Common Fixed-Point Theorems in the Partial <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="M1" 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 and an Application to the System of Boundary Value Problems

Nazam, Muhammad; Hamid, Zahida; Al Sulami, Hamed; Hussain, Aftab

doi:https://doi.org/10.1155/2021/7777754

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Volume 2021 | Article ID 7777754 | https://doi.org/10.1155/2021/7777754

Common Fixed-Point Theorems in the Partial -Metric Spaces and an Application to the System of Boundary Value Problems

Muhammad Nazam,¹Zahida Hamid,¹Hamed Al Sulami,²and Aftab Hussain²

Academic Editor: Rich Avery

Received14 Jun 2021

Accepted17 Aug 2021

Published23 Sept 2021

Abstract

In this paper, we investigate the conditions for the existence of the common fixed points of generalized contractions in the partial -metric spaces endowed with an arbitrary binary relation. We establish some unique common fixed-point theorems. The obtained results may generalize and improve earlier fixed-point results. We provide examples to illustrate our findings. As an application, we discuss the common solution to the system of boundary value problems.

1. Introduction, Preliminaries, and Motivations

The -metric space was introduced by Czerwik [1]. It is obtained by modifying the triangle property of the metric space. Every metric is a -metric, but the converse is not true. Almost all the fixed-point theorems in the metric spaces have been proved true in the -metric spaces; for example, see [2–10] and references therein.

Matthews [11] introduced the notion of the partial metric space as a part of the study of denotational semantics of the dataflow network. In this space, the usual metric is replaced by a partial metric having a property that the self-distance of any point of the space may not be zero. Every metric is a partial metric, but the converse is not true. Matthews [11] also initiated the fixed-point theory in the partial metric space. He proved the Banach contraction principle in this space to be applied in program verification. We can find so many fixed-point theorems in the metric spaces which have been proved in the partial metric spaces by many fixed-point theorists ([12, 26] and references therein).

Shukla [13] introduced the concept of partial -metric by modifying the triangle property of the partial metric and investigated fixed points of Banach contraction and Kannan contraction in the partial -metric spaces. Mustafa et al. [14] modified the triangle property of partial -metric and established a convergence criterion and some working rules in partial -metric spaces. Moreover, Mustafa et al. [14] investigated common fixed-point results for -weakly contractive mappings. Dolicanin-Ðekic [15] obtained the fixed-point theorems for Ciric-type contractions in the partial -metric spaces. Singh et al. [16] investigated some conditions to show the existence of the common fixed points of power graphic -contractions defined on the partial -metric space endowed with directed graphs. More results on -contractions can be seen in [8, 17, 18].

Let be a nonempty set, then the nonempty binary relation is a subset of The set itself is known as universal relation, and the empty set is known as an empty relation; both are trivial relations. If any two elements are related with respect to , then we shall write . We shall use the notation if either or . is reflexive if for all is symmetric if implies for all is antisymmetric if and implies for all is transitive if and implies for all . The inverse, transpose, or dual of binary relation is denoted by and defined as follows: . Let , then it is easy to prove that if and only if .

Definition 1 (see [19]). Let be a self-mapping on a nonempty set . A binary relation on is said to be -closed if for all ,

Definition 2 (see [19]). Let be a binary relation on . A path in from is a sequence such that (1)(2) for all The set of all paths from to in is denoted by . The path of length involves element of

Definition 3 (see [19]). A metric space equipped with the binary relation is called -regular (or -self-closed) if for each sequence , whenever and , we have , for all .

Alam and Imdad [19] used nonempty binary relation on the nonempty set to obtain the following relation-theoretic contraction principle.

Theorem 4 (see [19]). Let be a complete metric space and be a binary relation on . Let be a self-mapping defined on satisfying the following conditions: (a)There exists such that and is -closed(b)Either is continuous or is (c)There exists such that for with

Then admits a fixed point in Moreover, if is a nonempty set for all , then the fixed point is unique.

al-Sulami et al. [20] generalized Theorem 4 by replacing Banach contraction with -contraction as follows.

Theorem 5 (see [20]). Let be a complete metric space and be a binary relation on Let be a self-mapping defined on satisfying the following conditions: (a)There exists such that and is -closed(b)Either is continuous or is (c)There exists such that for with

Then admits a fixed point in Moreover, if is a nonempty set for all , then the fixed point is unique.

