Existence of Periodic Fixed Point Theorems in the Setting of Generalized Quasi-Metric Spaces

Abstract

We introduce the notions of ( α - [varphi] - ψ ) -weaker Meir-Keeler contractive mappings and ( α - [straight phi] ) -stronger Meir-Keeler contractive mappings. We discuss the existence of periodic points in the setting of generalized quasi-metric spaces. Our results improve, extend, and generalize several results in the literature.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Chi-Ming Chen 1 and Erdal Karapinar 2,3 and Vladimir Rakocevic 4

Academic Editor:Lai-Jiu Lin

1, Department of Applied Mathematics, National Hsinchu University of Education, Taiwan
2, Atilim University Department of Mathematics, Incek, 06586 Ankara, Turkey
3, Nonlinear Analysis and Applied Mathematics Research Group (NAAM), King Abdulaziz University, Jeddah, Saudi Arabia
4, Faculty of Sciences and Mathematics, University of Nis, Visegradska 33, 18000 Nis, Serbia

Received 9 July 2014; Accepted 16 August 2014; 26 August 2014

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.

1. Introduction and Preliminaries

Very recently, Lin et al. [1] introduced the notion of generalized quasi-metric inspired from the notion of generalized metric, defined by Branciari [2]. It is a very well-known fact that the concept of generalized metric can be derived from the definition of metric by replacing the triangle inequality with a weaker condition, namely, quadrilateral inequality. In spite of the analogy between the definitions of metric and generalized metric, the topological structure of these spaces is completely different. It was proved that the topologies of these two spaces are incomparable [3].

In what follows that we recall the basic definitions and results on the topics for the sake of completeness. Throughout the paper, the symbols R , N , and _{N 0} denote the real numbers, the natural numbers, and the positive integers, respectively.

A quite natural generalization of the notion of a metric was introduced by Branciari [2] in 2000 by replacing the triangle inequality assumption of a metric with a weaker condition, quadrilateral inequality.

Definition 1 (see [2]).

Let X be a nonempty set and let d : X × X [arrow right] [ 0 , ∞ ) be a mapping such that for all x , y ∈ X and for all distinct point u , v ∈ X each of them different from x and y , one has

(d1) d ( x , y ) = 0 if and only if x = y ;

(d2) d ( x , y ) = d ( y , x ) ;

(d3) d ( x , y ) ...4; d ( x , u ) + d ( u , v ) + d ( v , y ) (quadrilateral inequality).

Then ( X , d ) is called a generalized metric space (or shortly g . m . s ).

The following example illustrates that not every generalized metric on a set X is a metric on X .

Example 2 (see e.g. [1, 4]).

Let X = { t , 2 t , 3 t , 4 t , 5 t } with t > 0 is a constant, and we define d : X × X [arrow right] [ 0 , ∞ ) by

(1) d ( x , x ) = 0 , for all x ∈ X ;

(2) d ( x , y ) = d ( y , x ) , for all x , y ∈ X ;

(3) d ( t , 2 t ) = 3 γ ;

(4) d ( t , 3 t ) = d ( 2 t , 3 t ) = γ ;

(5) d ( t , 4 t ) = d ( 2 t , 4 t ) = d ( 3 t , 4 t ) = 2 γ ;

(6) d ( t , 5 t ) = d ( 2 t , 5 t ) = d ( 3 t , 5 t ) = d ( 4 t , 5 t ) = ( 3 / 2 ) γ ,

where γ > 0 is a constant. Then ( X , d ) is a generalized metric space, but it is not a metric space, because [figure omitted; refer to PDF] Now, we will mention that some standard properties can not be possesed by generalized metric: more precisely,

(P1) open ball need not be open set,

(P2) a convergent sequence in generalized metric space needs not to be Cauchy,

(P3) generalized metric needs not to be continuous,

(P4) generalized metric space needs not to be Hausdorff, and hence the uniqueness of limits can not be guaranteed.

Several authors noticed these weak points of the generalized metric space and inserted some additional assumptions to get the analog of celebrated fixed point theorems in the context of generalized metric space. In particular, generalized metric space assumed Hausdorff. Later, several authors proved that this assumption is superfluous; see for example [5-9].

