Some Fixed Point Results in Partial Rectangular Metric-Like Spaces

Sila, Eriola; Duraj, Sidite; Hoxha, Elida

doi:https://doi.org/10.1155/2021/5577776

Journal of Mathematics

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Research Article | Open Access

Volume 2021 | Article ID 5577776 | https://doi.org/10.1155/2021/5577776

Some Fixed Point Results in Partial Rectangular Metric-Like Spaces

Eriola Sila,¹Sidite Duraj,²and Elida Hoxha¹

Academic Editor: Ljubisa Kocinac

Received19 Jun 2021

Revised02 Sept 2021

Accepted04 Oct 2021

Published20 Oct 2021

Abstract

In this paper, we introduce a new concept of partial rectangular metric-like space and prove some results on the existence and uniqueness of a fixed point of a function , defined on a partial rectangular metric-like space , which fulfills a nonlinear contractive condition using a comparison function and the diameter of the orbits. The obtained results generalize some previously acknowledged results in partial metric spaces, partial rectangular metric spaces, and rectangular metric-like spaces. The examples presented prove the usefulness of the introduced generalizations.

1. Introduction

Fixed point theory has a wide application in functional analysis. Various authors [1–5] have contributed to this field expanding the metric spaces by changing their axioms or by improving the contractive conditions.

Partial metric spaces were introduced by Mathew [6] in 1994. He studied their topologies and presented a result relying on the fixed points in these spaces related to Banach contraction. These results had been generalized subsequently by many other studies [7–11].

In 2000, Branciari [12] introduced a new metric space defined as ‘rectangular metric space,’ replacing the triangle inequality in metric spaces with a new inequality called ‘the rectangular inequality.’ In his work, Branciari [12] studied fixed points for several types of functions satisfying certain contraction conditions. Many articles [13–16] were published in rectangular metric spaces inspired by this work, where the importance of these spaces and the applications in them are highlighted.

Later in 2014, Shukla [17] devised the ‘partial rectangular metric spaces.’ He associated his work with applications of fixed point theory to the contractions of Banach, Kannan, Reich, Hardy-Rogers, and Chatterjea, respectively [18].

In 2012, Amini-Harandi [19] defined the metric-like spaces and found some interesting results of fixed point theory in these spaces using the concepts given by Branciari [12, 19], on rectangular metric spaces and metric-like spaces respectively, Mlaiki et al. [20] in their work proposed a generalization of them, called “rectangular metric-like space” associated with respective theorems on the fixed points’ existence on them.

Several distinguished results of fixed point theory on contractions including the diameter of the orbits had been extended on generalized metric spaces [21–23].

Continuing in this way and trying to find an essential connection on these ideas, we propose the concept of partial rectangular metric-like spaces. As a first attempt, some results on the existence and uniqueness of fixed points in these new spaces using the diameter of orbits and comparison functions are displayed. Furthermore, some presented examples show that our results really generalize the previously known concepts and results in partial rectangular space and rectangular metric-like space.

2. Preliminaries

Let be a nonempty set.

Definition 1. (see [6]). The function is a partial metric on if for every , it satisfies the following conditions:(i) if and only if (ii) (iii) (iv) The pair is called the partial metric space.

Definition 2. (see [12]). A function is a rectangular metric on X if for every and for all distinct points it satisfies the following conditions:(i) if and only if (ii) (iii) The pair is called the rectangular metric space.

Definition 3. (see [17]). A function satisfying the conditions(i) for all (ii), if and only if for all (iii) for all (iv) for all (v) for all and for all distinct points is called a partial rectangular metric on . The pair is called a partial rectangular metric space.

Definition 4. (see [19]). A function is called a metric-like on if for each , it satisfies the following conditions:(i) If then (ii) (iii) The pair is called metric-like space.

Definition 5. (see [20]). The function is called a partial rectangular metric-like in if for every and for all distinct points , it satisfies the following conditions:(1) then (2)(3)The pair is called a rectangular metric-like space.

Definition 6. The function is called a partial rectangular metric-like if for all , the following conditions are satisfied:(1) then (2)(3)(4) for all The pair is called a partial rectangular metric-like space.

Remark 1. We see that in a partial rectangular metric-like space , if and then . The converse does not hold.
A rectangular metric-like space is not necessarily a partial rectangular metric-like space.

