<span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2724395pt" id="M1" height="8.14084pt" version="1.1" viewBox="-0.0657574 -7.8684 8.77813 8.14084" width="8.77813pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-111" d="M495 86L479 114C446 82 419 66 409 66C401 66 401 72 406 97C420 166 436 231 453 297C489 435 454 448 428 448C406 448 384 439 354 422C305 394 222 327 161 247H159L183 345C200 415 194 448 173 448C143 448 82 410 23 351L38 325C64 349 95 371 105 371C111 371 116 365 109 336L25 -4L31 -12C50 -4 77 3 107 9C119 69 132 122 145 168C197 254 321 381 370 381C387 381 393 374 378 305L329 95C309 17 320 -12 345 -12C372 -12 430 19 495 86Z"/></g></svg>-</span>Tupled Common Fixed Point Result in Fuzzy <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2723999pt" id="M2" height="12.533pt" version="1.1" viewBox="-0.0657574 -12.2606 8.20973 12.533" width="8.20973pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-99" d="M452 333C452 398 425 448 375 448C329 448 235 404 140 292H138L229 683C233 701 234 712 225 712C208 712 149 686 66 677L64 652H97C135 652 140 651 129 598L38 167C23 96 29 57 44 33C60 7 98 -12 132 -12C169 -12 232 3 290 41C383 102 452 223 452 333ZM365 316C365 199 308 87 239 49C221 39 200 35 183 35C137 35 99 70 112 163C116 191 123 215 129 233C190 316 290 392 330 392C348 392 365 372 365 316Z"/></g></svg>-</span>Metric Spaces

Rome, Badshah-E.; Sarwar, Muhammad; Jarad, Fahd

doi:https://doi.org/10.1155/2022/4097444

Journal of Function Spaces

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Fuzzy Sets and Their Applications in Mathematics

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Research Article | Open Access

Volume 2022 | Article ID 4097444 | https://doi.org/10.1155/2022/4097444

-Tupled Common Fixed Point Result in Fuzzy -Metric Spaces

Badshah-E. Rome,¹Muhammad Sarwar,¹and Fahd Jarad^2,3

Academic Editor: Muhammad Gulzar

Received01 Dec 2021

Accepted26 Jan 2022

Published28 Mar 2022

1. Introduction

One of the several important settings in which fixed point theory has been explored is the fuzzy context [1, 2]. The introduction of the fuzzy set by Zadeh [3] was a turning point in the landscape of fuzzy mathematics. Fuzzy mathematics has improved enormously in the last two decades. Fuzzy set theory has many important applications in various fields of applied sciences such as neural network theory, stability theory, mathematical programming, modeling theory, engineering sciences, medical sciences (medical genetics and nervous system), image processing, control theory, and communication [4–9]. Kramosil [10] introduced the notion of fuzzy metric space. George and Veeramani [11] later on slightly modified Kramosil’s definition of fuzzy metric space and also proved that every metric induces a fuzzy metric and every fuzzy metric induces Hausdorff topology. Sedghi and Shobe [12] introduced the concept of fuzzy -metric space. Following this idea, many researchers analysed fixed point theory in fuzzy -metric space via various contractive conditions [13–17].

Sessa [18] explored common fixed points of set-valued as well as single-valued mappings on a complete metric space under a contractive condition along with the commutativity concept. The concept of a common fixed point was later on generalized by many researchers [19–23]. Lakshmikantham and Bhaskar [24] initiated the concept of coupled fixed point. Ćirić and Lakshmikantham [25] established coupled coincidence and coupled common fixed point theorems in partially ordered metric spaces. Subsequently, many researchers explored coupled fixed point theory in various spaces [2, 12, 19, 25–28]. Hu [27] and Zhu and Xiao [2] gave a coupled fixed point theorem for contractions in fuzzy metric spaces. Vasuki [29] obtained common fixed point results in fuzzy metric spaces. Berinde and Borcut [30] presented the idea of a tripled fixed point and obtained some new tripled fixed point results using mixed -monotone mapping. Their results generalize and extend the Bhaskar and Lakshmikantham’s research for nonlinear mappings. Roldán et al. [31] investigated the multidimensional coincidence points between mappings. Roldán et al. [1] modified the concept of tripled fixed points and generalized the results of Berinde and Borcut [30] and Zhu and Xiao [2].