Example 3 (see [10], Example 1.1).

Let X = A ∪ B where A = { 0,2 } and A = { 1 / n : n ∈ N } . Define d : X × X [arrow right] [ 0 , ∞ ) in the following way: [figure omitted; refer to PDF] Notice that d ( a , b ) = d ( b , a ) = b whenever a ∈ A and b ∈ B . Furthermore, ( X , d ) is a complete generalized metric space. Clearly, we have (P1)-(P4). Indeed, the sequence { 1 / n : n ∈ N } converges to both 0 and 2 . There is no r > 0 such that _{B r} ( 0 ) ∩ _{B r} ( 2 ) = ∅ and hence it is not Hausdorff. It is clear that the ball _{B 2 / 3} ( 1 / 3 ) = { 0 , 1 / 3 , 2 } since there is no r > 0 such that _{B r} ( = ) ⊂ _{B 2 / 3} ( 1 / 3 ) ; that is, open balls may not be an open set. The function d is not continuous since _{lim ... n [arrow right] ∞} d ( 1 / n , 1 / 2 ) ...0; d ( 0 , 1 / 2 ) although _{lim ... n [arrow right] ∞} ( 1 / n ) = 0 . For more details see, for example, [4, 8, 10].

Regarding the weakness of the topology of generalized metric space, mentioned above, the authors add some additional conditions to get the analog of existing fixed point results in the literature; see, for example, [11-17].

The following is the definition of the notion of generalized quasi-metric space defined by Lin et al. [1].

Definition 4.

Let X be a nonempty set and let d : X × X [arrow right] [ 0 , ∞ ) be a mapping such that, for all x , y ∈ X and for all distinct point u , v ∈ X each of them different from x and y , one has

(i) d ( x , y ) = 0 if and only if x = y ;

(ii) d ( x , y ) ...4; d ( x , u ) + d ( u , v ) + d ( v , y ) .

Then ( X , d ) is called a generalized quasi-metric space (or shortly g . q . m . s ).

It is evident that any generalized metric space is a generalized quasi-metric space, but the converse is not true in general. We give an example to show that not every generalized quasi-metric on a set X is a generalized metric on X .

Example 5 (see [1]).

Let X = { t , 2 t , 3 t , 4 t , 5 t } with t > 0 being a constant, and we define d : X × X [arrow right] [ 0 , ∞ ) by

(1) d ( x , x ) = 0 , for all x ∈ X ;

(2) d ( t , 2 t ) = d ( 2 t , t ) = 3 γ ;

(3) d ( t , 3 t ) = d ( 2 t , 3 t ) = d ( 3 t , t ) = d ( 3 t , 2 t ) = γ ;

(4) d ( t , 4 t ) = d ( 2 t , 4 t ) = d ( 3 t , 4 t ) = d ( 4 t , t ) = d ( 4 t , 2 t ) = d ( 4 t , 3 t ) = 2 γ ;

(5) d ( t , 5 t ) = d ( 2 t , 5 t ) = d ( 3 t , 5 t ) = d ( 4 t , 5 t ) = ( 3 / 2 ) γ ;

(6) d ( 5 t , t ) = d ( 5 t , 2 t ) = d ( 5 t , 3 t ) = d ( 5 t , 4 t ) = ( 5 / 4 ) γ ,

where γ > 0 is a constant. Then ( X , d ) is a generalized quasi-metric space, but it is not a generalized metric space, because [figure omitted; refer to PDF]

We next give the definitions of convergence and completeness on generalized quasi-metric spaces.

Definition 6 (see [1]).

Let ( X , d ) be a g . q . m . s , and let { _{x n} } be a sequence in X and x ∈ X . We say that { _{x n} } is g . q . m . s convergent to x if and only if [figure omitted; refer to PDF]

Definition 7 (see [1]).

Let ( X , d ) be a g . q . m . s and let { _{x n} } be a sequence in X . We say that { _{x n} } is left-Cauchy if and only if for every [varepsilon] > 0 there exits k ∈ N such that d ( _{x n} , _{x m} ) < [varepsilon] for all n ...5; m > k .

Definition 8 (see [1]).