Example 1. Let and be such thatThen, is a rectangular metric-like in , but it is not a partial rectangular metric-like because it does not satisfy the second condition of Definition 6.
If , then .
Moreover, is not a rectangular metric since for every , , and it is clearly not a partial rectangular metric.
Example 2 shows that a partial rectangular metric-like space may not be a rectangular metric space.

Example 2. Let , then such thatis a partial rectangular metric-like in X, but it is not a rectangular metric.

Definition 7. Let be a partial rectangular metric-like space, a sequence in and .(1)If , then is called -convergent to the point (2)If is finite, then is called -Cauchy in (3)If each -Cauchy sequence in converges to a point (that is, if ), then the space is called -complete

Definition 8. (see [24]). A function is called a comparison function if the following conditions hold:(i) is an nondecreasing function(ii)For each , (iii)For each , The set of comparison functions is denoted by .

3. Main Results

Let be a partial rectangular metric-like space, and let be a function of .

If is a subset of , then is called a diameter of . By Definition 5, if then may not be zero.

For any , we call the set by orbit of .

For each , the union of the orbits isand the diameter can be expressed as

Theorem 1. Let be a partial rectangular metric-like space. If for every , the orbit is bounded and for the function , there is a , such that for every ,where at least one of the numbers or is equal to 1, and then has a unique fixed point.

Proof. Without any restriction, we assume that . Therefore, condition (5) of the theorem becomesWe take any arbitrary . Firstly, we show that the sequence is a -Cauchy one.
For that we have to show that is finite. We denote by and applying condition (6)
in which we get the inequality as follows:Since and , thenSince the function is nondecreasing, then we getUsing the following inequality:Condition 2 of Definition 6, and the relation (9), we getApplying this inequality, we obtain the following ones for any :Since the orbit is bounded, we use the properties of to see thatUsing (12), it follows that .
The sequence is a subsequence of , which is decreasing and bounded below by 0. Hence, the sequence converges to zero.
So, we getConsequently, the sequence is -Cauchy in . Since is complete, then there is any , such thatMeanwhile,For , we haveFrom relationships (16)–(19), we get the inequalitySo,Since and are bounded in by using the third property of the function we obtain
The sequence is a subsequence of the decreasing and bounded below by zero sequence , and it converges to zero; then,Taking into account the definition of and its limit given by (22), the following limits are also obtained:In the case when , then the function has evidently a fixed point.
Otherwise, if , thenFurthermore, taking the limit of for using (23), (14), and (15), we haveNow, we have to prove that is a fixed point of the function .
Indeed, since , then for any ; also,Using the obtained results (23) and (25), we see thatThus,Meanwhile,Since , thenFrom (25), we see that and using the obtained equation (30), we get the inequalitywhich contradicts the main property of the comparison function.
So, finally we get thatand is a fixed point of .
To prove the uniqueness of z, let us suppose that there is , such that .
We see thatconsequently , and finallyPutting we have the following theorem.

Corollary 1. Let be a complete partial rectangular metric-like space. If a function satisfies the condition, there exists such that , where and , and for each , is bounded,
Then, it has a unique fixed point. In the case of partial rectangular metric spaces, Theorem 1 yields.

Theorem 2. Let be a complete partial rectangular metric space and a function such that for each , the orbit is bounded andwhere at least or is equal to 1, and then the function has a unique fixed point.

The proof is analogous to that of Theorem 1.

Remark 2. Theorem 2 generalizes Theorem 19 in [17] and Corollary 1 of Theorem 4 in [25] because the contractive condition (5) is more general than the quasi-contractive condition with any constant since and for all The following example shows that Theorem 2 is a proper generalization of Theorem 19 in [17].

Example 3. Let us take . We determine the function , puttingThe function is a partial rectangular metric in satisfying the conditions of Definition 1.
Let be the function:We see that satisfies the conditions of Theorem 2 with .
Firstly, we see that for all , .
Furthermore, is a monotone nondecreasing function because for any , examining all the cases for x and y, we see that .
For the last condition, if , then , , , , if , then , , ,, if , for thenConsidering all cases, we get , so .
In the following, we show that T satisfies the contractive condition in Theorem 2 for . Firstly, we determine the orbits and their diameters:(i)if , then and(ii)if , then and ;(iii)if then and ;(iv)if , then and .Secondly, we show the fulfillment of the contractive property.

