Research Article | Open Access

# Property *P* and Some Fixed Point Results on -Weakly Contractive *G*-Metric Spaces

**Academic Editor:**Charles Chidume

#### Abstract

We prove some fixed point results for -weakly contractive maps in *G*-metric spaces, we show that these maps satisfy property *P*. The results presented in this paper generalize several well-known comparable results in the literature.

#### 1. Introduction

Metric fixed point theory is an important mathematical discipline because of its applications in areas such as variational and linear inequalities, optimization, and approximation theory. Generalizations of metric spaces were proposed by Gähler [1, 2] (called 2 metricspaces). In 2005, Mustafa and Sims [3] introduced a new structure of generalized metric spaces, which are called G-metric spaces as a generalization of metric space , to develop and introduce a new fixed point theory for various mappings in this new structure. Many papers dealing with fixed point theorems for mappings satisfying different contractive conditions on *G*-metric spaces can be found in [4–16]. Let be a self-map of a complete metric space with a nonempty fixed point set . Then is said to satisfy property if for each . However, the converse is false. For example, consider and defined by . Then has a unique fixed point at , but every even iterate of is the identity map, which has every point of as a fixed point. On the other hand, if , , then every iterate of has the same fixed point as (see [17, 18]). Jeong and Rhoades [17] showed that maps satisfying many contractive conditions have property . An interesting fact about maps satisfying property is that they have no nontrivial periodic points. Papers dealing with property are those in [17–19]. In this paper, we will prove some general fixed point theorems for -weakly contractive maps in G-metric spaces, and then we show that these maps satisfy property .

Now we give first in what follows preliminaries and basic definitions which will be used throughout the paper.

#### 2. Preliminaries

Consistent with Mustafa and Sims [3], the following definitions and results will be needed in the sequel.

*Definition 2.1. *Let be a nonempty set and satisfy the following properties:() if (coincidence),(), for all , where ,(), for all , with ,(), where is a permutation of (symmetry),(), for all (rectangle inequality).

Then, the function *G* is called the *G*-metric on , and the pair is called the *G*-metric space.

*Definition 2.2. * A *G*-metric is said to be symmetric if for all .

Proposition 2.3. *Every G -metric space will define a metric space by , for all .*

*Definition 2.4. *Let be a *G*-metric space, and be a sequence of points in . Then,(i)a point is said to be the limit of the sequence if
and we say that the sequence is *G* convergent to (we say ,(ii)A sequence is said to be *G*-Cauchy if
(iii) is called a complete *G*-metric space if every *G*-Cauchy sequence in *X* is *G* converge in .

Proposition 2.5. *Let be a G-metric space, then the following are equivalent:*(1)*,
*(2)*,
*(3)*,
*(4)*. *

Proposition 2.6. *Let be a G-metric space, then the following are equivalent:*(1)* is be G-Cauchy in ,*(2)*. *

Proposition 2.7. *Let be a G-metric space. Then, for any , it follows that:*(i)*If , then ,*(ii)*,
*(iii)*,*(iv)*,
*(v)*. *

#### 3. Main Results

Throughout the paper, denotes the set of all natural numbers.

*Definition 3.1 (see [20]). *A function is called altering distance if the following properties are satisfied:(1) is continuous and increasing,(2) if and only if .

The altering distance functions alter the metric distance between two points and enable us to deal with relatively new classes of fixed points problems.

Theorem 3.2. *Let be a complete G-metric space. Let be a self-map on satisfying the following:
**
for all , where , is an altering distance function, and is a continuous function with if and only if . Then, has a unique fixed point (say ), where is continuous at . *

*Proof. *Fix . Then construct a sequence by . We may assume that for each . Since, if there exist such that , then is a fixed point of .

From (3.1), substituting , then, for all ,
Let . Then, (3.2) gives
We have two cases, either or . Suppose that, for some . Then, we have
Therefore, . Hence . This is a contradiction since the 's are distinct.