Let ( X , d ) be a g . q . m . s and let { _{x n} } be a sequence in X . We say that { _{x n} } is right-Cauchy if and only if for every [varepsilon] > 0 there exits k ∈ N such that d ( _{x n} , _{x m} ) < [varepsilon] for all m ...5; n > k .

Definition 9 (see [1]).

Let ( X , d ) be a g . q . m . s and let { _{x n} } be a sequence in X . We say that { _{x n} } is Cauchy if and only if for every [varepsilon] > 0 there exits k ∈ N such that d ( _{x n} , _{x m} ) < [varepsilon] for all m , n > k .

Remark 10.

A sequence { _{x n} } in a g . q . m . s is Cauchy if and only if it is left-Cauchy and right-Cauchy.

Definition 11 (see [1]).

Let ( X , d ) be a g . q . m . s . We say that

(1) ( X , d ) is left-complete if and only if each left-Cauchy sequence in X is convergent;

(2) ( X , d ) is right-complete if and only if each right-Cauchy sequence in X is convergent;

(3) ( X , d ) is complete if and only if each Cauchy sequence in X is convergent.

In this paper, we examine the existence of ( α - ψ ) -contractive mappings in the context of generalized quasi-metric space without the assumption of being a Hausdorff. Consequently, our results extend, improve, and generalize several results in the literature.

2. Periodic Points of Weaker Meir-Keeler Contractive Mappings

In this section, we recall the weaker Meir-Keeler function and a weaker Meir-Keeler function, as follows.

Definition 12 (see [18]).

A function ψ : [ 0 , ∞ ) [arrow right] [ 0 , ∞ ) is said to be a Meir-Keeler type function, if, for each η ∈ [ 0 , ∞ ) , there exists δ > 0 such that for t ∈ [ 0 , ∞ ) with η ...4; t < η + δ , we have ψ ( t ) < η .

Definition 13.

We call [varphi] : [ 0 , ∞ ) [arrow right] [ 0 , ∞ ) a weaker Meir-Keeler function if the function [varphi] satisfies the following condition: [figure omitted; refer to PDF]

In the sequel, we need the following classes of auxiliary functions. Let Φ denote the set of the nondecreasing functions [varphi] : [ 0 , ∞ ) [arrow right] [ 0 , ∞ ) satisfying the following conditions:

( _{[varphi] 1} ): [varphi] : [ 0 , ∞ ) [arrow right] [ 0 , ∞ ) is a weaker Meir-Keeler function;

( _{[varphi] 2} ): [varphi] ( t ) > 0 for t > 0 and [varphi] ( 0 ) = 0 ;

( _{[varphi] 3} ): for all t ∈ ( 0 , ∞ ) , { ^{[varphi] n} ( t ) _{} n ∈ N} is decreasing;

( _{[varphi] 4} ): for t > 0 , if _{lim ... n [arrow right] ∞} ^{[varphi] n} ( t ) = 0 , then _{lim ... n [arrow right] ∞} ^{∑ i = n m} ... ^{[varphi] i} ( t ) = 0 , where m > n .

Furthermore, let Ψ denote the set of functions ψ : [ 0 , ∞ ) [arrow right] [ 0 , ∞ ) satisfying the following conditions:

( _{ψ 1} ): ψ is continuous;

( _{ψ 2} ): ψ ( t ) > 0 for t > 0 and ψ ( 0 ) = 0 .

The following lemma plays a crucial role in the proof of the main result that were inspired from [5, 8], proved first in [4].

Lemma 14 (see [4]).

Let ( X , d ) be a generalized quasi-metric space and let { _{x n} } be a Cauchy sequence in X such that _{x m} ...0; _{x n} whenever m ...0; n . Then the sequence { _{x n} } can converge to at most one point.

Proof.

Given [varepsilon] > 0 . Since { _{x n} } is a Cauchy sequence, there exists _{k 0} ∈ N such that [figure omitted; refer to PDF] We use the method of Reductio ad absurdum . Suppose, on the contrary, that there exist two distinct points x and y in X such that the sequence { _{x n} } converges to x and y , that is, [figure omitted; refer to PDF] By assumption for any n ∈ N , _{x n} ...0; _{x m} and since x ...0; y , there exists _{k 1} ∈ N such that _{x n} ...0; x and _{x n} ...0; y for any n > _{k 1} ...5; _{k 0} . Due to quadrilateral inequality, we have [figure omitted; refer to PDF] Letting n , m [arrow right] ∞ , we can obtain that d ( x , y ) = 0 by regarding (6) and (7). Hence, we get x = y which is a contradiction.

