∗ is a G_b-metric with b = 2^p-1.

Also in the above example, (X, G_∗) is not necessarily a G-metric space. For example, let X = R and G-metric G be defined by G(x, y, z) =1/3(|x -y| + |y-z| + |x -z|), for all x, y, z ∈ R. Then G(x, y, z)² = 1/9(|x-y|+|y-z|+|x-z|)² is a G_b-metric on R with b = 2^2-1 = 2, but it is not a G-metric on R. To see this, let x = 3, y = 5, z = 7, a = 7/2 we get, G_∗(3, 5, 7) = 64/9, G_∗(3, 7/2, 7/2) = 1/9, G_∗(7/2, 5, 7)= 49/9, so G_∗(3, 5, 7) = 64/9 ≤50/9 = G_∗(3, 7/2, 7/2) + G_∗(7/2, 5, 7).

Following definitions and properties are given in Aghajani et. al. ¹.

Definition 1.5 ¹

A G_b-metric G is said to be symmetric if G(x, y, y) = G(y, x, x) for all x, y ∈ X.

Definition 1.6 ¹

Let (X, G) be a G_b-metric space then for x₀∈X, r > 0, the G_b-ball with centre x₀ and radius r is B_G(x₀, r) = {y ∈ X|G(x₀, y, y)<r}.

For example, let X = R and consider the G_b-metric G defined by

for all x, y, z ∈ R. Then

By some straight forward calculations, we can establish the following.

Proposition 1.7 ¹

Let X be a G_b-metric space, then for each x, y, z, a ∈ X it follows that:

(1) if G(x, y, z) = 0 then x = y = z,

(2) G(x, y, z) ≤ b(G(x, x, y) + G(x, x, z)),

(3) G(x, y, y)≤ 2bG(y, x, x),

(4) G(x, y, z)≤b(G(x, a, z) + G(a, y, z))

Definition 1.8 ¹

Let X be a G_b-metric space, we define d_G(x, y) = G(x, y, y)+G(x, x, y), it is easy to see that d_G defines a b-metric on X, which we call it b-metric associated with G.

Proposition 1.9 ¹

Let X be a G_b-metric space, then for any x₀ ∈ X and r > 0, if y ∈ B_G(x₀, r) then there exists a δ > 0 such that B_G(y, δ) ⊆ B_G(x₀, r).

Definition 1.10 ¹

Let X be a G_b-metric space. A sequence {x_n} in X is said to be:

1. G_b-Cauchy sequence if, for each ε >0, there exists a positive integer n₀ such that, for all m, n, l ≥ n₀, G(x_n, x_m, x_l) < ε;

2. G_b-convergent to a point x ∈ X if, for each ε > 0, there exists a positive integer n₀ such that, for all m, n ≥ n₀, G(x_n, x_m, x) < ε.

Proposition 1.11 ¹ Let (X, G) be a G_b-metric space, then the following are equivalent:

1. the sequence {x_n} is G_b-Cauchy.

2. for any ε > 0, there exists n₀∈ N such that G(x_n, x_m, x_m) < ε, for all m, n ≥ n₀.

Proposition 1.12 ¹ Let (X, G) be a G_b-metric space, then following are equivalent:

1. {x_n} is G_b-convergent to x.

2. G(x_n, x_n, x)→ 0 as n → ∞.

3. G(x_n, x, x)→ 0 as n → ∞.

Definition 1.13 ¹ A G_b-metric space X is called G_b-complete if every G_b-Cauchy sequence is G_b-convergent in X.

Definition 1.14 ¹ Let (X, G) and (X, G') be two G_b-metric spaces. Then a function f : X → X' is G_b-continuous at a point x ∈ X if and only if it is G_b-sequentially continuous at x, that is, whenever {x_n} is G_b-convergent to x,{f (x_n)} is G'_b-convergent to f (x).

Lemma 1.15 ¹ Let (X, G) be a G_b-metric space with b≥1, and suppose that {x_n}, {y_n} and {z_n} are G_b-convergent to x, y and z respectively. Then we have

In particular, if x = y = z, then we have G(x_n,y_n,z_n) = ε.

Jungck ¹¹ introduced the concept of compatible mappings in metric spaces. Jungck, Murthy and Cho ¹² introduced the concept of compatible mappings of type (A) on metric spaces and proved some common fixed point theorems for compatible mappings of type (A). In 1995, Pathak, Kang and Beak ¹³ introduced the concept of weak compatible mapping of type (A) and proved some common fixed point theorems for weak compatible mappings of type (A) on Menger spaces. Readers can see about various forms of compatible mappings in the research papers in ^{14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33} and references therein. We state the following definitions in the setting of G_b-metric space.

Definition 1.16 Let (X, G) be a G_b-metric space. A pair {f, g} is said to be compatible mappings if G(fgx_n, gfx_n, gfx_n)=0, whenever {x_n} is a sequence in X such that fx_n=gx_n=t for some t in X.

Definition 1.17 Let (X, G) be a G_b-metric space. A pair {f, g} is said to be compatible mappings of type (A) if G(fgx_n, ggx_n, ggx_n)=0 and G(gfx_n, ffx_n, ffx_n)=0, whenever {x_n} is a sequence in X such that fx_n=gx_n=t for some t in X.

Definition 1.18 Let (X, G) be a G_b-metric space. A pair {f, g} is said to be weak compatible mappings of type (A) if G(fgx_n, ggx_n, ggx_n)=0, whenever {x_n} is a sequence in X such that fx_n=gx_n=t for some t in X.