Definition 6 (see [21]). Let and be two self-mappings on a nonempty set A binary relation on is said to be -closed if for all ,

Zada and Sarwar [21] generalized Theorem 4 by replacing Banach contraction with -contraction as follows.

Theorem 7 (see [21]). Let be a complete metric space and be a binary relation on . If the self-mappings and defined on satisfy the following conditions: (a)There exists such that and is -closed(b)Either are continuous or is (c)There exists , such that for all with ,

Then and have a unique common fixed point in . Moreover, if is nonempty for all , then the common fixed point is unique.

Liu et al. [22] introduced the -contractions where the mapping maps positive real numbers to positive real numbers and satisfies the conditions

() is nondecreasing

() , for each positive sequence

() is continuous

is a comparison function; that is, it satisfies the following conditions: (i) is monotone increasing, that is,(ii) for all , where stands for the iterate of

Let If is defined by then belongs to

Note that if is a comparison function, then for every . The mappings , are comparison functions

Definition 8 ([22], -contraction). Let be a self-mapping defined on the metric space . Let The mapping is called -contraction if it satisfies the following condition:

Definition 9 ([22], generalized -contraction). Let be a self-mapping defined on the metric space . If the mapping satisfies the condition where is defined by Then it is called generalized -contraction.

Liu et al. established the following theorem for -contractions.

Theorem 10 (see [22]). Every generalized -contraction has a unique fixed point in a complete metric space .

In this paper, in Section 3, we investigate common fixed-point results for generalized contractions in the partial -metric spaces endowed with binary relation . The obtained results generalize Theorems 4, 5, 7, 10. We support the results with a nontrivial example and counter the remarks given in [23].

2. Basic Notions in the Partial -Metric Spaces

Let be a nonempty set, and the mapping satisfies the following axioms: (1)(2)(3)(4)(5)There exists a real number such that

According to Matthews [11], if the mapping satisfies axioms (1-4), we say that it is a partial metric on the set and is called partial metric space. According to Shukla [13], if satisfies axioms (1, 2, 3, and 5), then it is a partial-metric on the set and is called partial-metric space. For convenience, we denote partial -metric by

Every partial -metric induces a -metric defined by

It is called induced-metric on .

Let , then is a collection of -balls which forms a base for partial-metric topology.

The following relation can be observed.

Remark 11. (1)In , , but the converse is not true (in this case, reduces to a -metric space) Figure 1.

Example 1 (see [13]). Let , be a constant and be defined by Then is a partial -metric space with coefficient , but it is neither a -metric space nor a partial metric space.

Example 2 (see [13]). Let and be the partial metric and -metric on , respectively. Then the mapping defined by for all defines a partial -metric on .

Example 3 (see [13]). Let be a partial metric. Then the mapping defined by for all is a partial -metric on provided .

Definition 12 (see [13]). A sequence in the partial -metric space is called a convergent sequence if there exists such that

The uniqueness of the limit of a convergent sequence may not be guaranteed in the partial -metric spaces (see [23]).

Definition 13 (see [13]). A sequence in a partial -metric space is called the Cauchy sequence if The partial -metric space is said to be complete if every Cauchy sequence in converges to a point .

Lemma 14 (see [14]). (1)Every Cauchy sequence in the -metric space is also Cauchy in the partial -metric space and vice versa(2)The partial -metric space is complete if and only if -metric space (induced -metric space) is complete(3)For any sequence in , if and only if

3. Common Fixed-Point Theorems in the Partial -Metric Spaces

This section is the main part of this paper. It contains some new common fixed-point theorems in the partial -metric spaces. The existence theorems given in [12, 15, 19–22, 24, 27] can be seen as a special case of the results proved in this section.

The results in this paper are based on the following contractive condition.

Definition 15. Let and be two self-mappings on the partial -metric space and be a binary relation on . Let

The mappings and form a -contraction if there exists a continuous comparison function such that

In [23], it was remarked that some contraction conditions on partial -metric spaces imply contraction conditions on -metric spaces (see Theorem 2.6 in [23]). In the following example, we show that the contraction condition (14) is independent of these remarks.

Example 4. Let and . Let be defined by

Then is a partial -metric space with coefficient The associated -metric is given by

Define by and Consider

This implies,

a contradiction to the definition of mapping On the other hand, for partial -metric, we have

Note that we have taken . Similarly, it can be shown that the above conclusion holds for all other cases.