In this paper, we aim to generalize and extend the notion of coupled/tripled common fixed point by introducing the concept of -tupled fixed point in fuzzy -metric space. We established an existence and uniqueness theorem for contractive mapping in fuzzy -metric space. Our main result generalizes and extends coupled and tripled fixed point theorems appearing in [1, 2, 13] to -dimensional common fixed points in fuzzy -metric space. Moreover, it can be particularized to complete metric spaces to obtain an -tupled Brinde–Borcut type coincidence/fixed point result in a nonfuzzy domain.

The paper is organized as follows: Section 2 is devoted to recall the basic definitions and lemmas that will be crucial throughout the paper. In Section 3, we introduce the notions of an -tupled common fixed point and an -tupled coincidence point. Moreover, an existence and uniqueness theorem for mappings satisfying certain contractive conditions in fuzzy -metric space has been proved. Section 4 is devoted to generalize the construction presented in Section 3 to nonfuzzy settings. Moreover, the idea is elaborated via a nontrivial example. In section 5, some applications of the main result of the paper are discussed, and a kind of Lipschitzian system and an integral system both for variables have been solved.

2. Preliminaries

In this section, some terms and definitions are provided which will be used in the main work of this manuscript. Henceforth, and will denote the set of real numbers and positive integers, respectively, while will stand for an arbitrary nonempty set. Arguments of a metric and fuzzy metric will be represented by subscripts. For example, and will be represented by and , respectively.

Definition 1 (see [32]). A map , such that is an ordered abelian topological monoid with unit 1, is called a continuous -norm.
, , and are examples of some frequently used continuous -norm that satisfy .

Definition 2 (see [33]). A continuous -norm is said to be of -type if the sequence is equicontinuous at . That is, for all there exists such that implies that for all , where the sequence is defined as and .
An important and most commonly used continuous -norm of -type is which satisfies . The following lemma characterizes continuous -norm to be of the -type.

Lemma 1 (see [1]). Let be a -norm and . IfThen, is a -norm of the-type.

Definition 3 (see [34]). The 3-tuple is called fuzzy metric space if is a nonempty set, is a continuous -norm, and is a fuzzy set on which satisfies the following conditions, for all and .(i) (ii) (iii) (iv) (v) is continuous(vi)

Definition 4 (see [12]). The 3-tuple is called fuzzy -metric space (F MS for short) if is a nonempty set, is a continuous -norm, and is a fuzzy set on which satisfies the following conditions, for all and and a given real number .(i) (ii) (iii) (iv) (v) is continuous(vi)

Note that represents the degree of closeness between and with respect to . The fuzzy -metric reduces to a fuzzy metric for . Therefore, the class of fuzzy -metric spaces is larger than the class of fuzzy metric spaces. The following example shows that a fuzzy -metric on a nonempty set need not be a fuzzy metric.

Example 1 (see [13]). Let and and with . It can be easily verified that is a F MS with . But for , is not a fuzzy metric space.

Remark 1 ( see [14]). For , it is always true that .

Lemma 2 (see [35]). is nondecreasing for all .

Definition 5 (see [36, 37]. In a fuzzy -metric space :(1)A sequence converges to if for every and there exists such that (2) is said to be Cauchy sequence if for every positive real number and there exists such that (3)A F MS is said to be complete if every Cauchy sequence converges in it

Remark 2 (see [14]). In general, a fuzzy b-metric is not continuous.
In a fuzzy -metric space, we have the following proposition.

