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Symmetry 2019, 11(12), 1457; https://doi.org/10.3390/sym11121457

Article
Some New Observations and Results for Convex Contractions of Istratescu’s Type
1
Nonlinear Analysis Research Group, Ton Duc Thang University, Ho Chi Minh City 758307, Vietnam
2
Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City 758307, Vietnam
3
Institut Supérieur d’Informatique et des Techniques de Communication, Université de Sousse, Hammam Sousse 4000, Tunisia
4
China Medical University Hospital, China Medical University, Taichung 40402, Taiwan
5
Department of Mathematics and General Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia
6
School of Computing, Union University, 11000 Belgrade, Serbia
7
Faculty for Trafic and Transport Engineering, University of Belgrade, 11000 Belgrade, Serbia
8
Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11120 Belgrade, Serbia
9
Institute of Research and Development of Processes IIDP, University of the Basque Country, Campus of Leioa, P.O. Box 48940, Leioa, Bizkaia, Spain
*
Authors to whom correspondence should be addressed.
Received: 30 October 2019 / Accepted: 23 November 2019 / Published: 27 November 2019

Abstract

:
The purpose is to ensure that a continuous convex contraction mapping of order two in b-metric spaces has a unique fixed point. Moreover, this result is generalized for convex contractions of order n in b-metric spaces and also in almost and quasi b-metric spaces.
Keywords:
convex contraction; fixed point; b-metric space; almost b-metric space; coincidence point
MSC:
Primary 47H10; Secondary 54H25

1. Introduction

In [1,2], the notion of a b-metric space was initiated and some usual fixed point results have been provided. Many new results in this space were obtained over the past ten years (see for example [3,4,5,6]). Istratescu [7] considered convex contraction mappings in metric spaces and showed that each convex contraction mapping of order two admits a unique fixed point. The Istratescu’s result has recently caused the attention and was the object of examination in b-metric spaces (see [8]). Our paper is a generalization of the Istratescu’s result for convex contractions of order n in b-metric spaces (and also in almost b-metric spaces and in quasi b-metric spaces).