Thus, , and (3.2) becomes
But is an increasing function. Thus, from (3.5), we get
Therefore, is a positive nonincreasing sequence. Hence there exists such that
Letting , and using (3.5) and the continuity of and , we get
Hence, , therefore , which implies that
Consequently, for a given , there is an integer such that
For with , we claim that
To show (3.11), we use induction on . Inequality (3.11) holds for from (3.10). Assume (3.11) holds for , that is,
For all , take . Using in Definition 2.1 and inequalities (3.10), (3.12), we get
By induction on , we conclude that
We conclude from Proposition 2.6 that is a *G*-Cauchy sequence in . From the completeness of , there exists in such that . For , we have
Letting , and using the fact that is continuous and is continuous on its variables, we get that . Hence . So is a fixed point of . Now, to show uniqueness, let be another fixed point of with . Therefore,
Hence,
This implies that , then and .

Now to show that is continuous at , let be a sequence in with limit . Using (3.1), we have
But is an increasing function, thus from (3.18), we get
Therefore, .

Corollary 3.3. *Let be a self-map on a complete G-metric space satisfying the following for all **
where is an altering distance function, and is a continuous function with if and only if . Then has a unique fixed point (say ), and is continuous at . *

*Proof. *We get the result by taking and , then apply Theorem 3.2.

Corollary 3.4. *Let be a complete G-metric space. Let be a self-map on satisfying the following:
**
for all where is an altering distance function and, is a continuous function with if and only if . Then has a unique fixed point (say ), and is continuous at . *

*Proof. *We get the result by taking and , in Theorem 3.2.

Corollary 3.5. *Let be a complete G-metric space. Let be a self-map on satisfying the following:
**
for all , where is an altering distance function, and is a continuous function with if and only if . Then has a unique fixed point (say ) and is continuous at . *

*Proof. *We get the result by taking and , in Theorem 3.2.

Theorem 3.6. *Under the condition of Theorem 3.2, has property . *

*Proof. *From Theorem 3.2, has a fixed point. Therefore for each . Fix , and assume that . We claim that . To prove the claim, suppose that . Using (3.1), we have
Letting , we deduce form (3.23),
If , then
hence, . By a property of , we deduce that , therefore, . This is a contradiction. On the other hand, if , then (3.1) gives that
Therefore,
which implies that , for all . Thus, , and by a property of , we have . This is a contradiction.

Therefore, , and has property .
where .

*Example 3.7. *Let and be a *G*-metric on . Define by . We take and , for and . So that
We have

#### 4. Applications

Denote by the set of functions satisfying the following hypotheses.(1) is a Lebesgue integral mapping on each compact of .(2)For every , we have .It is an easy matter to see that the mapping , defined by , is an altering distance function. Now, we have the following result.

Theorem 4.1. *Let be a complete G-metric space. Let be a self-map on satisfying the following:
**
where , and . Then has a unique fixed point. *

*Proof. *It follows from Theorem by taking and .

