Unique Common Fixed Points for Expansive Maps

Alzumi, Hadeel Z.; Bouhadjera, Hakima; Abdo, Mohammed S.

doi:https://doi.org/10.1155/2023/6689743

International Journal of Mathematics and Mathematical Sciences

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Research Article | Open Access

Volume 2023 | Article ID 6689743 | https://doi.org/10.1155/2023/6689743

Unique Common Fixed Points for Expansive Maps

Hadeel Z. Alzumi,¹Hakima Bouhadjera,²and Mohammed S. Abdo³

Academic Editor: Nawab Hussain

Received22 Jul 2023

Revised15 Aug 2023

Accepted28 Aug 2023

Published11 Sept 2023

Abstract

In this work, we establish three common fixed point results for expansive maps satisfying implicit relations in metric and dislocated metric spaces. We do this by utilizing recently developed concept of occasionally weakly biased maps of type . These studies about the theory of common fixed points refine several earlier ones. Some illustrative examples are offered to support our theorems, and even better, a pertinent application is supplied to demonstrate the viability and applicability of one of these results.

1. Introduction and Preliminary Notes

In fixed point theory, many researchers investigated the existence and uniqueness of fixed and common fixed points for contractive maps, while several authors concentrated their investigations on expansive maps, and of course, some other mathematicians focused their inquiries on both contractive and expansive maps simultaneously. According to Chouhan and Malviya [1], the research about fixed points of expansive maps was initiated in 1967 by Machuca [2]. Later, many works searched fixed and common fixed points for expansive maps in different spaces.

On the other hand, in 1985, in his thesis, Matthews [3] proposed the kind of metric domains and he observed that there is a bijection between the family of metric domains and the one of metric spaces. He introduced this new space to show that fixed points can exist in other spaces under various contractive conditions.

Definition 1. (see [3]). A metric domain is a pair where is a nonempty set, and is a function from to such that(1),: .(2),: .(3),,: .In 2001, in his thesis, Hitzler [4] used metric domains under the name of dislocated metrics. In 2012, Amini-Harandi [5] suggested a new generalization of the metric space which is called a metric-like space. In fact, the notions of metric domains, metric-like spaces, and dislocated metric spaces are exactly the same, and these spaces are sometimes called as d-metric spaces.

Definition 2. (see [4, 5]). Let be a nonempty set. A function is said to be a dislocated metric on if for any , ,, the following conditions hold:(1).(2).(3).The pair is then called a dislocated metric space.
Recently, in 2019, Markin and Sichel [6] introduced the notion of expansive maps and their types as follows.

Definition 3. (see [6]). Let be a metric space. A map on such thatis called an expansive map (or expansion).

Definition 4. (see [6]). Let be a metric space.(1)An expansion such that is called an isometry, which is the weakest form of expansive maps.(2)An expansion such that and we call it a proper expansion.(3)An expansion such that and we call it a strict expansion.(4)An expansion such thatand we call it an anti-contraction with expansion constant .
Now, to combine the existing extensions and generalizations of Banach fixed-point theorem, different methods were used by many authors. Among them, Popa, in 1997 and 1999, in his articles [7, 8], established the implicit relation’s idea. Afterwards, several researchers used this good combination for proving fixed and common fixed point theorems for single and multi-valued maps in various spaces (see for instance [9–26]). In this paper, we will introduce new kinds of implicit relations in order to use them to prove unified common fixed point theorems in metric and dislocated metric spaces.
In the sequel, our principal results will be presented and proved.

2. Unique Common Fixed Points for Quadruple Maps in Metric Spaces

In this section, we will present our new definitions and introduce some implicit relations in order to prove our first result.

2.1. New Concepts

In 2022, in [27], we initiated the notions of occasionally weakly -biased (respectively, -biased) maps of type , and we revealed that these definitions coincide with our concepts: occasionally weakly -biased (respectively, -biased) maps given in [28]. Note that several authors proved the existence of fixed points for occasionally weakly biased, subweakly biased, and biased maps (see for instance [29–32]).

