Research Article | Open Access
Property P and Some Fixed Point Results on -Weakly Contractive G-Metric Spaces
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.
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  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  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.
Consistent with Mustafa and Sims , 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 ). 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
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
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 .
The authors thank the referees for their valuable comments and suggestions.
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