2. Preliminaries

Definition 1.
Given a nonempty set Υ   and   s 1 . Let η : Υ × Υ [ 0 , ) satisfy:
(1) 
η ( ω , υ ) = 0 if ω = υ ;
(2) 
η ( ω , υ ) = η ( υ , ω ) ;
(3) 
η ( ω , ς ) s [ η ( ω , υ ) + η ( υ , ς ) ] ,
for all ω , υ , ς Υ , then d is a b-metric. Here, ( Υ , η , s ) is called a b-metric space.
Definition 2.
Let { ς n } be a sequence in a b-metric space ( Υ , η , s ) . Take ς Υ .
(a) 
{ ς n } is convergent to ς, if for each ε > 0 there is n 0 N so that η ( ς n , ς ) < ε for all n n 0 ;
(b) 
{ ς n } is Cauchy if for every ε > 0 there is n 0 N so that η ( ς n , ς m ) < ε for all n , m n 0 ;
(c) 
( Υ , η , s ) is complete if every Cauchy sequence is convergent.
Miculescu and Mihail [9] (Lemma 2.2) and Suzuki [10] (Lemma 6) gave the following result (see also [11]).
Lemma 1.
Let { ς n } be a sequence in the b-metric space ( Υ , η , s ) so that there is γ [ 0 , 1 ) in order that for every n 1 , η ( ς n + 1 , ς n ) γ η ( ς n , ς n 1 ) . Then { ς n } is Cauchy.
The above lemma is an important tool to get variant results in b-metric spaces since it facilitates many proofs concerning various contraction conditions. The following is a consequence of the proof of Lemma 1.3 in [9]).
Lemma 2.
Let { ς n } be a sequence in the b-metric space ( Υ , η , s ) so that there are γ [ 0 , 1 ) and C > 0 in order that for each n 0 ,
η ( ς n + 1 , ς n ) C γ n .
Then { ς n } is Cauchy.
Definition 3.
Let ( Υ , η , s ) be a b-metric space. The mapping Ω : Υ Υ is a convex Reich type contraction if
η ( Ω 2 ς , Ω 2 ω ) α η ( ς , ω ) + β η ( Ω ς , Ω ω ) + γ [ η ( ς , Ω ς ) + η ( ω , Ω ω ) ] + δ [ η ( Ω ς , Ω 2 ς ) + η ( Ω ω , Ω 2 ω ) ] ,
for all ς , ω Υ , where α , β , γ , δ 0 with α + β + 2 ( γ + δ ) < 1 .
Definition 4.
Let ( Υ , η , s ) be a b-metric space. A mapping Ω : Υ Υ is a convex contraction of Reich type of order n ( n 2 ) if
η ( Ω n ς , Ω n ω ) a 0 η ( ς , ω ) + a 1 η ( Ω ς , Ω ω ) + i = 0 n 1 a i + 2 [ η ( Ω i ς , Ω i + 1 ς ) + η ( Ω i ω , Ω i + 1 ω ) ] ,
for all ς , ω Υ , where a 0 , a 1 , a 2 , , a n + 1 are nonnegative constants with a 0 + a 1 + 2 i = 0 n 1 a i + 2 < 1 .
A slight modification of the definition of contraction mappings of order n was as follows:
Definition 5. 
([7]) Let ( Υ , η ) be a complete metric space. A mapping Ω : Υ Υ is a convex contraction of order n if there are a 0 , , a n 1 in ( 0 , 1 ) so that for all ς , ω Υ ,
η ( Ω n ( ς ) , Ω n ( ω ) ) a 0 η ( ς , ω ) + a 1 η ( Ω ( ς ) , Ω ( ω ) ) + + a n 1 η ( Ω n 1 ( ς ) , Ω n 1 ( ω ) ) ,
where Ω n ( ς ) = Ω ( Ω n 1 ( ς ) ) and a 0 + a 1 + + a n 1 < 1 .
If we exclude in Definition 1 the symmetry condition, ( Υ , η , s ) is said to be a quasi b-metric space. Now, we recall the definition of almost b-metric spaces, which relies on some symmetry-type limiting conditions of defined almost b-metrics.
Definition 6. 
([12]) Let Υ be a nonempty set and s 1 . Let η a b : Υ × Υ [ 0 , + ) and ς , ω , σ , ς n Υ so that:
  • (bM1) η a b ( ς , ω ) = 0 if and only if ς = ω ,
  • (bM2l) η a b ( ς n , ς ) 0 , n implies η a b ( ς , ς n ) 0 , n ,
  • (bM2r) η a b ( ς , ς n ) 0 , n implies η a b ( ς n , ς ) 0 , n ,
  • (bM3) η a b ( ς , ω ) s ( η a b ( ς , σ ) + η a b ( σ , ω ) ) .
  • If (bM1), (bM2l) and (bM3) are verified, then η a b is said to be an l-almost b-metric on Υ;
  • If (bM1), (bM2r) and (bM3) are verified, then η a b is said to be an r-almost b-metric on Υ;
  • If (bM1), (bM2l), (bM2r) and (bM3) are verified, then η a b is said to be an almost b-metric on Υ.
We refer readers to see more about those spaces in the papers [1,2,7,8,9,10,11,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33]. For all contractive type conditions, see [34,35]. One of applications of contractive mappings was used for maximum likelihood estimation of the multiple linear regression parameters in the generalized Gauss–Laplace distribution assumption of the measurement’s errors [36].

3. Main Results

3.1. Some Lemmas

The first lemma is an auxiliary result. We use it to be ensured that in a convex contraction, the Cauchyness holds. The same result was obtained for metric and b-metric spaces.
Lemma 3.
Let k N and { ς n } be a sequence in a b-metric space ( Υ , η , s ) so that
η ( ς n + k , ς n + k 1 ) i = 0 k 1 a i η ( ς n + i , ς n + i 1 ) ,
for all n N , where a i 0 such that i = 0 k 1 a i < 1 . Then
η ( ς n + k , ς n + k 1 ) c γ n ,
for all n N , where c = max { η ( ς 1 , ς 0 ) , , η ( ς k , ς k 1 ) } and γ = i = 0 k 1 a i 1 / k .
Proof. 
From (2), we obtain
η ( ς k + 1 , ς k ) i = 0 k 1 a i η ( ς i + 1 , ς i ) c i = 0 k 1 a i = c γ k c γ .
Similarly,
η ( ς k + 2 , ς k + 1 ) i = 0 k 1 a i η ( ς i + 2 , ς i + 1 ) c i = 0 k 2 a i + c a k 1 γ c i = 0 k 2 a i + c a k 1 = c γ k c γ 2 .
η ( ς 2 k , ς 2 k 1 ) i = 0 k 1 a i η ( ς i + k , ς i + k 1 ) c i = 0 k 1 a i γ i c i = 0 k 1 a i = c γ k .
So, by induction for n k we have that
η ( ς n + k , ς n + k 1 ) i = 0 k 1 a i η ( ς n + i , ς n + i 1 ) c i = 0 k 1 a i γ n + i k c γ n k i = 0 k 1 a i = c γ n .
 □
Lemma 4.
Let k N and { ς n } be a sequence in a b-metric space ( Υ , η , s ) so that
η ( ς n + k , ς n + k 1 ) i = 0 k 1 a i η ( ς n + i , ς n + i 1 ) ,
for all n N , where a i 0 such that i = 0 k 1 a i < 1 . Then { ς n } is Cauchy.
Proof. 
It follows directly from Lemma 2 and Lemma 3. □
Remark 1.
Lemma 4 with k = 1 corresponds to Lemma 1, while for k > 1 , Lemma 4 is more general.