Definition 5. (see [27]). Let and be maps from a nonempty set into itself. Maps and are called occasionally weakly -biased (respectively, -biased) of type , iff there is an element in such that impliesrespectively.

2.2. Implicit Relations

As we said above, in his papers, Popa [7, 8] unified several explicit contractions under the so-called implicit contraction. Motivated by Popa’s technique, we instigate the following.

Let be a set of functions such that is nondecreasing in , , , , and and satisfies the next condition:are negative for all positive.

Example 1. , where , and .(i)Trivially, is nondecreasing in ,,,, and .(ii).(iii).(iv).

Example 2. , where , and .(i)Clearly, is nondecreasing in variables ,,,, and .(ii).(iii).(iv).

Example 3. , where , and .(i)It is evident to see that is nondecreasing in variables ,,,, and .(ii).(iii).(iv).

Example 4. , where , and .(i)Evidently, is nondecreasing in variables ,,,, and .(ii).(iii).(iv).

Theorem 6. Let , , , and be maps from a metric space into itself, such that, for all , , we havewhere . Assume that maps and (respectively, and ) are occasionally weakly -biased (respectively, -biased) of type ; then, ,,, and admit only one common fixed point.

Proof. According to the assumptions, we have the existence of two elements and in which verify implies and implies .
Firstly, we will show that . Let us assume that , and the use of inequality (8) yieldsa contradiction, and thus .
Secondly, we assure that . Imagine we have the opposite; then, using assumption (8), we getAs maps and are occasionally weakly -biased of type , is nondecreasing in , , and ; using the triangle inequality, we obtainand this contradiction implies that and so .
Thirdly, suppose that . Using inequality (8), we obtainAgain, as is nondecreasing in , , and and maps and are occasionally weakly -biased of type , by the triangle inequality, we findwhich is a contradiction, and hence and so , i.e., and . Put ; therefore, is a common fixed point of maps , , , and .
Fourthly, assume the existence of another common fixed point (say ). From (8), we havewhich implies that .

The following example supports our result.

Example 5. Equip with the usual metric and set up the following maps:Trivially, maps and (respectively, and ) are occasionally weakly -biased (respectively, -biased) of type . Putting , we get(1)Firstly, for , , we have , , , and(2)Secondly, for , , we have , , , and(3)Thirdly, for , , we have , , , , and(4)Fourthly, for , , we have , , , , andThereby, all the theorem’s conditions are fulfilled, and the four maps admit 1 as the sole common fixed point.

Remark 7. Note that Theorem 2 of [33] is inapplicable because the space is incomplete and the four maps are discontinuous. Also, we mention that Theorems 4.1 and 4.4 of [34] are not applicable because and .

3. Unique Common Fixed Points for Four Maps in Dislocated Metric Spaces

In this part, we will present a new type of implicit relations in order to use them for proving the existence and uniqueness of a common fixed point for two pairs of occasionally weakly biased maps of type .

3.1. Implicit Relations

Now, let be a set of functions such that is nondecreasing in , , , , and and satisfies the next condition:for all .

Example 6. , where .(i)It is trivial that is nondecreasing in , , , , and .(ii).

Example 7. , where .(i)It is evident to see that is nondecreasing in variables , , , , and .(ii).

Example 8. , where , , , , and .(i)Clearly, is nondecreasing in variables , , , , and .(ii).

Theorem 8. Let , , , and be four maps from a dislocated metric space into itself satisfyingfor all , , where . Suppose that maps and (respectively, and ) are occasionally weakly -biased (respectively, -biased) of type ; then, maps , , , and possess only one common fixed point.

Proof. As in the demonstration of the first theorem, since maps and as well as and are occasionally weakly -biased (respectively, -biased) of type , there are two points and in such that implies and implies . We need four steps to prove the existence and uniqueness of the common fixed point, as follows. First step: We claim that . Suppose that we have the contrary; using inequality (21), we get Since is nondecreasing in the third and fourth variables, we get a contradiction, which implies that . Second step: If , the use of condition (21) gives Using the properties of , we get Again by the nondecreasing assumption of and using the relationship between maps and , we find i.e., which is a contradiction, and hence ; consequently, . Third step: Now, assume that is positive; then, Using our hypotheses, we get Then, we have a contradiction, which implies that ; consequently, which implies that . Fourth step: Put and assume the existence of another common fixed point (say ); utilizing (21), we obtainAs is nondecreasing in the third and the fourth variables, we getwhich is a contradiction; hence, , and this completes the proof.