Proposition 1 (see [37]). Let be a F MS and suppose a sequence converges to , then

3. Main Results

Definition 6. Let and , a point is said to be the following:(1)-tupled fixed point of if (2)-tupled coincidence point of and if (3)-tupled common fixed point of and if

Definition 7. Let and be two mappings. Then, and are said to be commuting if .
In our proof of main result, we will use the following lemmas.

Lemma 3. Let and be mappings on a F MS , such that is constant and is continuous and commuting with . Then, is a unique -tupled common fixed point of and .

Proof. As is constant on , therefore, there exists such that for all . Then, from and being commuting, it can be deduced thatTherefore, . That is, is an n-tupled common fixed point of and . Let be another -tupled common fixed point of and such that for . Then,Which is a contradiction. Therefore, is a unique -tupled common fixed point of and .
Let denote the class of all increasing and continuous functions such that for all with and . Then, the following lemma is used.

Lemma 4. Let be a complete F MS with , where for all . Let and be such that For some , for all and . Suppose is an -tupled coincidence point of and . Then,

Proof. Suppose on the contrary that there are at least two distinct integers and in such that . Let . Then,which is a contradiction. Hence,

Theorem 1. Let be a complete F MS, where for all . Let and be such that and is continuous and commuting with . Suppose for all and ,where , then and have a unique -tupled common fixed point.

Proof. If is constant, then the proof of the theorem follows from Lemma 2. Suppose is not constant on . In this case, the proof is divided into three steps.

Step 1. Definition of the sequences , , .
Let be arbitrary points of . As , therefore, there exist such that and .
Again, as , therefore, there exists such thatContinuing in the same way, sequences , , can be constructed such that

Step 2. , , are Cauchy sequences.Using (9), for all , we have(13) implies that for all and ,Obviously,It means is an increasing sequence in , and therefore, for all . If , then by letting in , we get a contradiction . Therefore, , that is, for all ,To show that the sequences , , , where , are Cauchy, first we prove that for every there exist such that . Suppose it is not true. Then, there exists some such that for each , there exist integers and with such thatLet be the least such positive integer which exceeds and satisfies (17). Then,Let . Using (18) and (19) and properties and , we haveLetting we get the contradictionHence, , , are Cauchy sequences.

Step 3. and have an -tupled common fixed point.
As is complete, so there will be some such that . Due to continuity of , . Also, commutes with , therefore, .
Using (9), we haveUsing Proposition 1 and (16), we haveHence, . Similarly, it can be shown thatLemma 4 implies thatFrom (9) and (24) along with Proposition 1 and continuity of , it comes out thatapplying Proposition 1 again, we getTherefore, . Similarly, it can be shown thatFrom (24), it follows thatTo show uniqueness, let be another -tupled common fixed point of and such that . Then,which is a contradiction. Thus, and have a unique -tupled common fixed point. □
Taking in the abovementioned theorem, we get the following corollary, which is the main result of [13].

Corollary 1. Let be a complete F MS, where for all . Let and be such that and is continuous and commuting with . Suppose for all and ,where , then there exists a unique such that .

Corollary 2. Let be a complete F MS, where for all . Let and be such that and is continuous and commuting with . Suppose for all and ,where , and , then and have a unique -tupled common fixed point.

Proof. Proof follows from Theorem 1, by setting .

Corollary 3. Let be a complete F MS, where for all . Let and be such that and is continuous and commuting with . Suppose for all and ,where , then and have a unique -tupled common fixed point.

Proof. It is adequate to set in Theorem 1,

Remark 3. Coincidence point of and is not necessarily unique. For example, if , where is constant, then every point is coincidence point of and .

Example 2. Let and and with where . It can be easily verified that is a complete F MS with . Let be such that . Define as , and for all and . Obviously is continuous, and are commuting and . Also,That is, all the conditions of Theorem 1 are fulfilled. Therefore, is a unique quadrupled common fixed point of and .