3.2. On Convex Contractions of Order k in b-Metric Spaces

Our next theorem is a generalization of Istratescu’s result about convex contractions. We generalize the result of [7] in two directions by proving it for any k N and by considering the class of b-metric spaces. What distinguishes our obtained result is the fact that it is the same as in usual metric spaces and in b-metric spaces. There are two reasons for our new result: The first is due to Lemma 4 and the other is adding the assumption that the considered mapping is continuous.
Theorem 1.
Let Ω : Υ Υ be a continuous convex contraction of order k, on a complete b-metric space ( Υ , η ) , so that
η ( Ω k ς , Ω k ω ) i = 0 k 1 a i η ( Ω i ς , Ω i ω ) ,
for all ς , ω Υ , where a i 0 such that i = 0 k 1 a i < 1 . Then there is a unique fixed point of Ω.
Proof. 
Let t 0 be in Υ . Consider t n = Ω n ( t 0 ) .
η ( t n + k , t n + k 1 ) = η ( Ω k ( t n ) , Ω k ( t n 1 ) ) i = 0 k 1 a i η ( Ω i ( t n ) , Ω i ( t n 1 ) ) = i = 0 k 1 a i η ( t n + i , t n + i 1 )
and directly from Lemma 4, we conclude that { t n } is a Cauchy sequence in ( Υ , η ) (which is complete). Hence, there is t Υ so that lim n t n = t . Since Ω is continuous, we obtain that
Ω ( t ) = lim n Ω ( t n ) = lim n t n + 1 = t ,
i.e., t is a fixed point of Ω . Its uniqueness follows from (5). □
Example 1.
The space Υ = l p = { ( x n ) R : n = 1 + | x n | p < + } , p ( 0 , 1 ) , together with the function d : l p × l p R ,
d ( x , y ) = n = 1 + | x n y n | p 1 p ,
where x = ( x n ) , y = ( y n ) l p , is a b-metric space with s = 2 1 p , [1]. Indeed, by an elementary calculation, we obtain
d ( x , z ) 2 1 p [ d ( x , y ) + d ( y , z ) ] .
Let Ω : l p l p be a mapping defined by
Ω ( x 1 , x 2 , x 3 , x 4 , ) = ( 0 , x 1 , x 2 2 , x 3 2 , x 4 2 , ) .
Note that Ω has a unique fixed point, which is ( 0 , 0 , 0 , ) (see also [37]). We have
Ω 2 ( x 1 , x 2 , x 3 , x 4 , ) = ( 0 , 0 , x 1 2 , x 2 2 2 , x 3 2 2 , x 4 2 2 , ) ,
Ω 3 ( x 1 , x 2 , x 3 , x 4 , ) = ( 0 , 0 , 0 , x 1 2 2 , x 2 2 3 , x 3 2 3 , x 4 2 3 , ) ,
Ω n ( x 1 , x 2 , x 3 , x 4 , ) = ( 0 , , 0 n , x 1 2 n 1 , x 2 2 n , x 3 2 n , x 4 2 n , ) .
Further, for x , y l p , we have
d ( Ω n x , Ω n y ) = | x 1 y 1 | p 2 p ( n 1 ) + | x 2 y 2 | p 2 p n + | x 3 y 3 | p 2 p n + 1 p 1 2 p ( n 1 ) | x 1 y 1 | p + | x 2 y 2 | p + | x 3 y 3 | p + 1 p 1 2 n 1 d ( x , y ) .
So, the mapping Ω : l p l p is a convex contraction of order n. On the other hand, Ω is not a contraction. Indeed, for x = ( 1 , 0 , 0 , ) and y = ( 2 , 0 , 0 , ) , we have Ω x = ( 0 , 1 , 0 , 0 , ) , Ω y = ( 0 , 2 , 0 , 0 , ) , d ( x , y ) = 1 and d ( Ω x , Ω y ) = 1 . Consequently,
d ( Ω x , Ω y ) > λ d ( x , y ) ,
for each λ ( 0 , 1 ) .
Example 2. 
(a) Let Υ = [ 1 , 4 ] be endowed with η ( ς , ω ) = | ς ω | . Consider Ω : Υ Υ defined by Ω ς = 2 ς . Here, Ω is not a contraction on the metric space ( Υ , η ) . Also, Ω is a convex contraction mapping of order 2. Namely, we have
