Journal of Applied Mathematics
Volume 2012, Issue 1 891713
Research Article
Open Access

Some Fixed Point Results in GP-Metric Spaces

Hassen Aydi, 

Hassen Aydi

Institut Supérieur d′Informatique et des Technologies de Communication de Hammam Sousse, Université de Sousse, Route GP1-4011, Hammam Sousse, Tunisia uc.rnu.tn

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Erdal Karapınar, 

Corresponding Author

Erdal Karapınar

Department of Mathematics, Atilim University, Incek, 06836 Ankara, Turkey atilim.edu.tr

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Peyman Salimi, 

Peyman Salimi

Department of Mathematics, Sahand University of Technology, Tabriz, Iran sut.ac.ir

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First published: 18 October 2012
https://doi.org/10.1155/2012/891713
Citations: 2
Academic Editor: Yuantong Gu

Abstract

Following a recent paper of Zand and Nezhad (2011), we establish some fixed point results in GP-metric spaces. The presented theorems generalize and improve several existing results in the literature. Also, some examples are presented.

1. Introduction

Partial metric space is a generalized metric space introduced by Matthews [1] in which each object does not necessarily have to have a zero distance from itself. A motivation is to introduce this space to give a modified version of the Banach contraction principle [2]. Subsequently, several authors studied the problem of existence and uniqueness of a fixed point for mappings satisfying different contractive conditions, see [3–23].

On the other hand, in 2006 Mustafa and Sims [24] introduced a new notion of generalized metric spaces called G-metric spaces. Based on the notion of a G-metric space, many fixed point results for different contractive conditions have been presented, for more details see [25–42].

Recently, based on the two above notions, Zand and Nezhad [43] introduced a new generalized metric space as both a generalization of a partial metric space and a G-metric space. It is given as follows.

Definition 1.1 (see [43].)Let X be a nonempty set. A function Gp : X × X × X → [0, +∞) is called a GP-metric if the following conditions are satisfied:

  •  

    (GP1) x = y = z if Gp(x, y, z) = Gp(z, z, z) = Gp(y, y, y) = Gp(x, x, x);

  •  

    (GP2) 0 ≤ Gp(x, x, x) ≤ Gp(x, x, y) ≤ Gp(x, y, z) for all x, y, z ∈ X;

  •  

    (GP3) Gp(x, y, z) = Gp(x, z, y) = Gp(y, z, x) = ⋯, symmetry in all three variables;

  •  

    (GP4) Gp(x, y, z) ≤ Gp(x, a, a) + Gp(a, y, z) − Gp(a, a, a) for any x, y, z, a ∈ X.

Then the pair (X, G) is called a GP-metric space.

Example 1.2 (see [43].)Let X = [0, ∞) and define Gp(x, y, z) = max {x, y, z}, for all x, y, z ∈ X. Then (X, Gp) is a GP-metric space.

Proposition 1.3 (see [43].)Let (X, Gp) be a GP-metric space, then for any x, y, z and a ∈ X it follows that

  • (i)

    Gp(x, y, z) ≤ Gp(x, x, y) + Gp(x, x, z) − Gp(x, x, x);

  • (ii)

    Gp(x, y, y) ≤ 2Gp(x, x, y) − Gp(x, x, x);

  • (iii)

    Gp(x, y, z) ≤ Gp(x, a, a) + Gp(y, a, a) + Gp(z, a, a) − 2Gp(a, a, a);

  • (iv)

    Gp(x, y, z) ≤ Gp(x, a, z) + Gp(a, y, z) − Gp(a, a, a).

Proposition 1.4 (see [43].)Every GP-metric space (X, Gp) defines a metric space , where

(1.1)

Definition 1.5 (see [43].)Let (X, Gp) be a GP-metric space and let {xn} be a sequence of points of X. A point x ∈ X is said to be the limit of the sequence {xn} or xn → x if

(1.2)

Proposition 1.6 (see [43].)Let (X, Gp) be a GP-metric space. Then, for any sequence {xn} in X, and a point x ∈ X the following are equivalent:

  • (A)

    {xn} is GP-convergent to x;

  • (B)

    Gp(xn, xn, x) → Gp(x, x, x) as n → ∞;

  • (C)

    Gp(xn, x, x) → Gp(x, x, x) as n → ∞.

Definition 1.7 (see [43].)Let (X, Gp) be a GP-metric space.

  •  (S1)

    A sequence {xn} is called a GP-Cauchy if and only if lim m,n→∞ Gp(xn, xm, xm) exists (and is finite).