In this study, we also recall the following notions of α -admissible mappings.

Definition 15 (see [19]).

Let f : X [arrow right] X be a self-mapping of a set X and α : X × X [arrow right] ^{R +} . Then f is called a α -admissible if [figure omitted; refer to PDF]

We now introduce the notion of ( α - [varphi] - ψ ) -weaker Meir-Keeler contractive mappings in the following way.

Definition 16.

Let ( X , d ) be a g . q . m . s , let α : X × X [arrow right] ^{R +} , and let f : X [arrow right] X be a function satisfying [figure omitted; refer to PDF] for all x , y ∈ X . Then f is said to be a ( α - [varphi] - ψ ) -weaker Meir-Keeler contractive mapping.

We state two main periodic point theorems of ( α - [varphi] - ψ ) -weaker Meir-Keeler contractive mapping, as follow.

Theorem 17.

Let ( X , d ) be a complete g . q . m . s , and let α : X × X [arrow right] ^{R +} . Suppose f is a ( α - [varphi] - ψ ) -weaker Meir-Keeler contractive mapping which satisfies

(i) f is α -admissible;

(ii) there exists _{x 0} ∈ X such that α ( _{x 0} , f _{x 0} ) ...5; 1 , α ( f _{x 0} , _{x 0} ) ...5; 1 and α ( _{x 0} , ^{f 2} _{x 0} ) ...5; 1 , α ( ^{f 2} _{x 0} , _{x 0} ) ...5; 1 ;

(iii): f is continuous.

Then f has a periodic point in X .

Proof.

Regarding the assumption (ii) of theorem, we let _{x 0} ∈ X be an arbitrary point such that α ( _{x 0} , f _{x 0} ) ...5; 1 and α ( _{x 0} , f _{x 0} ) ...5; 1 . We will construct a sequence { _{x n} } in X by _{x n + 1} = f _{x n} = ^{f n + 1} _{x 0} for all n ...5; 0 . If we have _{x n 0} = _{x n 0 + 1} for some _{n 0} , then u = _{x n 0} is a fixed point of f . Hence, for the rest of the proof, we presume that [figure omitted; refer to PDF] Since f is α -admissible, we have [figure omitted; refer to PDF] Utilizing the expression above, we obtain that [figure omitted; refer to PDF] By repeating the same steps with starting with the assumption α ( _{x 1} , _{x 0} ) = α ( f _{x 0} , _{x 0} ) ...5; 1 , we conclude that [figure omitted; refer to PDF] In a similar way, we derive that [figure omitted; refer to PDF] Recursively, we get that [figure omitted; refer to PDF] Analogously, we can easily derive that [figure omitted; refer to PDF] In the sequel, we prove that the sequence { _{x n} } is Cauchy; that is, { _{x n} } is both right-Cauchy and left-Cauchy.

Step 1. We will prove that [figure omitted; refer to PDF] Since f is a ( α - [varphi] - ψ ) -weaker Meir-Keeler contractive mapping, we have that, for each n ∈ N ∪ { 0 } , [figure omitted; refer to PDF] Since [varphi] is nondecreasing, by iteration, we derive the following inequality: [figure omitted; refer to PDF] Due to fact that [varphi] is weak Meir-Keeler function, we find that [figure omitted; refer to PDF] Since { ^{[varphi] n} ( d ( _{x 0} , _{x 1} ) ) _{} n ∈ N} is decreasing, it must converge to some η ...5; 0 . We claim that η = 0 . Suppose, on the contrary, that η > 0 . Then by the definition of weaker Meir-Keeler function [varphi] , corresponding to the given η , there exists δ > 0 such that for _{x 0} , _{x 1} ∈ X with η ...4; d ( _{x 0} , _{x 1} ) < δ + η , and _{n 0} ∈ N such that ^{[varphi] _{n 0}} ( d ( _{x 0} , _{x 1} ) ) < η . Since _{lim ... n [arrow right] ∞} ^{[varphi] n} ( d ( _{x 0} , _{x 1} ) ) = η , there exists _{p 0} ∈ N such that η ...4; ^{[varphi] p} ( d ( _{x 0} , _{x 1} ) ) < δ + η , for all p ...5; _{p 0} . Thus, we conclude that ^{[varphi] _{p 0} + _{n 0}} ( d ( _{x 0} , _{x 1} ) ) < η , which is a contradiction. Therefore _{lim ... n [arrow right] ∞} ^{[varphi] n} ( d ( _{x 0} , _{x 1} ) ) = 0 , that is, [figure omitted; refer to PDF]