| Ω 2 ς Ω 2 ω | 2 2 | ς ω | ,
for all ς , ω Ω and ς = 4 is the unique fixed point of Ω.
(b) Let Υ = [ 1 , 4 ] . Consider Ω ς = 2 ς and η ( ς , ω ) = ( ς ω ) 2 . Here, ( Υ , η , s = 2 ) is a b-metric space. Note that Ω is not contraction, but since
η ( Ω 2 ς , Ω 2 , ω ) = 8 ( ς 4 ω 4 ) 2 1 4 η ( Ω ς , Ω ω ) + η ( ς , ω )
Ω is also a convex contraction of order 2.
Example 3.
Let Υ = R . Take Ω ζ = ζ + 3 4 and η ω , v = ω v 2 . Then Υ , η is a complete b —metric space with the coefficient s = 2 . Given Ω : Υ Υ as Ω ζ = ζ + 3 4 for all ζ Υ . We obtain that
d Ω 2 ω , Ω 2 v = ω + 15 16 v + 15 16 2 = ω v 2 256 a n d d Ω ω , Ω v = ω v 2 16 ,
that is,
d Ω 2 ω , Ω 2 v a · d Ω ω , Ω v + b · d ω , v ,
i.e.,
ω v 2 256 a · ω v 2 16 + b · ω v 2 .
Or, equivalently 1 256 a 16 + b where a , b 0 and a + b < 1 . Putting, for example a = b we get that for a = b 1 272 , all the conditions of Theorem 1 are satisfied, i.e., Ω is a convex contraction of order 2 and therefore has a unique fixed point (which is, ζ = 2 ) .
Remark 2.
Theorem 2.1 of [7] and Theorem 4 of [8] follow from Theorem 1. Also Theorem 1 improves the contraction condition (in b-metric spaces) stated for convex contraction mappings of order 2 in [8].
In the previous lemmas and in Theorem 1, we did not use the symmetry of the b-metric η . Thus, with minor changes in the contraction conditions, the main results are similarly valid in the setting of almost b-metric spaces (and also in quasi b-metric spaces). Notice that previous formulations can also applied in l-almost b-metric spaces. Next, we state results for r-almost b-metric spaces.
Remark 3.
Let k N and { ς n } be a sequence in the r-almost b-metric space ( Υ , η , s ) so that
η ( ς n + k 1 , ς n + k ) i = 0 k 1 a i η ( ς n + i 1 , ς n + i ) ,
for all n N , where a i 0 such that i = 0 k 1 a i < 1 . Then
η ( ς n + k , ς n + k 1 ) c γ n ,
for all n N , where c = max { η ( ς 0 , ς 1 ) , , η ( ς k 1 , ς k ) } and γ = i = 0 k 1 a i 1 / k .
Since the result is similar to Lemma 2 and is valid in r-almost b-metric spaces (see [12]), the r-almost version of Lemma 4 is as follows: let k N and { ς n } be a sequence in the r-almost b-metric space ( Υ , η , s ) so that
η ( ς n + k 1 , ς n + k ) i = 0 k 1 a i η ( ς n + i 1 , ς n + i ) ,
for all n N , where a i 0 such that i = 0 k 1 a i < 1 . Then { ς n } is right-Cauchy sequence.
Next, we extend Lemma 3 to quasi b-metric spaces.
Remark 4.
Let ( Υ , η , s ) be a quasi b-metric space and let { ς n } be a sequence in Υ and k N such that (4) is satisfied for all n N , where a i 0 such that i = 0 k 1 a i < 1 . Then
η ( ς n + k , ς n + k 1 ) c γ n ,
for all n N , where c = max { η ( ς 1 , ς 0 ) , η ( ς 0 , ς 1 ) , η ( ς k , ς k 1 ) , η ( ς k 1 , ς k ) } and γ = i = 0 k 1 a i 1 / k .

