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Some Common Fixed Point Theorems for Weakly Contractive Maps in G-Metric Spaces

Manoj Kumar

Department of Mathematics Guru Jambheshwar University of Science and Technology, Hisar, India

Abstract: In this paper, first we prove a common fixed point theorem for a pair of weakly compatible maps under weak contractive condition. Secondly, we prove common fixed point theorems for weakly compatible mappings along with E.A. and (CLRf) properties.

Keywords: weakly compatible maps, weak contraction, generalized weak contraction, altering distance functions, E.A. property, (CLRf) property

Received December 05, 2014; Revised January 20, 2015; Accepted January 27, 2015

Citation: Manoj Kumar, Some Common Fixed Point Theorems for Weakly Contractive Maps in G-Metric Spaces. Turkish Journal of Analysis and Number Theory, 3 (1). 17-20. January 2015. doi:10.12691/tjant-3-1-4

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] 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 {xn} be a sequence of points in X, then {xn} is said to be G-convergent to x in X, if G(x, xn, xm) → 0, as n, m → ∞.

G-Cauchy sequence in X, if G(xn, xm, xl) → 0, as n, m, l → ∞.

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

{xn} is G-convergent to x.

G(xn, xn, x) → 0, as n → ∞.

G(xn, x, x) → 0, as n → ∞.

G(xn, xm, x) → 0, as n, m → ∞.

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

the sequence {xn} is G-Cauchy.

for any ε > 0 there exists k in such that G(xn, xm, xm) < 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 {xn} is G-convergent to x, {f(xn)} 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 {xn} in X such that for some t in X.

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

Definition 1.11. Two self-mappings f and g of a metric space (X, d) are said to satisfy (CLRf) property if there exists a sequence {xn} 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:

[picture] | (2.1)

[picture] | (2.2)

[picture] | (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 {xn} and {yn} in X by yn = fxn+1 = gxn, n = 0, 1, 2,…

From (2.3), we have

[picture] | (2.4)

Since is non-decreasing, therefore we have

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

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

[picture] | (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

[picture] | (2.6)

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

[picture] | (2.7)

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

[picture] | (2.8)

Then, we have

[picture] | (2.9)

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

From (2.8), we get

[picture] | (2.10)

Moreover, we have

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

[picture] | (2.11)

Taking x = xn(k), y = xm(k) and z = xm(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 {yn} is a G-Cauchy sequence.

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

[picture] | (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:

[picture] | (3.1)

[picture] | (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 {xn} in X such that

gxn = fxn = x₀ for some x₀ in X.

Now, fX is closed subset of X, therefore, by (3.1), we have fxn = 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. (CLRf) 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:

[picture] | (4.1)

Then f and g have a unique common fixed point.

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

gxn = fxn = 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 {xn} = {} so that , hence the pair (f, g) satisfy the (CLRf) 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.

References

[1]   | Aamri M. and Moutawakil D. El., Some new common fixed point theorems under strict contractive conditions, J. Math. Anal. Appl. 270, 181-188, 2002.
[2]   | Aydi H., A fixed point result involving a generalized weakly contractive condition in G-metric spaces, Bulletin of Mathematical Analysis and Applications, vol. 3 Issue 4 (2011), 180-188.
[3]   | Jungck G., Common fixed points for non-continuous non-self mappings on non-metric spaces, Far East J. Math. Sci. 4 (2), (1996), 199-212.
[4]   | Khan M.S., Swaleh M. and Sessa S., Fixed point theorems for altering distances between the points, Bull. Aust. Math. Soc. 30 (1) (1984), 1-9.
[5]   | Mustafa Z. and Sims B., Some remarks concerning D-metric spaces, in Proceedings of the International Conference on Fixed Point Theory and Applications, pp. 189-198, Yokohama, Japan, 2004.
[6]   | Mustafa Z. and Sims B., A new approach to generalized metric spaces, J. Nonlinear and Convex Anal. 7 (2) (2006), 289-297.
[7]   | Mustafa Z., Obiedat H., and Awawdeh F., Some fixed point theorem for mapping on complete G-metric spaces, Fixed Point Theory Appl. Volume 2008, Article ID 189870, 12 pages, 2008.
[8]   | Mustafa Z., Shatanawi W. and Bataineh M., Existence of fixed point results in G-metric spaces, International J. Math. Math. Sciences, vol. 2009, Article ID 283028, 10 pages, 2009.
[9]   | Mustafa Z. and Sims B., Fixed point theorems for contractive mappings in complete G-metric spaces, Fixed Point Theory Appl. vol. 2009, Article ID 917175, 10 pages, 2009.
[10]   | Sintunavarat W. and Kumam P., Common Fixed Point Theorems for a Pair of Weakly Compatible mappings in Fuzzy Metric Spaces, Hindawi Publishing Corporation, Journal of Applied mathematics, vol. 2011, Article ID 637958, 14 pages.