To support our result, we furnish the next example.

Example 9. Endow with the dislocated metric and establish the following maps:First of all, the occasionally weakly biased of type assumption is satisfied. Define , and we get(1)For , , we have , , , , and(2)For , , we have , , , , and(3)For , , we have , , , , and(4)Finally, for , , we have , , , , andThereby, all hypotheses of the theorem are satisfied; the four maps accept 0 as the only common fixed point.

Remark 9. Mention that and .
Using Theorem 8 and Example 6, we gain the next result.

Corollary 10. Suppose that , , , and are four maps from a dislocated metric space into itself such that for all , , the following condition holds:where . If and , and are occasionally weakly -biased (respectively, -biased) of type , then, there exists only one point (say ) which verifies .

Proof. Indeed, by assumptions, we have the existence of two elements and in which verifyThe proof needs four cases. Case one: assume that ; the use of inequality (38) gives a contradiction, and hence . Case two: suppose that ; using condition (38), we get which is a contradiction, and hence ; consequently, . Case three: now, if is positive, then a contradiction, which implies that ; consequently, which implies that . Case four: put and assume the existence of another element (say ) which satisfies ; by inequality (38), we obtaina contradiction; hence, , and the uniqueness of the common fixed point is satisfied; this achieves the proof.

3.2. Unique Common Fixed Points for a Sequence of Maps

Theorem 11. Suppose that , , and are maps from a metric space into itself such that for all , , the next condition holds:where . In addition, assume that maps and (respectively, and ) are occasionally weakly -biased (respectively, -biased) of type ; then, there exists one element which satisfies for .

Theorem 12. Let , , and be maps from a dislocated metric space into itself such that for all , , the following inequality holds:where is a function, nondecreasing in variables , , , , and , and satisfies for all . If and (respectively, and ) are occasionally weakly -biased (respectively, -biased) of type , then there is only one element which verifies for .

4. Application to an Integral Equation

Consider the next integral equation:for all , where(1), , are continuous.(2), are continuous functions.(3), , , are continuous functions.

Let be the set of real continuous functions on , endowed with the dislocated metricfor all , . It is evident that is a dislocated metric space.

Theorem 13. Integral equation (46) has only one solution in for and if the next assumptions hold:(1).(2).(3)The functions commute at their each coincidence point.(4)There is such that for all and , for .(5)There is such that for all and , for .(6)There is such that for all , .

Proof. Define , , , , , bywhere is the identity function on .
By the virtue of condition 3, we can see that and as well as and are occasionally weakly -biased (respectively, -biased) of type .
Now, we prove that condition (38) of Corollary 10 is satisfied.which implies thatIt follows that for all , ,Similarly, we havewhich implies thatIt follows that for all , ,Hence, we havewhich implies thatFrom (51) and (56), we getAs a result, the assumptions of Corollary 10 are fulfilled. Thus, there exists only one element which satisfies ; consequently, is a unique solution of (46).

5. Conclusion

Three common fixed point results for expansive maps meeting implicit relations are proved in this article. Additionally, we have determined the basic characteristics of these maps in metric and dislocated metric spaces. Our recently created concept of occasionally weakly biased maps of type served as the foundation for this. These findings improve a number of previous results about the notion of common fixed points. Our theorems have been supported by some illustrated examples; also, a relevant application has been provided to show the viability and usefulness of one of these results. The relevant work shown and discussed in [4, 5] is expanded upon and improved by our results.

Data Availability

No real data were used to support this study. The data used in this study are hypothetical.