3.3. On Convex Reich Type Contractions of Order k in b-Metric Spaces

Our next result is about convex Reich type contractions of order k ( k 2 ), (see [18,35,38,39]).
Theorem 2.
Let ( Υ , η , s ) be a complete b-metric space and Ω : Υ Υ be a continuous convex Reich type contraction mapping of order k ( k 2 ) so that
η ( Ω k ς , Ω k ω ) a 0 η ( ς , ω ) + a 1 η ( Ω ς , Ω ω ) + i = 0 k 1 a i + 2 [ η ( Ω i ς , Ω i + 1 ς ) + η ( Ω i ω , Ω i + 1 ω ) ] ,
for all ς , ω Υ , where a 0 , a 1 , a 2 , , a k + 1 0 with a 0 + a 1 + 2 i = 0 k 1 a i + 2 < 1 . Then there is a unique fixed point of Υ.
Proof. 
Let ς 0 be Υ . Take ς n = Ω n ( ς 0 ) for all n N . Now, since Ω is a convex Reich type contraction of order k we obtain
η ( ς n + k , ς n + k 1 ) a 0 η ( ς n , ς n 1 ) + a 1 η ( ς n + 1 , ς n ) + i = 0 k 1 a i + 2 [ η ( ς n + i , ς n + i + 1 ) + η ( ς n + i 1 , ς n + i ) ] .
Since
i = 0 k 1 a i + 2 η ( ς n + i 1 , ς n + i ) = a 2 η ( ς n 1 , ς n ) + i = 0 k 2 a i + 3 η ( ς n + i , ς n + i + 1 ) ,
we obtain
η ( ς n + k , ς n + k 1 ) ( a 0 + a 2 ) η ( ς n , ς n 1 ) + a 1 η ( ς n + 1 , ς n ) + i = 0 k 2 ( a i + 2 + a i + 3 ) η ( ς n + i , ς n + i + 1 ) + a k + 1 η ( ς n + k 1 , ς n + k ) .
That is,
( 1 a k + 1 ) η ( ς n + k , ς n + k 1 ) ( a 0 + a 2 ) η ( ς n , ς n 1 ) + a 1 η ( ς n + 1 , ς n ) + i = 0 k 2 ( a i + 2 + a i + 3 ) η ( ς n + i , ς n + i + 1 ) .
Thus,
η ( ς n + k , ς n + k 1 ) i = 0 k 1 b i η ( ς n + i , ς n + i 1 ) .
for all n N , where b 0 = a 0 + a 2 1 a k + 1 , b 1 = a 1 + a 2 + a 3 1 a k + 1 , b i = a i + 1 + a i + 2 1 a k + 1 , i = 2 , , k 1 . Since a 0 + a 1 + 2 i = 0 k 1 a i + 2 < 1 , we obtain that i = 0 k 1 b i < 1 , therefore using Lemma 4 we conclude that ς n is a Cauchy sequence, and so there is ς Υ such that lim n ς n = ς . It further shows that ς is the unique fixed point of Ω (as in Theorem 1). □
Remark 5.
From Theorem 2, we obtain Theorem 2.3 of [7] (case k = 2 ).
Finally, as an open problem, we may ask the following question:
Let ( Υ , η , s ) be a b-metric space. Find the conditions for the constants a 0 , a 1 , b i , c i such that the mapping Ω : Υ Υ has a unique fixed point, if the next condition (Hardy–Rogers type contraction order of k) is fulfilled:
η ( Ω k ς , Ω k ω ) a 0 η ( ς , ω ) + a 1 η ( Ω ς , Ω ω ) + i = 0 k 1 b i [ η ( Ω i ς , Ω i + 1 ς ) + η ( Ω i ω , Ω i + 1 ω ) ] + i = 0 k 1 c i [ η ( Ω i ς , Ω i + 1 ω ) + η ( Ω i ω , Ω i + 1 ς ) ] ,
for all ς , ω Υ .

4. Conclusions

We generalized the Istratescu’s result for convex contractions. Particulary, we considered convex contractions of order k in the setting of b-metric spaces. Some related observations have been made for the classes of almost and quasi b-metric spaces. The presented results have been supported by some examples.

Author Contributions

All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

Funding

This research has been partially funded by Basque Government with Grant IT1207-19.

Conflicts of Interest

The authors declare no conflict of interest.

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