1. Introduction

In 2006, Mustafa and Sims ^[6] introduced a new notion of generalized metric space called G-metric space. In fact, Mustafa et. al. [5-9]^[5] studied many fixed point results for a self-mapping in G-metric space under certain conditions.

In the present work, we study some fixed point results for a pair of self mappings in a complete G-metric space X under weakly contractive conditions related to altering distance functions.

In 1984, Khan et. al. ^[4] introduced the notion of altering distance function as follows:

Definition 1.1. A mapping f: [0, ∞) → [0, ∞) is called an altering distance function if the following properties are satisfied:

f is continuous and non-decreasing.

f(t) = 0 t = 0.

Definition 1.2. Let X be a nonempty set, and let G : X × X × X → be a function satisfying the following properties:

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

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

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

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

(G5) G(x, y, z) ≤ G(x, a, a) + G(a, y, z), for all x, y, z, a in X, (rectangular inequality).

Then the function G is called a generalized metric, or specially a G-metric on X, and the pair (X, G) is called a G-metric space.

Definition 1.3. Let (X, G) be a G-metric space and let {x_n} be a sequence of points in X, then {x_n} is said to be G-convergent to x in X, if G(x, x_n, x_m) → 0, as n, m → ∞.

G-Cauchy sequence in X, if G(x_n, x_m, x_l) → 0, as n, m, l → ∞.

Proposition 1.4. Let (X, G) be a G-metric space. Then, the following are equivalent

{x_n} is G-convergent to x.

G(x_n, x_n, x) → 0, as n → ∞.

G(x_n, x, x) → 0, as n → ∞.

G(x_n, x_m, x) → 0, as n, m → ∞.

Proposition 1.5. Let (X, G) be a G-metric space. Then, the following are equivalent

the sequence {x_n} is G-Cauchy.

for any ε > 0 there exists k in such that G(x_n, x_m, x_m) < ε, for all m, n ≥ k.

Proposition 1.6. Let (X, G) be a G-metric space. Then f : X → X is G-continuous at x in X if and only if it is G-sequentially continuous at x, that is, whenever {x_n} is G-convergent to x, {f(x_n)} is G-convergent to f(x).

Proposition 1.7. 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 1.8. A G-metric space (X, G) is called G-complete if every G-Cauchy sequence is G-convergent in (X, G).

In 1996, Jungck ^[3] introduced the concept of weakly compatible maps as follows:

Definition 1.9. Two self maps f and g are said to be weakly compatible if they commute at coincidence points.

In 2002, Aamri et. al. ^[1] introduced the notion of E.A. property as follows:

Definition 1.10. Two self-mappings f and g of a metric space (X, d) are said to satisfy E.A. property if there exists a sequence {x_n} in X such that for some t in X.

In 2011, Sintunavarat et. al. ^[10] introduced the notion of (CLR_f) property as follows:

Definition 1.11. Two self-mappings f and g of a metric space (X, d) are said to satisfy (CLR_f) property if there exists a sequence {x_n} in X such that for some x in X.

In 2011, Aydi H. ^[2] introduced the concept of weak contraction in G-metric space as follows:

Definition 1.12. Let (X, G) be a G-metric space. A mapping f : X → X is said to be a -weak contraction, if there exists a map : [0, ∞) → [0, ∞) with (0) = 0 and (t) > 0 for all t > 0 such that

G(fx, fy, fz) ≤ G(x, y, z) – (G(x, y, z)), for all x, y, z in X.

In 2011, Aydi H. ^[2] proved the following result:

Theorem 1.13. Let X be a complete G-metric space. Suppose the map f : X → X satisfies the following:

(G(fx, fy, fz)) ≤ (G(x, y, z)) – (G(x, y, z)), for all x, y, z in X,

where and are altering distance functions.

Then f has a unique fixed point (say u) and f is G-continuous at u.

2. Weakly Compatible Maps

Theorem 2.1. Let (X, G) be a G-metric space and let f and g be self mappings on X satisfying the followings:

(2.1)

(2.2)

(2.3)

Then, f and g have a point of coincidence in X.

Moreover, if f and g are weakly compatible, then f and g have a unique common fixed point.

Proof. Let x₀ X. From (2.1), we can construct sequences {x_n} and {y_n} in X by y_n = fx_n+1 = gx_n, n = 0, 1, 2,…

From (2.3), we have

(2.4)

Since is non-decreasing, therefore we have

Let u_n = G(y_n, y_n+1, y_n+1), then 0 ≤ u_n ≤ u_n-1 for all n > 0.

It follows that the sequence {u_n} is monotonically decreasing and bounded below. So, there exists some r ≥ 0 such that

(2.5)

From (2.4) and (2.5) and letting n → ∞, we have

(r) ≤ (r) – (r), since and are continuous.