Disclosure

This work was carried out as part of the authors’ duties at Hodeidah University.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References

P. Chouhan and N. Malviya, “Fixed points of expansive type mappings in 2-Banach spaces,” Int. J. Anal. Appl., vol. 3, no. 1, pp. 60–67, 2013.
View at: Google Scholar
R. Machuca, “A coincidence theorem,” The American Mathematical Monthly, vol. 74, no. 5, p. 569, 1967.
View at: Publisher Site | Google Scholar
S. G. Matthews, Metric Domains for Completness, University of Warwick, Coventry, UK, 1985.
P. Hitzler, Generalized Metrics and Topology in Logic Programming Semantics, National University Ireland, Galway, Ireland, 2001.
A. Amini-Harandi, “Metric-like spaces, partial metric spaces and fixed points,” Fixed Point Theory and Applications, vol. 2012, pp. 204–210, 2012.
View at: Publisher Site | Google Scholar
M. V. Markin and E. S. Sichel, “On expansive mappings,” Mathematics, vol. 7, no. 11, pp. 1004–1010, 2019.
View at: Publisher Site | Google Scholar
V. Popa, “Fixed point theorems for implicit contractive mappings,” Stud. Cercet. Ştiinţ. Ser. Mat. Univ. Bacău., vol. 7, pp. 127–133, 1997.
View at: Google Scholar
V. Popa, “Some fixed point theorems for compatible mappings satisfying an implicit relation,” Demonstratio Mathematica, vol. 32, no. 1, pp. 157–163, 1999.
View at: Publisher Site | Google Scholar
D. M. Abd-Elhamed, “Fixed point theorems for functions satisfying implicit relations in generalized b-metric spaces,” International Journal of Mathematics and Computer Science, vol. 15, no. 1, pp. 1–10, 2020.
View at: Google Scholar
W. M. Alfaqih, A. Aldurayhim, M. Imdad, and A. Perveen, “Relation-theoretic fixed point theorems under a new implicit function with applications to ordinary differential equations,” AIMS Mathematics, vol. 5, no. 6, pp. 6766–6781, 2020.
View at: Publisher Site | Google Scholar
W. M. Alfaqih, M. Imdad, and F. Rouzkard, “Unified common fixed point theorems in complex valued metric spaces via an implicit relation with applications,” Boletim da Sociedade Paranaense de Matemática, vol. 38, no. 4, pp. 9–29, 2019.
View at: Publisher Site | Google Scholar
J. Ali and M. Imdad, “Unifying a multitude of common fixed point theorems employing an implicit relation,” Communications of the Korean Mathematical Society, vol. 24, no. 1, pp. 41–55, 2009.
View at: Publisher Site | Google Scholar
A. Arif, M. Nazam, A. Hussain, and M. Abbas, “The ordered implicit relations and related fixed point problems in the cone b-metric spacesb-metric spaces,” AIMS Mathematics, vol. 7, no. 4, pp. 5199–5219, 2022.
View at: Publisher Site | Google Scholar
G. V. R. Babu and P. D. Sailaja, “Coupled fixed point theorems with new implicit relations and an application,” Journal of Operators, vol. 2014, Article ID 389646, 16 pages, 2014.
View at: Publisher Site | Google Scholar
I. Beg and A. Rashid Butt, “Common flxed point for generalized set valued contractions satisfying an implicit relation in partially ordered metric spaces,” Mathematical Communications, vol. 15, no. 1, pp. 65–76, 2010.
View at: Google Scholar
V. Berinde, “Approximation fixed points of implicit almost contractions,” Hacet. J. Math. Stat., vol. 41, no. 1, pp. 93–102, 2012.
View at: Google Scholar
S. Chauhan, M. A. Khan, Z. Kadelburg, and M. Imdad, “Unified common fixed point theorems for a hybrid pair of mappings via an implicit relation involving altering distance function,” Journal of Function Spaces, vol. 2014, Article ID 718040, 8 pages, 2014.
View at: Publisher Site | Google Scholar
R. Jain, H. K. Nashine, and S. Kumar, “An Implicit Relation Approach in Metric Spaces under -Distance and Application to Fractional Differential Equation,” Journal of Function Spaces, vol. 2021, Article ID 9928881, 12 pages, 2021.
