#### Abstract

We establish a tripled fixed point result for a mixed monotone mapping satisfying nonlinear contractions in ordered generalized metric spaces. Also, some examples are given to support our result.

#### 1. Introduction and Preliminaries

The study of fixed points of mappings satisfying certain contractive conditions has been at the center of rigorous research activity, see [13]. The notion of -metric space is a generalization of usual metric spaces and it is introduced by Dhage [47]. Recently, Mustafa and Sims [8, 9] have shown that most of the results concerning Dhage’s -metric spaces are invalid. In [8, 9], they introduced an improved version of the generalized metric space structure which they called -metric spaces. For more results on -metric spaces, one can refer to the papers [1026].

Now, we give some preliminaries and basic definitions which are used throughout the paper. In 2006, Mustafa and Sims [9] introduced the concept of -metric spaces as follows.

Definition 1.1 (see [9]). Let be a nonempty set, be a function satisfying the following properties:(G1) if ,(G2) for all with ,(G3) for all with ,(G4) (symmetry in all three variables),(G5) for all (rectangle inequality).Then the function is called a generalized metric or, more specially, a -metric on , and the pair is called a -metric space.
Every -metric on will define a metric on by

Example 1.2. Let be a metric space. The function , defined by or for all , is a -metric on .

Definition 1.3 (see [9]). Let be a -metric space, and let be a sequence of points of ; therefore, we say that is -convergent to if , that is, for any , there exists such that , for all . One calls the limit of the sequence and writes or .

Proposition 1.4 (see [9]). Let be a -metric space. The following are equivalent:(1) is -convergent to ,(2) as ,(3) as ,(4) as .

Definition 1.5 (see [9]). Let be a -metric space. A sequence is called a -Cauchy sequence if, for any , there is such that for all , that is, as .

Proposition 1.6 (see [9]). Let be a -metric space. Then the following are equivalent:(1)the sequence is -Cauchy,(2)for any , there exists such that , for all .

Definition 1.7 (see [9]). A -metric space is called -complete if every -Cauchy sequence is -convergent in .

Definition 1.8. Let be a -metric space. A mapping is said to be continuous if for any three -convergent sequences , , and converging to , , and , respectively, is -convergent to .
Recently, Berinde and Borcut [27] introduced these definitions.

Definition 1.9. Let be a partially ordered set and . The mapping is said to have the mixed monotone property if, for any ,

Definition 1.10. Let . An element is called a tripled fixed point of if
Very recently, Berinde and Borcut [28] proved some tripled coincidence theorems for contractive type mappings in partially ordered metric spaces. Also, Samet and Vetro [29] introduced the notion of fixed point of -order as natural extension of that of coupled fixed point and established some new coupled fixed point theorems in complete metric spaces, using a new concept of -invariant set.
Berinde and Borcut [27] proved the following theorem.

Theorem 1.11. Let be a partially ordered set and suppose there is a metric on such that is a complete metric space. Suppose such that has the mixed monotone property and for any for which , and . Suppose either is continuous or has the following properties:(1)if a nondecreasing sequence , then for all ,(2)if a nonincreasing sequence , then  for all ,(3)if a nondecreasing sequence , then for all .If there exist such that , , and , then there exist such that that is, has a tripled fixed point.

In this paper, we establish a tripled fixed point result for a mapping having a mixed monotone property in -metric spaces. Also, we give some examples to illustrate our result.

#### 2. Main Results

Let be the set of all non-decreasing functions such that for all . If , then following Matkowski [30], we have(1) for all ,(2).

The aim of this paper is to prove the following theorem.

Theorem 2.1. Let be partially ordered set and a -metric space. Let be a continuous mapping having the mixed monotone property on . Assume there exists such that for , with , , and , one has If there exist such that , , and , then has a tripled fixed point in , that is, there exist such that

Proof. Suppose are such that , , and . Define , , and . Then , , and . Again, define , , and . Since has the mixed monotone property, we have , , and . Continuing this process, we can construct three sequences , , and in such that If, for some integer , we have , then , , and ; that is, is a tripled fixed point of . Thus we will assume that for all ; that is, we assume that either or or . For any , we have from (2.1) From (2.4), it follows that By repeating (2.5) -times and using the fact that is non-decreasing, we get that
Now, we shill show that is a -Cauchy sequence in . Let . Since and , there exists such that By (2.6), this implies that For , we prove by induction on that Since , then by using (2.9) and the property (G4), we conclude that (2.10) holds when . Now suppose that (2.10) holds for . For , we have Similarly, we show that Hence, we have Thus (2.10) holds for all . Hence , , and are -Cauchy sequences in . Since is a -complete metric space, there exist such that , , and converge to , , and , respectively. Finally, we show that is a tripled fixed point of . Since is continuous and , we have . By the uniqueness of limit, we get that . Similarly, we show that and . So is a tripled fixed point of .

Corollary 2.2. Let be partially ordered set and a -metric space. Let be a continuous mapping having the mixed monotone property on . Suppose that there exists such that for , with , , and one has If there exist such that , , and , then has a tripled fixed point in , that is, there exist such that

Proof. It follows from Theorem 2.1 by taking .

Corollary 2.3. Let be partially ordered set and be a -metric space.
Let be a continuous mapping having the mixed monotone property on . Suppose that there exists such that for , with , , and one has If there exist such that , , and , then has a tripled fixed point in , that is, there exist such that

Proof. Note that Then, the proof follows from Corollary 2.2.

By adding an additional hypothesis, the continuity of in Theorem 2.1 can be dropped.

Theorem 2.4. Let be a partially ordered set and a complete metric space. Let be a mapping having the mixed monotone property. Assume that there exists such that for all with , , and . Assume also that has the following properties:(i)if a nondecreasing sequence , then for all ,(ii)if a nonincreasing sequence , then for all .If there exist such that , , and , then has a tripled fixed point.

Proof. Following proof of Theorem 2.1 step by step, we construct three -Cauchy sequences , , and in with such that , , and . By the hypotheses on , we have , , and for all . If for some , , , and , then which implies that , , and ; that is, is a tripled fixed point of . Now, assume that, for all , . Thus, for each , From (2.19), we have Letting in (2.23) and using (2.22) in the fact that for all , it follows that , , and . Hence is a tripled fixed point of .

Now we give some examples illustrating our results.

Example 2.5. Take endowed with the complete -metric: for all . Set and defined by . The mapping has the mixed monotone property. We have for all , , and , that is, (2.14) holds. Take , then all the hypotheses of Corollary 2.2 are verified, and is the unique tripled fixed point of .

Example 2.6. As in Example 2.5, take and for all . Set and defined by . The mapping has the mixed monotone property. For all , , and , we have that is, (2.16) holds. Take , then all the hypotheses of Corollary 2.3 hold, and is the unique tripled fixed point of .

Remark 2.7. In our main results (Theorems 2.1 and 2.4), the considered contractions are of nonlinear type. Then, inequality (2.1) does not reduce to any metric inequality with the metric (this metric is given by (1.1)). Hence our theorems do not reduce to fixed point problems in the corresponding metric space .