Step 2. We will prove that [figure omitted; refer to PDF] Since f is a ( α - [varphi] - ψ ) -weaker Meir-Keeler contractive mapping, we have that, for each n ∈ N ∪ { 0 } , [figure omitted; refer to PDF] Inductively, we find that [figure omitted; refer to PDF] by using the fact that [varphi] is nondecreasing. Since { ^{[varphi] n} ( d ( _{x 0} , _{x 2} ) ) _{} n ∈ N} is decreasing, it must converge to some η ...5; 0 . We claim that η = 0 . Suppose, on the contrary, that η > 0 . Then by the definition of weaker Meir-Keeler function [varphi] , corresponding to the given η , there exists δ > 0 such that for _{x 0} , _{x 2} ∈ X with η ...4; d ( _{x 0} , _{x 2} ) < δ + η , and _{n 0} ∈ N such that ^{[varphi] _{n 0}} ( d ( _{x 0} , _{x 2} ) ) < η . Since _{lim ... n [arrow right] ∞} ^{[varphi] n} ( [straight phi] ( d ( _{x 0} , _{x 2} ) ) ) = η , there exists _{p 0} ∈ N such that η ...4; ^{[varphi] p} ( d ( _{x 0} , _{x 2} ) ) < δ + η , for all p ...5; _{p 0} . Thus, we conclude that ^{[varphi] _{p 0} + _{n 0}} ( d ( _{x 0} , _{x 2} ) ) < η , which is a contradiction. Therefore _{lim ... n [arrow right] ∞} ^{[varphi] n} ( d ( _{x 0} , _{x 2} ) ) = 0 ; that is, [figure omitted; refer to PDF]

Step 3. We will prove that the sequence { _{x n} } is right-Cauchy by standard technique. For this purpose, it is sufficient to examine two cases.

Case (I). Suppose that k > 2 and k is odd. Let k = 2 m + 1 , k ...5; 1 . Then, by using the quadrilateral inequality, we have [figure omitted; refer to PDF] Letting n [arrow right] ∞ , then, by using the condition ( _{[varphi] 4} ) , we have [figure omitted; refer to PDF]

Case (II). Suppose that k > 2 and k is even. Let k = 2 m , k ...5; 1 . Then, by using the quadrilateral inequality, we also have [figure omitted; refer to PDF] Letting n [arrow right] ∞ . Then, by using the condition ( _{[varphi] 4} ) , we have [figure omitted; refer to PDF] By above argument, we get that { _{x n} } is a right-Cauchy sequence.

Analogously, we derive that the sequence { _{x n} } is left-Cauchy. Consequently, the sequence { _{x n} } is Cauchy. Since X is a complete g . q . m . s , there exists u ∈ X such that [figure omitted; refer to PDF]

Step 4. We claim that f has a periodic point in X . Suppose, on the contrary, that f has no periodic point. Since f is continuous, we obtain from (31) that [figure omitted; refer to PDF] From (31) and (32), we get immediately that _{lim ... n [arrow right] ∞} ^{f n} _{x 0} = _{lim ... n [arrow right] ∞} f _{x n} = f u . Due to Lemma 14, we conclude that u = f u which contradicts the assumption that f has no periodic point. Therefore, there exists u ∈ X such that u = ^{f p} ( u ) for some p ∈ N . So f has a periodic point in X .

Theorem 18.