The following propositions are easy to prove and hence we omit their proofs.

Proposition 1.19 Let f, g: (X, G)→(X, G) be mappings. If f and g are weak compatible mappings of type (A) and ft=gt for some t in X, then fgt=ggt.

Proposition 1.20 Let f, g: (X, G)→(X, G) be mappings. If f and g are weak compatible mappings of type (A) and fx_n=gx_n=t for some t in X. Then we have gfx_n=ft, if f is continuous.

The aim of this paper is to prove a common fixed point theorem for two pairs of weak compatible mappings of type (A) in G_b-metric space.

2. Main Results

Our first result is the following common fixed point theorem.

Theorem 2.1 Suppose that f, g, M and T are self-mappings on a complete G_b-metric space (X, G) such that f(X) ⊆ T(X), g(X) ⊆ M(X). If

holds for each with 0 < q <1 and , then f, g, M and T have a unique common fixed point in X provided that M and T are continuous and pairs {f, M} and{g, T}are compatible.

Proof. Let x₀X. As f(X) ⊆ T(X), there exists x₁X such that fx₀ = Tx₁. Since gx₁ M(X), we can choose x₂X such that gx₁ = Mx₂. In general, x_2n+1 and x_2n+2are chosen in X such that fx_2n= Tx_2n+1 and gx_2n+1 = Mx_2n+2. Define a sequence y_n in X such that y_2n= fx_2n= Tx_2n+1, and y_2n+1 = gx_2n+1 = Mx_2n+2, for all n ≥ 0. Now, we show that y_n is a Cauchy sequence. Consider

Now, since G(y_2n-1, y_2n, y_2n) ≤ 2bG(y_2n-1, y_2n, y_2n) and G(y_2n, y_2n+1, y_2n+1) ≤2bG(y_2n, y_2n+1, y_2n+1) we have

If max = 2bG(y_2n, y_2n+1, y_2n+1), we obtain

which is a contradiction.

So, max = 2bG(y_2n-1, y_2n, y_2n) and we have

i.e., G(y_2n, y_2n+1, y_2n+1) ≤ 2q/b⁵G(y_2n-1, y_2n, y_2n).

Let λ = 2q/b⁵. Since b ≥ 3/2 we have that 0 < λ <1.

Now,

and so on.

Hence, for all n ≥ 2, we obtain

(2)

Using (G_b5), and (2) for all n >m, we have

On taking limit as m, n, we have G(y_m, y_n, y_n) as bλ<1. Therefore {y_n} is a Cauchy sequence. Since X is a complete G_b-metric space, there is some y in X such that

We show that y is a common fixed point of f, g, M and T. Since M is continuous, therefore

Since the pair {f, M} is weak compatible type (A), G(fMx_2n, Mfx_2n, Mfx_2n) = 0. So by proposition 1.20, we have

Putting x = Mx_2n and y = x_2n+1 in (1) we obtain

(3)

Taking the upper limit as n in (3) and using Lemma 1.15, we get

Consequently, G(My, y, y) ≤ qG(My, y, y). As 0 < q <1, so My = y. Using continuity of T, we obtain T²x_2n+1 = Ty and Tgx_2n+1 = Ty.

Since g and T are weak compatible type (A), G(gTx_n, Tgx_n, Tgx_n) = 0. So, by proposition 1.20, we have gTx_2n= Ty. Putting x = x_2n and y = Tx_2n+1 in (1), we obtain

(4)

Taking upper limit as n in (4) and using Lemma 1.15, we obtain

which implies that Ty = y. Also, we can apply condition (1) to obtain

(5)

Taking upper limit n in (5), and using My = Ty = y, we have

which implies that G(fy, y, y) = 0 and fy= y as 0 < q <1. Finally, from condition (1), and the fact My = Ty = fy = y, we have

which implies that G(y, gy, gy) = 0 and gy= y. Hence My = Ty = fy= gy= y. If there exists another common fixed point x in X for f, g, M and T, then

which further implies that G(x, y, y) = 0 and hence, x = y. Thus, y is a unique common fixed point of f, g, M and T.

Example 2.2 Let X = [0, 1] be endowed with G_b-metric G∗(x, y, z) = (|y +z -2x|+|y- z|)², where b = 4. Define f, g, M and T on X by

Obviously, f(X) ⊆ T(X) and g(X) ⊆ M(X). Furthermore, the pairs {f, M} and {g, T} are weak compatible mappings of type (A). For each x, y X, we have

where ≤q ≤ 1 and b = 4. Thus, f, g, M and T satisfy all condition of Theorem 2.1. Moreover 0 is the unique common fixed point of f, g, M and T.

Corollary 2.3 Let (X, G) be a complete G_b-metric space and f, g: X → X two mappings such that

holds for all x, y X with 0 < q <1 and b ≥ 3/2. Then, there exists a unique point y in X such that fy= gy = y.

Proof. If we take M = T = I_X (identity mapping on X), then theorem 2.1 gives that f and g have a unique common fixed point.

Note. If we take f and g as identity maps on X, then Theorem 2.1 gives that M and T have a unique common fixed point.

Corollary 2.4 Let (X, G) be a complete G_b-metric space and f: X → X mapping such that

holds for all x, y X with 0 < q <1 and b ≥ 3/2. Then f has a unique fixed point in X.

Proof. Take M and T as identity maps on X and f = g and then apply Theorem 2.1.

Acknowledgments

Authors are thankful to the referees for their valuable suggestions.

References