Thus, we get (r) = 0, i.e., r = 0, by property of , we have

(2.6)

Now, we prove that {y_n} is a G-Cauchy sequence. Let, if possible, {y_n} is not a G-Cauchy sequence. Then, there exists ε > 0, for which, we can find subsequences {y_m(k)} and {y_n(k)} of {y_n} with n(k) > m(k) > k such that

(2.7)

Let m(k) be the least positive integer exceeding n(k) satisfying (3.7) such that

(2.8)

Then, we have

(2.9)

Letting k → ∞, and using (2.6), we have

From (2.8), we get

(2.10)

Moreover, we have

Letting k → ∞ in the above two inequalities and using (2.6) – (2.10), we get

(2.11)

Taking x = x_n(k), y = x_m(k) and z = x_m(k) in (2.3), we get

Letting k → ∞, using (2.11) and the continuity of and , we get

, that is, , a contradiction, since > 0.

Thus {y_n} is a G-Cauchy sequence.

Since fX is complete subspace of X, so there exists a point u fX, such that

(2.12)

Now, we show that u is the common fixed point of f and g.

Since u fX, so there exists a point p X, such that, fp = u.

From (2.3), we have

Using (2.12) and the property of and , we have

(G(fp, gp, gp)) ≤ (0) – (0), implies that, G(fp, gp, gp) = 0, that is, fp = gp = u.

Hence u is the coincidence point of f and g.

Since, fp = gp, and f, g are weakly compatible, we have fu = fgp = gfp = gu.

Now, we claim that, fu = gu = u.

Let, if possible, gu ≠ u.

From (2.3), we have

Hence gu = u = fu, so u is the common fixed point of f and g.

For the uniqueness, let v be another common fixed point of f and g so that fv = gv = v.

We claim that u = v. Let, if possible, u ≠ v.

From (2.3), we have

Thus, we get, u = v.

Hence u is the common fixed point of f and g.

3. E.A. Property

Theorem 3.1. Let (X, G) be a G-metric space. Let f and g be weakly compatible self maps of X satisfying (2.3) and the followings:

(3.1)

(3.2)

Then f and g have a unique common fixed point.

Proof. Since f and g satisfy the E.A. property, there exists a sequence {x_n} in X such that

gx_n = fx_n = x₀ for some x₀ in X.

Now, fX is closed subset of X, therefore, by (3.1), we have fx_n = fz, for some z in X.

From (2.3), we have

Letting limit as n → ∞, we have

Using (2.3), and property of , , we have

(G(fz, gz, gz)) ≤ (0) –(0) = 0, implies that, G(fz, gz, gz) = 0, that is, fz = gz.

Now, we show that gz is the common fixed point of f and g.

Suppose that gz ≠ ggz. Since f and g are weakly compatible, gfz = fgz and therefore ffa = gga.

From (2.3), we have

Hence ggz = gz, so gz is the common fixed point of f and g.

Finally, we show that the fixed point is unique.

Let u and v be two common fixed points of f and g such that u ≠ v.

From (2.3), we have

Thus, we get, u = v.

Hence u is the unique common fixed point of f and g.

4. (CLR_f) Property

Theorem 4.1. Let (X, G) be a G-metric space. Let f and g be weakly compatible self maps of X satisfying (2.3) and the following:

(4.1)

Then f and g have a unique common fixed point.

Proof. Since f and g satisfy the (CLR_f) property, there exists a sequence {x_n} in X such that

gx_n = fx_n = fx for some x in X.

From (2.3), we have

Letting limit as n → ∞, we have

Using (2.3), and property of , , we have

(G(fz, gz, gz)) ≤ (0) –(0) = 0, implies that, G(fx, gx, gx) = 0, that is, fx = gx.

Let w = fx =gx.

Since f and g are weakly compatible, gfx = fgx, implies that, fw = fgx = gfx = gw.

Now, we claim that gw = w.

Let, if possible, gw ≠ w.

From (2.3), we have

Hence gw = w = fw, so w is the common fixed point of f and g.

Finally, we show that the fixed point is unique.

Let v be another common fixed point of f and g such that fv = v =gv.

From (2.3), we have

Thus, we get, w = v.

Hence w is the unique common fixed point of f and g.

Example 4.2. Let X = [0, 1] and G(x, y, z) =max{|x-y|, |y-z|, |x-z|}, for all x, y, z in X. Clearly (X, G) is a G-metric space.

Let and for each x X. Then

Without loss of generality, assume that x > y > z.

Then, G(x, y, z) =|x-z|.

Let (t) = 5t and (t) = t. Then

From here, we have

So (G(gx, gy, gz)) < (G(fx, fy, fz)) - (G(fx, fy, fz)).

From here, we conclude that f, g satisfy the relation (2.3).

Consider the sequence {x_n} = {} so that , hence the pair (f, g) satisfy the (CLR_f) property. Also, f and g are weakly compatible and 0 is the unique common fixed point of f and g.

From here, we also deduce that , where 0 X, implies that f and g satisfy E.A. property.

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