  •  (S2)

    A GP-partial metric space (X, Gp) is said to be GP-complete if and only if every GP-Cauchy sequence in X is GP-convergent to x ∈ X such that Gp(x, x, x) = lim m,n→∞Gp(xn, xm, xm).

Now, we introduce the following.

Definition 1.8. Let (X, Gp) be a GP-metric space.

  • (M1)

    A sequence {xn} is called a 0-GP-Cauchy if and only if lim m,n→∞Gp(xn, xm, xm) = 0;

  • (M2)

    A GP-metric space (X, Gp) is said to be 0-GP-complete if and only if every 0-GP-Cauchy sequence is GP-convergent to a point x ∈ X such that Gp(x, x, x) = 0.

Example 1.9. Let X = [0, +∞) and define Gp(x, y, z) = max {x, y, z}, for all x, y, z ∈ X. Then (X, Gp) is a GP-complete GP-metric space. Moreover, if X = ℚ∩[0, +∞) (where ℚ denotes the set of rational numbers), then (X, Gp) is a 0-GP-complete GP-metric space.

Lemma 1.10. Let (X, Gp) be a GP-metric space. Then

  • (A)

    if Gp(x, y, z) = 0, then x = y = z;

  • (B)

    if x ≠ y, then Gp(x, y, y) > 0.

Proof. By (GP2) we have

(1.3)
Then, by Proposition 1.4, we have , that is, x = y. Similarly, we can obtain that y = z. The assertion (A) is proved.

On the other hand, if x ≠ y and Gp(x, y, y) = 0, then by (A), x = y which is a contradiction and so (B) holds.

In this paper, we establish some fixed point results in GP-metric spaces analogous to results of Ilić et al. [44] which were proved in partial metric spaces. Also, some examples are provided to illustrate our results. To our knowledge, we are the first to give some fixed point results in GP-metric spaces, and so is the novelty and original contributions of this paper. This opens the door to other possible fixed (common fixed) point results.

2. Main Results

We start by stating a fixed point result of Ilić et al. [44].

Theorem 2.1 (see [44].)Let (X, p) be a complete partial metric space. Let f be a self-mapping on X. Suppose that for all x, y, z ∈ X the following condition holds:

(2.1)
where 0 ≤ α < 1. Then
  • (1)

    the set XP = {y ∈ X : p(x, x) = inf y∈Xp(y, y)} is nonempty;

  • (2)

    there is a unique u ∈ XP such that fu = u;

  • (3)

    for all x ∈ XP the sequence {fnx} converges to u with respect to the metric dp (where dp(x, y) = p(x, y) − p(x, x) − p(y, y) for x, y ∈ X).

The analog of Theorem 2.1 in GP-metric spaces is given as follows.

Theorem 2.2. Let (X, Gp) be a GP-complete GP-metric space. Let f be a self-mapping on X. Suppose that for all x, y, z ∈ X the following condition holds:

(2.2)
where 0 ≤ r < 1. Then
  • (T1)

    the set is nonempty;

  • (T2)

    there is a unique such that fx* = x*;

  • (T3)

    for all the sequence {fnx} converges to x* with respect to the metric .

Proof. Let x ∈ X. By (2.2), we have

(2.3)
Hence, {Gp(fnx, fnx, fnx)} n≥0 is a nonincreasing sequence. Put
(2.4)
(2.5)
We shall show that
(2.6)
Again, by (2.2), we have for all m > n ≥ 0
(2.7)
At first
(2.8)

Similarly

(2.9)

Then we have

(2.10)
By continuing this process, we get
(2.11)

Now, by Proposition 1.3 (ii), we have GP(fix, x, x) ≤ 2GP(x, fix, fix) ≤ 2Γx. Hence