#### Acknowledgment

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

#### References

- S. Gähler, “$G$-metrische Räume und ihre topologische Struktur,”
*Mathematische Nachrichten*, vol. 26, pp. 115–148, 1963. View at: Publisher Site | Google Scholar | Zentralblatt MATH - S. Gähler, “Zur geometric 2-metriche raume,”
*Revue Roumaine de Mathématiques Pures et Appliquées*, vol. 40, pp. 664–669, 1966. View at: Google Scholar - Z. Mustafa and B. Sims, “A new A pproach to Generalize Metric Spaces,”
*Journal of Nonlinear and Convex Analysis*, vol. 7, no. 2, pp. 289–297, 2006. View at: Google Scholar - Z. Mustafa, M. Khandagji, and W. Shatanawi, “Fixed point results on complete $G$-metric spaces,”
*Studia Scientiarum Mathematicarum Hungarica*, vol. 48, no. 3, pp. 304–319, 2011. View at: Publisher Site | Google Scholar - Z. Mustafa and B. Sims, “Fixed point theorems for contractive mappings in complete $G$-metric spaces,”
*Fixed Point Theory and Applications*, vol. 2009, Article ID 917175, 10 pages, 2009. View at: Publisher Site | Google Scholar - W. Shatanawi, “Fixed point theory for contractive mappings satisfying Φ-maps in $G$-metric spaces,”
*Fixed Point Theory and Applications*, vol. 2010, Article ID 181650, 9 pages, 2010. View at: Publisher Site | Google Scholar - H. Aydi, B. Damjanović, B. Samet, and W. Shatanawi, “Coupled fixed point theorems for nonlinear contractions in partially ordered $G$-metric spaces,”
*Mathematical and Computer Modelling*, vol. 54, no. 9-10, pp. 2443–2450, 2011. View at: Publisher Site | Google Scholar - B. S. Choudhury, “Unique fixed point theorem for weakly C-contractive mappings,”
*Kathmandu University Journal of Science, Engineering and Technology*, vol. 5, no. 1, pp. 6–13, 2009. View at: Google Scholar - W. Shatanawi, “Fixed point theorems for nonlinear weakly
*C*-contractive mappings in metric spaces,”*Mathematical and Computer Modelling*, vol. 54, no. 11-12, pp. 2816–2826, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH - M. Abbas, T. Nazir, and S. Radenović, “Common fixed points of four maps in partially ordered metric spaces,”
*Applied Mathematics Letters*, vol. 24, no. 9, pp. 1520–1526, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH - M. Abbas, T. Nazir, and S. Radenović, “Some periodic point results in generalized metric spaces,”
*Applied Mathematics and Computation*, vol. 217, no. 8, pp. 4094–4099, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH - Z. Mustafa, H. Aydi, and E. Karapinar, “On common fixed points in
*G*-metric spaces using (E, A) property,”*Computers & Mathematics with Applications*. In press. View at: Publisher Site | Google Scholar - N. Tahat, H. Aydi, E. Karapinar, and W. Shatanawi, “Common fixed point for single-valued and multi-valued maps satisfying a generalized contraction in
*G*-metric spaces,”*Fixed Point Theory and Applications*, vol. 2012, article 48, 2012. View at: Publisher Site | Google Scholar - M. Abbas, T. Nazir, and D.
*Đ*orić, “Common fixed point of mappings satisfying (E, A) property in generalized metric spaces,”*Applied Mathematics and Computation*, vol. 218, no. 14, pp. 7665–7670, 2012. View at: Publisher Site | Google Scholar - M. Abbas, S. H. Khan, and T. Nazir, “Common fixed points of R-weakly commuting maps in generalized metric space,”
*Fixed Point Theory and Applications*, vol. 2011, article 41, 2011. View at: Publisher Site | Google Scholar - L. Gajić and Z. L. Crvenković, “A fixed point result for mappings with contractive iterate at a point in
*G*-metric spaces,”*Filomat*, vol. 25, no. 2, pp. 53–58, 2011. View at: Publisher Site | Google Scholar - G. S. Jeong and B. E. Rhoades, “Maps for which $F(T)=F({T}^{n})$,” in
*Fixed Point Theory and Applications*, vol. 6, pp. 71–105, Nova Science Publishers, New York, NY, USA, 2007. View at: Google Scholar | Zentralblatt MATH - G. S. Jeong and B. E. Rhoades, “More maps for which $F(T)=F({T}^{n})$,”
*Demonstratio Mathematica*, vol. 40, no. 3, pp. 671–680, 2007. View at: Google Scholar | Zentralblatt MATH - B. E. Rhoades and M. Abbas, “Maps satisfying generalized contractive conditions of integral type for which $F(T)=F({T}^{n})$,”
*International Journal of Pure and Applied Mathematics*, vol. 45, no. 2, pp. 225–231, 2008. View at: Google Scholar | Zentralblatt MATH - M. S. Khan, M. Swaleh, and S. Sessa, “Fixed point theorems by altering distances between the points,”
*Bulletin of the Australian Mathematical Society*, vol. 30, no. 1, pp. 1–9, 1984. View at: Publisher Site | Google Scholar | Zentralblatt MATH

#### Copyright

Copyright © 2012 Mona Khandaqji et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.