Case 1. If , then we have , and , so

Case 2. If and , thenAlso,

Case 3. Taking such that and where , we haveSince and , thenAn analogous proof is valid for .

Case 4. If and , then , andTherefore, the function satisfies all the conditions of Theorem 2, so it has a fixed point.
Meanwhile, we see that the function does not meet the conditions of Theorem 19 in [17].
Indeed, for and , we haveSo, because and cannot be .
Thus, we already proved that for each , .
Since , there is any such that
Taking the limit for in the inequality , we get .
The last inequality is not true for any so it cannot be found any such that the condition of quasi-contraction of Theorem 19 in [17] for the function is satisfied.

Remark 3. Theorem 1 holds also in the rectangular metric-like space, and consequently it is a generalization of Theorem 2, Theorem 3, and Theorem 4 in [20].

Remark 4. Theorem 1 can also be formulated and proved in the metric-like space; furthermore, it is a generalization of Theorem 2.4 in [19]. To prove this, we show in the following that our contractive condition is more general than the contractive condition in Theorem 2.4 in [19].
The contractive condition in Theorem 2.4 in [19] iswhereand satisfies the following: is nondecreasing , for all , for all Sincewe have .
It remains to show that if the function satisfies the conditions , and then it fulfills also the conditions of Definition 7.
It is clear that if the conditions hold for the function , then the conditions and of Definition 7 also hold.
To show that the condition is fulfilled, we take for each , , so the sequence is a decreasing one and bounded below by 0. Consequently, it is a convergent sequence and ; hence, . If since it is monotone nondecreasing, we have that it has one-sided limits in and .
From the fulfillment of the condition , an can be found, such that . Therefore, for that , there is any such that for we have . For that , there is any such that for all naturals , , sowhich contradicts the supposition , being
Consequently, it remains that ; thus, , and we get that the condition of contractivity in Theorem 1 is more general than the one in Theorem 2.5 [19].

4. Conclusions

This paper defines the partial rectangular metric-like spaces, which generalize the rectangular metric-like spaces [20] and rectangular metric space [12]. In addition, they are generalizations of partial metric spaces.

The main result of this paper is Theorem 1, which provides the existence of a unique fixed point for the functions that satisfy condition (5) of contractivity with respect to a comparison function φ and the orbits’ diameters of .

It generalizes Theorems 2, 3, and 4 in [20] and Theorem 2.4 in [19], as it is shown in Remark 4, where we have proved that our condition of contractivity is really more general than the analogous one at Theorem 2.4 in [19]. Example 3.5 shows that Theorem 1 is a generalization of Theorem 19 in [17] and Corollary 1 of Theorem 4 in [25].

Data Availability

The data used to support the findings of the study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References