View at: Publisher Site | Google Scholar
K. S. Kim, “Coupled fixed point theorems under new coupled implicit relation in Hilbert spaces,” Demonstratio Mathematica, vol. 55, no. 1, pp. 81–89, 2022.
View at: Publisher Site | Google Scholar
H. K. Pathak, R. Rodríguez-López, and R. K. Verma, “A common fixed point theorem using implicit relation and property (E.A) in metric spacesE.Ain metric spaces,” Filomat, vol. 21, no. 2, pp. 211–234, 2007.
View at: Publisher Site | Google Scholar
H. K. Pathak, R. Tiwari, and M. S. Khan, “A common fixed point theorem satisfying integral type implicit relations,” Applied Mathematics E-Notes, vol. 7, pp. 222–228, 2007.
View at: Google Scholar
V. Popa, M. Imdad, and J. Ali, “Using implicit relations to prove unified fixed point theorems in metric and 2-metric spaces,” Bull. Malays. Math. Sci. Soc., vol. 33, no. 1, pp. 105–120, 2010.
View at: Google Scholar
R. A. Rashwan, H. A. Hammad, and M. G. Mahmoud, “Common fixed point results for weakly compatible mappings under implicit relations in complex valued G-metric spaces,” Inf. Sci. Lett., vol. 8, no. 3, pp. 1–10, 2019.
View at: Google Scholar
G. S. Saluja, “Fixed point theorems using implicit relation in partial metric spaces,” Facta Universitatis Series: Mathematics and Informatics, vol. 35, no. 3, pp. 857–872, 2020.
View at: Publisher Site | Google Scholar
G. S. Saluja, “Fixed point theorems on cone S-metric spaces using implicit relationS-metric spaces using implicit relation,” Cubo, vol. 22, no. 2, pp. 273–289, 2020.
View at: Publisher Site | Google Scholar
S. Sharma and B. Deshpande, “On compatible mappings satisfying an implicit relation in common fixed point consideration,” Tamkang J. Math., vol. 33, no. 3, pp. 245–252, 2002.
View at: Publisher Site | Google Scholar
H. Bouhadjera, “Fixed points for occasionally weakly biased mappings of type (A),” Mathematica Moravica, vol. 26, no. 1, pp. 113–122, 2022.
View at: Publisher Site | Google Scholar
H. Bouhadjera and A. Djoudi, “Fixed point for occasionally weakly biased maps,” Southeast Asian Bulletin of Mathematics, vol. 36, no. 4, pp. 489–500, 2012.
View at: Google Scholar
N. Hussain, M. A. Khamsi, and A. Latif, “Common fixed points for operators and occasionally weakly biased pairs under relaxed conditionsJH-operators and occasionally weakly biased pairs under relaxed conditions,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 6, pp. 2133–2140, 2011.
View at: Publisher Site | Google Scholar
N. Hussain and H. K. Pathak, “Subweakly biased pairs and Jungck contractions with applications,” Numerical Functional Analysis and Optimization, vol. 32, no. 10, pp. 1067–1082, 2011.
View at: Publisher Site | Google Scholar
Y. Mahendra Singh and M. R. Singh, “Fixed points of occasionally weakly biased mappings,” Adv. Fixed Point Theory, vol. 2, no. 3, pp. 286–297, 2012.
View at: Google Scholar
N. Shahzad and S. Sahar, “Some common fixed point theorems for biased mappings,” Archivum Mathematicum, vol. 36, pp. 183–194, 2000.
View at: Google Scholar
A. Djoudi, “A unique common fixed point for compatible mappings of type Bsatisfying an implicit relation,” Demonstratio Mathematica, vol. 36, no. 3, pp. 763–770, 2003.
View at: Google Scholar
A. Aliouche and A. Djoudi, “Common fixed point theorems for expansive mappings satisfying an implicit relation,” Matematicki Vesnik, vol. 67, no. 2, pp. 115–122, 2015.
View at: Google Scholar

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Copyright © 2023 Hadeel Z. Alzumi 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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