4. Consequences

It is known that if is a F MS and is a continuous -norm such that , then is F MS. Since for any -norm , it is always true that , therefore, is F MS implies is F MS. As F MS is a generalization of -metric space, therefore, from a given -metric space, a F MS can be considered in different ways.

Example 3 (see [1]). Let be a -metric space. For and , the following is defined: It is known that each of , , and is a F MS.
Moreover, the completeness of implies the completeness of any one of these F MSs and vice versa. With this approach, many important results for -metric space can be deduced from the corresponding results in a fuzzy setting. In the following theorem the -metric space is viewed as the crisp F MS .

Theorem 2. Let be a complete -metric space, and be given mappings such that and is continuous and commuting with . Suppose and satisfy some of the following conditions for all :

Then, there exists a unique such that .

Proof. Consider as defined in example 3. As is complete, is complete. We prove (2), for , and . In case any one of , is 0, then (2) obvious. SupposeNow (a)Hence, (2) is satisfied.
(b) AsHence, (2) is true.
(c) Let . Then,Hence, (2) is true.

Corollary 4 (see [1], Theorem 15). Let be a complete metric space and and be such that and is continuous and commuting with . Suppose and satisfy some of the following conditions for all :Then, there exists a unique such that .

Proof. Proof is similar to that of Theorem 2.

Example 4. Let , and for all . Let and be defined as and for all . Then,That is, the condition of Theorem 2 (c) is fulfilled.
Moreover, for -tupple , where , we haveHence, is a unique -tupled common fixed point of and .

5. Applications

5.1. Lipschitzian Systems

Let be Lipschitzian mappings on (equipped with the Euclidean metric) with Lipschitz constants , respectively, that is, for each where , there exists a corresponding real number where such that for all . Define as for all , where for . Then,

That is, is itself a Lipschitzian mapping with . If , then is contraction and by Banach contraction principle there is a unique such that . Define by

Obviously, for all . Moreover,

If , then by Theorem 2 (a), there exists a unique such that

Corollary 5. Let be Lipschitzian mappings furnished with the Euclidean metric, and be such that . Then, the systemhas a unique solution , where is a unique real solution of .

Example 5. Consider the following:Let . and are Lipschitzian mappings with and . Let and . Then,System has a unique solution , where is a unique solution of

5.2. Integral Systems

Let , where with and with where is Lebesgue integral. Then, is a complete metric space. Let and let be such that and satisfying

If , we will look for mappings such thatis satisfied for all (arguments of are represented by subscripts).

Let for all and all , be defined as follows:

Then,

If, then by 4.1 (c), (50) has a unique solution of the form .

6. Conclusion

We generalized the concept of tripled fixed point by introducing -tupled fixed points and established an -tupled unique fixed point result in fuzzy -metric space. This generalization may be helpful for further investigation and applications.

We conclude this paper by indicating, in the form of open questions, some directions for further investigation and work.(1)Can the condition of continuity of in Theorem 1 be relaxed?(2)If the answer to 1 is yes, then what hypotheses on and are needed to guarantee the existence of the -tupled common fixed points and ?(3)Can the concept of -tupled coincidence point be extended to more than two mappings?

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally and significantly in writing this article.