Let ( X , d ) be a complete g . q . m . s , and let α : X × X [arrow right] ^{R +} . Suppose f is a ( α - [varphi] - ψ ) -weaker Meir-Keeler contractive mapping which satisfies

(i) f is α -admissible;

(ii) there exists _{x 0} ∈ X such that α ( _{x 0} , f _{x 0} ) ...5; 1 , α ( f _{x 0} , _{x 0} ) ...5; 1 and α ( _{x 0} , ^{f 2} _{x 0} ) ...5; 1 , α ( ^{f 2} _{x 0} , _{x 0} ) ...5; 1 ;

(iii): if { _{x n} } is a sequence in X such that α ( _{x n} , _{x n + 1} ) ...5; 1 , α ( _{x n + 1} , _{x n} ) ...5; 1 for all n and _{x n} [arrow right] x ∈ X as n [arrow right] ∞ , then α ( _{x n} , x ) ...5; 1 , α ( x , _{x n} ) ...5; 1 for all n .

Then f has a periodic point in X .

Proof.

Following the proof of Theorem 17, we know that the sequence { _{x n} } defined by _{x n + 1} = f _{x n} for all n ...5; 0 , converges for some u ∈ X . From (31) and condition (iii), there exists a subsequence { _{x n ( k )} } of { _{x n} } such that α ( _{x n ( k )} , u ) ...5; 1 for all k . Applying (10), for all k , we get that [figure omitted; refer to PDF] Letting k [arrow right] ∞ in the above equality, we find that [figure omitted; refer to PDF] Therefoe, we have _{lim ... k [arrow right] ∞} ^{f n ( k )} _{x 0} = _{lim ... k [arrow right] ∞} f _{x n ( k )} = f u . Owing to Lemma 14, we conclude that u = f u which contradicts the assumption that f has no periodic point. Thus, there exists u ∈ X such that u = ^{f p} ( u ) for some p ∈ N . So f has a periodic point in X .

3. Periodic Points of Stronger Meir-Keeler Contractive Mappings

In this section, we recall the notion of stronger Meir-Keeler function, as follows.

Definition 19.

We call [straight phi] : [ 0 , ∞ ) [arrow right] [ 0,1 ) a stronger Meir-Keeler function if the function [straight phi] satisfies the following condition: [figure omitted; refer to PDF]

And, we let the function [straight phi] : [ 0 , ∞ ) [arrow right] [ 0,1 ) satisfy the following conditions:

( _{[straight phi] 1} ): [straight phi] : [ 0 , ∞ ) [arrow right] [ 0,1 ) is a stronger Meir-Keeler function;

( _{[straight phi] 2} ): [straight phi] ( t ) > 0 for t > 0 and [straight phi] ( 0 ) = 0 .

Next, we introduce the notion of ( α - [straight phi] ) -stronger Meir-Keeler contractive mappings via the stronger Meir-Keeler function [straight phi] and the α -admissible mapping α .

Definition 20.

Let ( X , d ) be a g . q . m . s , let α : X × X [arrow right] ^{R +} , and let f : X [arrow right] X be a function satisfying [figure omitted; refer to PDF] for all x , y ∈ X . Then f is said to be a ( α - [straight phi] ) -stronger Meir-Keeler contractive mapping.

We state two main periodic point theorms of ( α - [straight phi] ) -stronger Meir-Keeler contractive mapping, as follows.

Theorem 21.

Let ( X , d ) be a complete g . q . m . s , and let α : X × X [arrow right] ^{R +} . Suppose f is a ( α - [straight phi] ) -stronger Meir-Keeler contractive mapping which satisfies

(i) f is α -admissible;

(ii) there exists _{x 0} ∈ X such that α ( _{x 0} , f _{x 0} ) ...5; 1 , α ( f _{x 0} , _{x 0} ) ...5; 1 and α ( _{x 0} , ^{f 2} _{x 0} ) ...5; 1 , α ( ^{f 2} _{x 0} , _{x 0} ) ...5; 1 ;

(iii): f is continuous.

Then f has a periodic point in X .

Proof.

Following the proof of Theorem 17, we obtained that [figure omitted; refer to PDF] Next, we prove that the sequence { _{x n} } is Cauchy; that is, { _{x n} } is both right-Cauchy and left-Cauchy.

