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.
convex contraction; fixed point; b-metric space; almost b-metric space; coincidence point
Primary 47H10; Secondary 54H25
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  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 ). 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).
Given a nonempty set
. Let satisfy:
for all , then d is a b-metric. Here, is called a b-metric space.
Let be a sequence in a b-metric space . Take .
is convergent to ς, if for each there is so that for all ;
is Cauchy if for every there is so that for all ;
is complete if every Cauchy sequence is convergent.
Miculescu and Mihail  (Lemma 2.2) and Suzuki  (Lemma 6) gave the following result (see also ).
Let be a sequence in the b-metric space so that there is in order that for every , . Then 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 ).
Let be a sequence in the b-metric space so that there are and in order that for each ,
Then is Cauchy.
Let be a b-metric space. The mapping is a convex Reich type contraction if
for all , where with .
Let be a b-metric space. A mapping is a convex contraction of Reich type of order if
for all , where are nonnegative constants with .
A slight modification of the definition of contraction mappings of order n was as follows:
() Let be a complete metric space. A mapping is a convex contraction of order n if there are in so that for all ,
where and .
If we exclude in Definition 1 the symmetry condition, 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.
() Let Υ be a nonempty set and . Let and so that:
(bM1) if and only if ,
(bM2l) implies ,
(bM2r) implies ,
If (bM1), (bM2l) and (bM3) are verified, then is said to be an l-almost b-metric on Υ;
If (bM1), (bM2r) and (bM3) are verified, then is said to be an r-almost b-metric on Υ;
If (bM1), (bM2l), (bM2r) and (bM3) are verified, then 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 .
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 4 with corresponds to Lemma 1, while for , 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  in two directions by proving it for any 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.
Let be a continuous convex contraction of order k, on a complete b-metric space , so that
for all , where such that . Then there is a unique fixed point of Ω.
Let be in . Consider .
and directly from Lemma 4, we conclude that is a Cauchy sequence in (which is complete). Hence, there is so that . Since is continuous, we obtain that
i.e., t is a fixed point of . Its uniqueness follows from (5). □
The space , together with the function
where , is a b-metric space with , . Indeed, by an elementary calculation, we obtain
Let be a mapping defined by
Note that Ω has a unique fixed point, which is (see also ). We have
Further, for , we have
So, the mapping is a convex contraction of order n. On the other hand, Ω is not a contraction. Indeed, for and , we have , , and . Consequently,
for each .
(a) Let be endowed with . Consider defined by . Here, Ω is not a contraction on the metric space . Also, Ω is a convex contraction mapping of order 2. Namely, we have
for all and is the unique fixed point of Ω.
(b) Let . Consider and . Here, is a b-metric space. Note that Ω is not contraction, but since
Ω is also a convex contraction of order 2.
Let . Take and Then is a complete —metric space with the coefficient Given as for all We obtain that
Or, equivalently where and Putting, for example we get that for , 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,
Theorem 2.1 of  and Theorem 4 of  follow from Theorem 1. Also Theorem 1 improves the contraction condition (in b-metric spaces) stated for convex contraction mappings of order 2 in .
In the previous lemmas and in Theorem 1, we did not use the symmetry of the b-metric . 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.
Let and be a sequence in the r-almost b-metric space so that
for all , where such that . Then
for all , where and
Since the result is similar to Lemma 2 and is valid in r-almost b-metric spaces (see ), the r-almost version of Lemma 4 is as follows: let and be a sequence in the r-almost b-metric space so that
for all , where such that . Then is right-Cauchy sequence.
Next, we extend Lemma 3 to quasi b-metric spaces.
Let be a quasi b-metric space and let be a sequence in Υ and such that (4) is satisfied for all , where such that . Then
for all , where and
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 (), (see [18,35,38,39]).
Let be a complete b-metric space and be a continuous convex Reich type contraction mapping of order k so that
for all , where with . Then there is a unique fixed point of Υ.
Let be . Take for all Now, since is a convex Reich type contraction of order k we obtain
for all , where , , , Since , we obtain that , therefore using Lemma 4 we conclude that is a Cauchy sequence, and so there is such that . It further shows that is the unique fixed point of (as in Theorem 1). □
From Theorem 2, we obtain Theorem 2.3 of  (case ).
Finally, as an open problem, we may ask the following question:
Let be a b-metric space. Find the conditions for the constants such that the mapping has a unique fixed point, if the next condition (Hardy–Rogers type contraction order of k) is fulfilled:
for all .
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.
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
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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