(2.12)
that is, (2.6) holds. On the other hand, by (GP2), we have
(2.13)
Given any ϵ > 0, by (2.4), there exists n0 ∈ ℕ such that . Since 0 ≤ r < 1, so without loss of generality, we have . Therefore, for all m, n ≥ 2n0
(2.14)
Then, Sx = lim m,n→∞Gp(fnx, fmx, fmx) and so {fnx} is a GP-Cauchy sequence. Since (X, Gp) is GP-complete, then there exists x* ∈ X such that {fnx}  GP-converges to x*, that is,
(2.15)
Since {fnx}GP-converges to x*, then Proposition 1.6 yields that
(2.16)
We obtain that
(2.17)
(2.18)
For all n ∈ ℕ
(2.19)
By taking the limit as n → ∞ in the above inequality, we get
(2.20)
On the other hand, from (2.2), we have
(2.21)
Thus, Gp(x*, fx*, fx*) ≤ Gp(x*, x*, x*). By (GP2), we deduce that
(2.22)
Now we show that is nonempty. Let Ω = inf y∈XGp(y, y, y). For all k ∈ ℕ, pike xk ∈ X with Gp(xk, xk, xk) < Ω + 1/k. Define for all n ≥ 1. Let us show that
(2.23)
Given ϵ > 0, put n0 : = [3/ϵ(1 − r)] + 1. If k ≥ n0, then we have
(2.24)
Therefore, we have
(2.25)
On the other hand, if k ≥ n0, then . It follows that
(2.26)
By (GP4), (2.22), and (2.25), we can obtain
(2.27)
Thus
(2.28)
Now, by (2.25) and (2.26), we have
(2.29)
that is, (2.23) holds. Again, Since (X, Gp) is GP-complete, then there exists y ∈ X such that
(2.30)
This leads that , so is nonempty.

Let x ∈ X. By (2.22), we get

(2.31)
From (GP1), it follows that x* = fx*. By (2.17), we have
(2.32)
Therefore, for all x ∈ XP the sequence {fnx} converges with respect to the metric to x*. The uniqueness of the fixed point follows easily from (2.2).

We illustrate Theorem 2.2 by the following examples.

Example 2.3. Let X = [0, ∞) and define Gp(x, y, z) = max {x, y, z}, for all x, y, z ∈ X. Then (X, Gp) is a complete GP-metric space. Clearly, (X, G) is not a G-metric space. Consider f : X → X defined by fx = x2/(1 + x). Without loss of generality, take x ≤ y ≤ z. We have

(2.33)
for all r ∈ [0,1). So, (2.2) holds. Here, u = 0 is the unique fixed point of f.

Example 2.4. Let X = [0, ∞). Define Gp : X3 → [0, ∞) by Gp(x, y, z) = max {x, y, z}. Clearly, (X, Gp) is a GP-metric space. Define f : X → X by

(2.34)

Then, the inequality (2.2) of Theorem 2.2 holds. Here, u = 0 is the unique fixed point of f.

Proof. Clearly, M(x, y, z): = max {rGp(x, y, z), Gp(x, x, x), Gp(y, y, y), Gp(z, z, z)} = max {x, y, z}, for all r ∈ (0,1). We have the following cases.

Case  1 (0 ≤ x, y, z ≤ 1/3).

Consider the following:

(2.35)
Case  2 (1/3 < x, y, z ≤ 1/2).

Consider the following:

(2.36)
Case  3 (x, y, z > 1/2).

Consider the following:

(2.37)
Case  4 (0 ≤ x ≤ 1/3, 1/3 < y ≤ 1/2 and z > 1/2).

Consider the following:

(2.38)
Case  5 (0 ≤ x, y ≤ 1/3, 1/3 < z ≤ 1/2).

Consider the following:

(2.39)
Case  6 (0 ≤ x, y ≤ 1/3, z > 1/2).

Consider the following:

(2.40)
Case  7 (1/3 < x, y ≤ 1/2, 0 ≤ z ≤ 1/3).

Consider the following:

(2.41)
Case  8 (1/3 ≤ x, y ≤ 1/2, z > 1/2).

Consider the following:

(2.42)
Case  9 (x, y > 1/2, 0 ≤ z ≤ 1/3).

Consider the following:

(2.43)
Case  10 (x, y > 1/2, 1/3 < z ≤ 1/2).

Consider the following:

(2.44)

Thus, the inequality (2.2) holds. Applying Theorem 2.2, we get u = 0 is the unique point fixed point of f.

Also, Ilić et al. [44] proved the following result.

Theorem 2.5 (see [44].)Let (X, P) be a complete partial metric space. Let f be a self-mapping on X. Suppose that for all x, y, z ∈ X the following condition holds:

(2.45)
where 0 ≤ α < 1. Then
  • (i)

    the set XP = {y ∈ X : p(x, x) = inf y∈Xp(y, y)} is nonempty;

  • (ii)

    there is a unique x* ∈ XP such that fx* = x*;

  • (iii)

    for all x ∈ XP, the sequence {fnx} converges to x* with respect to the metric dp.

The analog of Theorem 2.5 in GP-metric spaces is stated as follows.

Theorem 2.6. Let (X, Gp) be a GP-complete GP-metric space. Let f be a self-mapping on X. Suppose that for all x, y, z ∈ X the following condition holds:

(2.46)
where 0 ≤ r < 1. Then
  • (R1)

    the set XP = {y ∈ X : Gp(x, x, x) = inf y∈XGp(y, y, y)} is nonempty;

  • (R2)

    there is a unique x* ∈ XP such that fx* = x*;

  • (R3)

    for all , the sequence {fnx} converges to x* with respect to the metric .