Step 1. First, we will prove that [figure omitted; refer to PDF] Taking into account (36) and the definition of stronger Meir-Keeler function [straight phi] , we have that, for each n ∈ N , [figure omitted; refer to PDF] Thus the sequence { d ( _{x n} , _{x n + 1} ) } is decreasing and bounded below and hence it is convergent. Let _{lim ... n [arrow right] ∞} d ( _{x n} , _{x n + 1} ) = η ...5; 0 . Then there exists _{n 0} ∈ N and δ > 0 such that for all n ∈ N with n ...5; _{n 0} [figure omitted; refer to PDF] Taking into account (40) and the definition of stronger Meir-Keeler function [straight phi] , corresponding to η use, there exists _{γ η} ∈ [ 0,1 ) such that [figure omitted; refer to PDF] Thus, we can deduce that for each n ∈ N with n ...5; _{n 0} + 1 [figure omitted; refer to PDF] and so [figure omitted; refer to PDF] Since _{γ η} ∈ [ 0,1 ) , we get [figure omitted; refer to PDF]

Step 2. We will prove that [figure omitted; refer to PDF] Taking into account (36) and the definition of stronger Meir-Keeler function [straight phi] , we have that for each n ∈ N [figure omitted; refer to PDF] Thus the sequence { d ( _{x n} , _{x n + 2} ) } is decreasing and bounded below and hence it is convergent. By the same above proof process of Step 1, we also conclude that [figure omitted; refer to PDF]

Step 3. We will prove that the sequence { _{x n} } is right-Cauchy by standard technique. For this purpose, it is sufficient to examine two cases.

Case (I). Suppose that k > 2 and k is odd. Let k = 2 m + 1 , k ...5; 1 . Then, by using the quadrilateral inequality, we have [figure omitted; refer to PDF] Letting n [arrow right] ∞ , then, we have [figure omitted; refer to PDF]

Case (II). Suppose that k > 2 and k is even. Let k = 2 m , k ...5; 1 . Then, by using the quadrilateral inequality, we also have [figure omitted; refer to PDF] Letting n [arrow right] ∞ , then, by using the condition _{[varphi] 4} , we have [figure omitted; refer to PDF] By above argument, we get that { _{x n} } is a right-Cauchy sequence.

Analogously, we derive that the sequence { _{x n} } is left-Cauchy. Consequently, the sequence { _{x n} } is Cauchy.

Since X is a complete g . q . m . s , there exists u ∈ X such that [figure omitted; refer to PDF]

Step 4. We claim that f has a periodic point in X . Suppose, on the contrary, that f has no periodic point. Since f is continuous, we obtain from (52) that [figure omitted; refer to PDF] From (52) and (53), we get immediately that _{lim ... n [arrow right] ∞} ^{f n} _{x 0} = _{lim ... n [arrow right] ∞} f _{x n} = f u . Regarding Lemma 14, we deduce that u = f u which contradicts the assumption that f has no periodic point. So, there exists u ∈ X such that u = ^{f p} ( u ) for some p ∈ N . So f has a periodic point in X .

Apply Theorems 18 and 21, and we can easily deduce the following theorem.

Theorem 22.

Let ( X , d ) be a complete g . q . m . s , and let α : X × X [arrow right] ^{R +} . Suppose f is a ( α - [straight phi] ) -stronger Meir-Keeler contractive mapping which satisfies

(i) f is α -admissible;

(ii) there exists _{x 0} ∈ X such that α ( _{x 0} , f _{x 0} ) ...5; 1 , α ( f _{x 0} , _{x 0} ) ...5; 1 and α ( _{x 0} , ^{f 2} _{x 0} ) ...5; 1 , α ( ^{f 2} _{x 0} , _{x 0} ) ...5; 1 ;

(iii): if { _{x n} } is a sequence in X such that α ( _{x n} , _{x n + 1} ) ...5; 1 , α ( _{x n + 1} , _{x n} ) ...5; 1 for all n and _{x n} [arrow right] x ∈ X as n [arrow right] ∞ , then α ( _{x n} , x ) ...5; 1 , α ( x , _{x n} ) ...5; 1 for all n .

Then f has a periodic point in X .

Acknowledgment

The authors thank the anonymous referees for their remarkable comments, suggestions, and ideas that helped to improve this paper.

Conflict of Interests

The authors declare that they have no competing interests.

References

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