References

A. Roldán, J. Martinez-Moreno, and C. Roldán, “Tripled fixed point theorem in fuzzy metric spaces and applications,” Fixed Point Theory and Applications, vol. 29, pp. 1687–1812, 2013.
View at: Publisher Site | Google Scholar
X.-H. Zhu and J.-Z. Xiao, “Note on “Coupled fixed point theorems for contractions in fuzzy metric spaces“,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 16, pp. 5475–5479, 2011.
View at: Publisher Site | Google Scholar
L. A. Zadeh, “Fuzzy sets,” Information and Control, vol. 8, no. 3, pp. 338–353, 1965.
View at: Publisher Site | Google Scholar
M. Alaei and F. Yazdanpanah, “A fuzzy-based routing scheme for network-on-chip with honeycomb topology,” Comput. Methods. Differential Equations, vol. 7, pp. 511–520, 2019.
View at: Google Scholar
S. Biswas and T. K. Roy, “A semianalytical method for fuzzy integro-differential equations under generalized Seikkala derivative,” Soft Computing, vol. 23, no. 17, pp. 7959–7975, 2019.
View at: Publisher Site | Google Scholar
S. Biswas and T. K. Roy, “Generalization of Seikkala derivative and differential transform method for fuzzy Volterra integro-differential equations,” Journal of Intelligent and Fuzzy Systems, vol. 34, no. 4, pp. 2795–2806, 2018.
View at: Publisher Site | Google Scholar
S. Biswas and T. Roy, “Adomian decomposition method for fuzzy differential equations with linear differential operator,” Journal of Information and Computational Science, vol. 11, no. 4, pp. 243–250, 2016.
View at: Google Scholar
M. R. Pidatella, G. Gallo, and M. Zeinali, “Pattern classification through fuzzy likelihood,” Le Matematiche, vol. 70, pp. 135–146, 2015.
View at: Google Scholar
M. Zeinali, S. Shahmorad, and K. Mirnia, “Hermite and piecewise cubic Hermite interpolation of fuzzy data,” Journal of Intelligent and Fuzzy Systems, vol. 26, no. 6, pp. 2889–2898, 2014.
View at: Publisher Site | Google Scholar
I. Kramosil and J. Michalek, “Fuzzy metric and statistical metric spaces,” Kybernetika, vol. 11, pp. 336–344, 1975.
View at: Google Scholar
R. Vasuki and P. Veeramani, “Fixed point theorems and Cauchy sequences in fuzzy metric spaces,” Fuzzy Sets and Systems, vol. 135, no. 3, pp. 415–417, 2003.
View at: Publisher Site | Google Scholar
S. Sedghi and N. Shobe, “Common fixed point theorem in -fuzzy metric space,” Journal of Nonlinear Functional Analysis, vol. 3, pp. 349–359, 2012.
View at: Google Scholar
T. Došenović, A. Javaheri, S. Sedghi, and N. Shobe, “Coupled fixed point theorem-metric spaces,” Journal of Mathematics, vol. 47, no. 1, pp. 77–88, 2017.
View at: Google Scholar
D. Raktić, A. Mukheimer, T. Došenović, D. Zoran, and S. Radenović, “Some new fixed point results in fuzzy -metric spaces,” Journal of Inequalities and Applications., vol. 70, 2020.
View at: Google Scholar
S. Nadan, “Fuzzy, -metric spaces,” International Journal of Computers, Communications & Control, vol. 2, pp. 273–281, 2016.
View at: Google Scholar
M. Sarwar and T. Abdeljawad, “-extended fuzzy-metric space and related fixed point result,” AIMS Mathematics, vol. 5, no. 5, pp. 5184–5192, 2020.
View at: Google Scholar
M. Sarwar and R. Rodríguez-López, “Fixed point results via-admissibility in extended fuzzy rectangular -metric spaces with applications to integral equations,” Mathematics, vol. 16, p. 9, 2021.
View at: Google Scholar
S. Sessa and B. Fisher, “On common fixed points of weakly commuting mappings and set-valued mappings,” International Journal of Mathematics and Mathematical Sciences, vol. 9, no. 2, pp. 323–329, 1986.
View at: Publisher Site | Google Scholar
W. Sintunavarat and P. Kumam, “Common fixed point theorems for a pair of weakly compatible mappings in fuzzy metric spaces,” Journal of Applied Mathematics, vol. 2011, Article ID 637958, p. 14, 2011.