Proof. Since

(2.47)
then
(2.48)
Thus
(2.49)
Then, the conditions of Theorem 2.1 hold. Hence, it follows that (R1), (R2), and (R3) hold.

Example 2.7. Let X = [0,1] and define Gp(x, y, z) = max {x, y, z}, for all x, y, z ∈ X. We have (X, Gp) is a complete GP-metric space. Take fx = x2/2 and r = 1/2. For all x ≤ y ≤ z, we have

(2.50)
that is, (2.2) holds. Here, u = 0 is the unique fixed point of f.

Similarly, we have the following.

Theorem 2.8. Let (X, Gp) be a GP-complete GP-metric space. Let f be a self-mapping on X. Suppose that for all x, y, z ∈ X the following condition holds:

(2.51)
where 0 ≤ r < 1. Then
  • (N1)

    the set is nonempty;

  • (N2)

    there is a unique such that fx* = x*;

  • (N3)

    for all , the sequence {fnx} converges with respect to the metric to x*.

The following lemma is useful.

Lemma 2.9. Let (X, Gp) be a GP-metric space and {xn} be a sequence in X. Assume that {xn}GP converges to a point x ∈ X with Gp(x, x, x) = 0. Then lim n→+∞Gp(xn, y, y) = Gp(x, y, y) for all y ∈ X. Moreover, lim m,n→+∞Gp(xn, xm, x) = 0.

Proof. By (GP4), we have

(2.52)
and so lim n→+∞Gp(xn, y, y) = P(x, y, y). Again by (GP4), we get
(2.53)
and hence lim m,n→+∞Gp(xn, xm, x) = 0.

Theorem 2.10. Let (X, Gp) be a 0-GP-complete GP-metric space and f be a self-mapping on X. Assume that (1/3)Gp(x, fx, fx) < Gp(x, y, y) implies

(2.54)
for all x, y ∈ X, where α, β, γ ≥ 0 with α + β + γ < 1. Then f has a unique fixed point.

Proof. If x = fx, then x is a fixed point for f. Assume that x ≠ fx. So by Lemma 1.10, it follows that Gp(x, fx, fx) > 0. Therefore, (1/3)Gp(x, fx, fx) < Gp(x, fx, fx) and so from (2.54), we have

(2.55)
Let x0 ∈ X and define a sequence {xn} by xn = fnx0 for all n ∈ ℕ. Now by (2.55), we can obtain that
(2.56)
Then, for any m > n, by (2.56), we get
(2.57)
It implies that lim m,n→∞Gp(fnx0, fmx0, fmx0) = 0; that is, {xn} is a 0-GP-Cauchy sequence. Since X is 0-GP-complete, so {xn}  GP converges to some point z ∈ X with Gp(z, z, z) = 0, that is,
(2.58)

Now, we suppose that the following inequality holds:

(2.59)
for some x, y ∈ X. Then, by Proposition 1.3 (iii) and (2.55), we have
(2.60)
which is a contradiction. Thus, for all x, y ∈ X, either
(2.61)
holds. Therefore, either
(2.62)
holds for every n ∈ ℕ.

On the other hand, by (2.54), it follows that

(2.63)
If we take the limit as n → ∞ in each of these inequalities, having in mind (2.58), (2.62), and applying Lemma 2.9, then we get (1 − γ)Gp(z, fz, fz) ≤ 0, that is, z = fz. The uniqueness of the fixed point follows easily from (2.54).

As a consequence of Theorem 2.10, we may state the following corollaries.

Corollary 2.11. Let (X, Gp) be a 0-GP-complete GP-metric space and f be a self-mapping on X. Assume that

(2.64)
for all x, y ∈ X, where 0 ≤ r < 1. Then f has a unique fixed point.

Corollary 2.12. Let (X, Gp) be a 0-GP-complete GP-metric space and f be a self-mapping on X. Assume that

(2.65)
for all x, y ∈ X, where 0 ≤ r < 1. Then f has a unique fixed point.

3. Conclusion

In [43], Zand and Nezhad initiated the notion of a GP-metric space. Also, they studied fully its topology. Based on this new space, in this paper we present some fixed point results for self mappings involving different contractive conditions. They are illustrated by some examples. The presented theorems are the first results in fixed point theory on GP-metric spaces.

Acknowledgments

The authors thank the editor and referees for their valuable comments and suggestions.

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