L. G. Huang and X. Zhang, “Cone metric spaces and fixed point theorems of contractive mappings,” Journal of Mathematical Analysis and Applications, vol. 332, no. 2, pp. 1468–1476, 2007.
View at: Publisher Site | Google Scholar
M. Jleli and B. Samet, “A generalized metric space and related fixed point theorems,” Fixed Point Theory and Applications, vol. 61, 2015.
View at: Publisher Site | Google Scholar
S. Czerwik, “Contraction mappings in b-metric spaces,” Acta Mathematica et Informatica Universitatis Ostraviensis, vol. 1, no. 5–11, 1993.
View at: Google Scholar
L. Chen, C. Li, R. Kaczmarek, and Y. Zhao, “Several fixed point theorems in convex b-metric spaces and applications,” Mathematics, vol. 8, p. 242, 2020.
View at: Publisher Site | Google Scholar
T. Došenović, Z. Kadelburg, D. Mitrović, and S. Radenović, “New fixed point results in bv(s)-metric spaces,” Mathematica Slovaca, vol. 70, no. 2, pp. 441–452, 2020.
View at: Google Scholar
S. G. Mathews, “Partial metric topology,” in Proceedings of the 8th Summer Conference on General Topology and Applications, vol. 782, pp. 188–197, Annals of the New York Academy of Sciences, Amsterdam, Netherland, August 1994.
View at: Google Scholar
D. Paesano and P. Vetro, “Fixed points and completeness on partial metric spaces,” Miskolc Mathematical Notes HU, vol. 16, no. 1, pp. 369–383, 2015.
View at: Google Scholar
M. A. Alghamdi, N. Shahzad, and O. Valero, “”On fixed point theory in partial metric spaces,” Fixed Point Theory and Applications, vol. 175, 2012.
View at: Publisher Site | Google Scholar
U. Y. Batsari, P. Kumam, and S. Dhompongsa, “Fixed points of terminating mappings in partial metric spaces,” Journal of Fixed Point Theory and Applications, vol. 21, no. 1, p. 39, 2019.
View at: Publisher Site | Google Scholar
A. Ishak, A. Mohammad, I. Mohammad, and A. M. Waleed, “Fixed point results for F-R generalized contractive mappings in partial metric spaces,” Mathematica Slovaca, vol. 69, no. 6, pp. 1413–1424, 2019.
View at: Publisher Site | Google Scholar
P. Dhawan and J. Kaur, “Some common fixed point theorems in ordered partial metric spaces via F-generalized contractive type mappings,” Mathematics, vol. 7, no. 2, p. 193, 2019.
View at: Publisher Site | Google Scholar
A. Branciari, “A fixed point theorem of Banach- Caccioppoli type on a class of generalized metric spaces,” Publicationes Mathematicae, vol. 57, no. 1-2, pp. 31–37, 2000.
View at: Google Scholar
I. M. Erhan, E. Karapınar, and T. Sekulić, “Fixed points of (ψ,ϕ) contractions on rectangular metric spaces,” Fixed Point Theory and Applications, vol. 138, 2012.
View at: Publisher Site | Google Scholar
Z. Mustafa, V. Parvaneh, M. M. M. Jaradat, and Z. Kadelburg, “Extended rectangular b-metric spaces and some fixed point theorems for contractive mappings,” Symmetry, vol. 11, no. 4, p. 594, 2019.
View at: Publisher Site | Google Scholar
N. Y. Özgür, N. Mlaiki, N. Taş, and N. Souayah, “A new generalization of metric spaces: rectangular M-metric spaces,” Mathematical Sciences, vol. 12, no. 3, pp. 223–233, 2018.
View at: Publisher Site | Google Scholar
Z. D. Mitrović, “A note on a Banach’s fixed point theorem in b-rectangular metric space and b-metric space,” Mathematica Slovaca, vol. 68, no. 5, pp. 1113–1116, 2018.
View at: Google Scholar
S. Shukla, “Partial rectangular metric spaces and fixed point theorems,” The Scientific World Journals, vol. 2014, Article ID 756298, 7 pages, 2014.
View at: Publisher Site | Google Scholar
B. E. Rhoades, “A comparison of various definitions of contractive mappings,” Transactions of the American Mathematical Society, vol. 226, p. 257, 1977.
View at: Publisher Site | Google Scholar
A. Amini-Harandi, “Metric-like spaces, partial metric spaces and fixed points,” Fixed Point Theory and Applications, 2012.
View at: Publisher Site | Google Scholar
N. Mlaiki, K. Abodayeh, H. Aydi, T. Abdeljawad, and M. Abuloka, “Rectangular metric-like type spaces and related fixed points,” Hindawi Journals of Mathematics, vol. 2018, Article ID 3581768, 7 pages, 2018.
View at: Publisher Site | Google Scholar
G. V. R. Babu, “Generalization of fixed point theorems relating to the diameter of orbits by using a control function,” Tamkang Journal of Mathematics, vol. 35, no. 2, pp. 159–168, 2004.
View at: Publisher Site | Google Scholar
A. Nicolae, “Generalized asymptotic pointwise contractions and nonexpansive mappings involving orbits,” Fixed Point Theory and Algorithms, vol. 2009, Article ID 458265, 20 pages, 2009.
View at: Publisher Site | Google Scholar
E. Sila and E. Hoxha, “Some generalizations of fixed point theorems in quasi-cone metric space relating to the diameter of orbits by using a comparison function,” Journal of Mathematical Analysis, vol. 7, no. 5, pp. 132–139, 2015.
View at: Google Scholar
N. Hussain, Z. Kadelburg, S. Radenović, and F. Al-Solamy, “Comparison functions and fixed point results in partial metric spaces,” Abstract and Applied Analysis, vol. 2012, Article ID 605781, 15 pages, 2012.
View at: Publisher Site | Google Scholar
N. V. Dung and V. T. Le Hang, “A note on partial rectangular metric spaces,” Mathematica Moravica, vol. 18, pp. 1–8, 2014.
View at: Google Scholar

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Copyright © 2021 Eriola Sila et al. 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.

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