View at: Publisher Site | Google Scholar
M. Aamri and D. El Moutawakil, “Some new common fixed point theorems under strict contractive conditions,” Journal of Mathematical Analysis and Applications, vol. 270, no. 1, pp. 181–188, 2002.
View at: Publisher Site | Google Scholar
G. Jungck, “Compatible mappings and common fixed points,” International Journal of Mathematics and Mathematical Sciences, vol. 9, no. 4, pp. 771–779, 1986.
View at: Publisher Site | Google Scholar
G. Jungck and B. E. Rhoades, “Fixed points for set valued functions without continuity,” Indian Journal of Pure and Applied Mathematics, vol. 29, no. 3, pp. 227–238, 1998.
View at: Google Scholar
K. P. R. Sastry and I. S. R. Krishna Murthy, “Common fixed points of two partially commuting tangential selfmaps on a metric space,” Journal of Mathematical Analysis and Applications, vol. 250, no. 2, pp. 731–734, 2000.
View at: Publisher Site | Google Scholar
T. G. Bhaskar and V. Lakshmikantham, “Fixed point theorems in partially ordered metric spaces and applications,” Nonlinear Analysis: Theory, Methods & Applications, vol. 65, no. 7, pp. 1379–1393, 2006.
View at: Publisher Site | Google Scholar
V. Lakshmikantham and L. Ćirić, “Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 12, pp. 4341–4349, 2009.
View at: Publisher Site | Google Scholar
S. Shakeri, L. ĆCirić, and R. Saadati, “Common fixed point theorem in partially ordered l-fuzzy metric spaces,” Fixed Point Theory Application, vol. 2010, Article ID 125082, 2010.
View at: Google Scholar
X. Hu, “Common coupled fixed point theorems for contractive mappings in fuzzy metric spaces,” Fixed Point Theory and Applications, vol. 2011, Article ID 363716, 2011.
View at: Publisher Site | Google Scholar
J.-X. Fang, “Common fixed point theorems of compatible and weakly compatible maps in Menger spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 5-6, pp. 1833–1843, 2009.
View at: Publisher Site | Google Scholar
R. Vasuki, “Common fixed points for R-weakly commuting maps in fuzzy metric space,” Indian Journal of Pure and Applied Mathematics, vol. 30, pp. 419–423, 1999.
View at: Google Scholar
V. Berinde and M. Borcut, “Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 15, pp. 4889–4897, 2011.
View at: Publisher Site | Google Scholar
A. Roldán, J. Martínez-Moreno, and C. Roldán, “Multidimensional fixed point theorems in partially ordered complete metric spaces,” Journal of Mathematical Analysis and Applications, vol. 396, no. 2, pp. 536–545, 2012.
View at: Publisher Site | Google Scholar
B. Schweizer and A. Sklar, “Statistical metric spaces,” Pacific Journal of Mathematics, vol. 10, no. 1, pp. 313–334, 1960.
View at: Publisher Site | Google Scholar
O. Hadžíc and E. Pap, Fixed Point Theory in Probabilistic Metric Spaces, Kluwer Academic, Dordrecht, Netherlands, 2001.
A. George and P. Veeramani, “On some results in fuzzy metric spaces,” Fuzzy Sets and Systems, vol. 64, no. 3, pp. 395–399, 1994.
View at: Publisher Site | Google Scholar
V. Gregori and A. Sapena, “On fixed-point theorems in fuzzy metric spaces,” Fuzzy Sets and Systems, vol. 125, no. 2, pp. 245–252, 2002.
View at: Publisher Site | Google Scholar
S. Sedghi, N. Shobe, and M. A. Selahshoor, “A common fixed point theorem for Four mappings in two complete fuzzy metric spaces,” Advances in Fuzzy Mathematics, vol. 1, no. 1, 2006.
View at: Google Scholar
S. Sedghi and N. Shobe, “Common fixed point theorem for -weakly commuting maps in -fuzzy metric space,” Nonlinear Functional Analysis and Applications, vol. 19, no. 2, pp. 285–295, 2014.
View at: Google Scholar

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