Abstract
We present a fixed point theorem for generalized -Meir-Keeler type contractions in the setting of generalized -metric spaces. The presented results improve, generalize, and unify many existing famous results in the corresponding literature.
1. Introduction and Preliminaries
The idea of a -metric has been introduced in the papers [1, 2]. Very recently, this idea was extended in [3] to a generalized -metric space in the following manner.
Definition 1. Let be a nonempty set and be a fixed constant. A function is called a generalized -metric space (in brief, gbms) if and only if for the conditions are satisfied:(d1) if and only if .(d2) .(d3) .
A triple is called a generalized -metric space.
On the other hand, Meir and Keeler [4] have proved the following very general result on the existence of fixed points of Meir-Keeler contraction mappings in metric spaces.
Theorem 2 (see [4]). Let be a complete metric space and satisfy the following condition: (d)Given , there exists such that has a unique fixed point . Moreover, for any , where denotes the th iteration of at a point .
This result has been generalized and extended in many directions; see [5–15]. Using some auxiliary functions, the main purpose of this paper is to extend and generalize this result on generalized -metric spaces.
For the sake of explicitness, we recall some notations. The symbols denote the natural and real numbers, respectively. Furthermore, and .
Berinde [16] characterized comparison functions to define the contraction mappings in the setting of -metric spaces.
Definition 3. Let be a real number. A function is called a -comparison function if (1) is increasing;(2)there exist , , and a convergent nonnegative series such that , for and any .
Denote as the set of -comparison functions. We will need the following essential properties in our further discussion.
Lemma 4 (see [16–18]). For a -comparison function , the following statements hold: (1)The series converges for any .(2)The function defined by , , is increasing and continuous at .(3)Each iterate of for is also a -comparison function.(4) is continuous at .(5) for any .
Inspired by Popescu [19], we introduce the concept of generalized -orbital admissible mappings.
Definition 5. Let be a mapping and be a function. We say that is a generalized -orbital admissible if
Notice that each -orbital admissible mapping [19] is generalized -orbital admissible.
Based on the concept of generalized -orbital admissibility, we are the first who establish a fixed point result for a Meir-Keeler type contraction in the setting of generalized -metric spaces.
2. Main Results
We start with this definition.
Definition 6. For an arbitrary constant , let be a self-mapping defined on a generalized -metric space . Then is called an -Meir-Keeler contractive mapping if there exist two auxiliary mappings and such that
Remark 7. For and with , from (4) we derive that
Our main result is as follows.
Theorem 8. Let be a fixed constant and be a complete generalized -metric space. Suppose that a self-mapping is an -Meir-Keeler type contraction. Assume also that (i) is generalized -orbital admissible;(ii)there exists such that ;(iii) is continuous. Then for such , one of the following statements holds: ()For every , ()There exists such that and . In this case, there exists such that .
Proof. On account of assumption (ii), there exists such that . We suppose that case () is not satisfied. Consequently, we have to examine case (). Consequently, there exists such that and . If , the proof is completed. Assume that . By property of and Remark 7, we haveSince is a generalized -orbital admissible mapping, by (ii), we derive that Recursively, we obtain thatApplying (10) in (8), we getThusAgain, on account of (10) and (12) in (8), by induction, one getsConsequently, for , by (13) we haveFinally,for all . By (15) and the fact that , it follows that is a Cauchy sequence of elements of .
Since is complete, there exists with Since is continuous, we get and is a fixed point of , which ends the proof.
Definition 9. Let be a fixed constant. We say that a generalized -metric space is regular if is a sequence in such that for all and as ; then there exists a subsequence of such that and for all .
Theorem 10. Let be a fixed constant and be a complete generalized -metric space. Suppose that a self-mapping is an -Meir-Keeler type contraction. Assume also that (i) is a generalized -orbital admissible mapping;(ii)there exists such that ;(iii) is regular. Then for such , one of the following statements holds: ()For every , ()There exists such that and . In this case, there exists such that .
Proof. In case (), following the proof of Theorem 8, we know that the sequence converges to some . By Definition 9 and condition (iii), there exists a subsequence of such that and for all . Applying (5) for all , we get thatLetting in the above equality, we get ; that is, .
For the uniqueness of a fixed point of an -Meir-Keeler type contraction mapping in , we shall consider the following condition:()For all , we have , where denotes the set of fixed points of .
Theorem 11. By adding condition to the hypotheses of Theorem 8 (resp., Theorem 10), has at most one fixed point in .
Proof. Let be an -Meir-Keeler type contraction. Owing to Theorem 8 (resp., Theorem 10), has a fixed point .
Now, we shall show that has at most one fixed point in . We argue by contradiction. For this, assume that there exist two distinct fixed points and of , where ; that is, We deduce By condition , and since , in view of (5), one writes which is a contradiction, so . This completes the proof.
3. Consequences
3.1. Meir-Keeler Contraction Mappings in gbms
In this section, we present our main result. By letting and , we get the following result.
Theorem 12. Let be a generalized complete -metric space and satisfy the following: given , there exists such that
Let . Then one of the following alternatives holds:()For every ( being the set of all nonnegative integers), ()There exists such that .
In case , we assert the following:(i)The sequence is Cauchy in .(ii)There exists a point such that and .(iii) is the unique fixed point of in .(iv)For every ,
Remark 13. Unfortunately, if is a metric space, we do not get the result of Meir-Keeler [4].
Conflicts of Interest
The authors declare that they have no conflicts of interest regarding the publication of this paper.