Since, in general, -metric is discontinuous mapping (see [5]), so by Example 2, the partial -metric is not continuous in general. The following lemma is necessary for the upcoming results.

Lemma 16 (see [14]). Let be a partial -metric space. If there exists a in and such that . Then

3.1. Main Results

We state our main results which describe the conditions for the existence of the common fixed points of -contraction in the partial -metric spaces.

Theorem 17. Let be a complete partial -metric space and be a transitive binary relation on . Let and form a -contraction. Then and have a common fixed point in if the following conditions are satisfied. (a)There exists such that (b)-closed(c) are continuous

Proof. By assumption (a), there exists such that . Taking as the initial point, we define the sequence in by ☐

Moreover, by assumptions (a) and (b), we have

In general, we have and

Case 1. If for some then

Indeed, on the contrary, if , then , and by contractive condition (14), we have

Since for every we obtain

Since the function is nondecreasing, so This contradicts the second condition of partial -metric spaces . Hence, implies Consequently, is a common fixed point of , and that is .

Case 2. If for all . We have for all Since , so . Setting and in (14), we get for all

Similarly, setting and in (14), we get

In general, for all , either even or odd, we have

Taking limit in the above inequality, we get

This implies , and by we have

This implies (by (8)) that

By axiom (2), we have Thus, for all we have

We claim that is a Cauchy sequence in , for this is sufficient to prove that is Cauchy sequence. On the contrary, if is not Cauchy, then for some subsequences and , there exist and a positive integer , such that for all , we have and ; thus,

As in the above inequality, we have

By using triangular inequality (axiom (5)), we get

Taking limit , we have

Also, we have the following information:

Taking limit , we have

By axiom (5), we have

Taking limit and using (35), we have

By using (31), we have the following information from (33), (35), (37), and (39):

Since , by (14), we have

This is a contradiction to the definition of function . Thus, is a Cauchy sequence in . By Lemma 14 (1), is a Cauchy sequence in . Since is a complete Partial -metric space, so by Lemma 14 (2), is also a complete metric space. Thus, there exists such that that is, 0. By Lemma 14 (3), we get

Since , so that . Thus, converges to in

Now, we claim that . By (40), we have

Since and are continuous, we have

By Lemma 16, we have

Thus,

This implies Similar arguments lead us to have Hence,

; that is, and have a common fixed point .

If , then we have the following theorem.

Theorem 18. Let be a complete partial -metric space and be a transitive binary relation on . Let and form a -contraction. Suppose that and statement of Theorem 17 holds, then the mappings and admit a unique common fixed point in .

Proof. We have proved the existence of a common fixed point in Theorem 17 On the contrary, suppose that and are two distinct common fixed points of and in . Then the class of paths of finite length in from to is . Let one of the paths be in from to with By transitivity of , we have It is given that and form a -contraction, that is, This implies . This is a contradiction to the definition of . Hence, . This shows that is a unique common fixed point of and ☐

Remark 19. If the mappings and are discontinuous, then we have the following theorem.

Theorem 20. Let be an complete partial -metric space. Let and form a -contraction. Suppose that is an antisymmetric relation then and admit a common fixed point in if they meet the conditions (a) and (b): (a)There exists such that (b)-closed

Proof. By Theorem 17, we know that and as . It is given that is so for all . There are two possible cases.☐

Case 1. If the sequence is constant. Let for each so that and Since is , so . We know that ; thus, . As is an antisymmetric relation, so , by the same arguments we have as required.

Case 2. If is not constant and arbitrary, we claim that . Let . It is proved in Theorem 17 that so there must be an integer , such that

It is assumed that is , and by Theorem 17, we know that . By contractive condition (2.1), monotonicity of , and Lemma 16, we have

This is a contradiction to the definition of mapping Thus, . Also, we have the following information:

Thus, By interchanging roles of and , we have

Hence, ; that is, is a common fixed point of and in

The following is the most general theorem of this section.

Theorem 21. Let be an complete partial -metric space and be a transitive and antisymmetric binary relation on . Let and form a -contraction. Suppose that , and assumptions (a) and (b) in Theorem 17 hold. Then the mappings and admit a unique common fixed point in

Proof. See the proofs of Theorems 17, 18, and 20, respectively.☐

Remark 22. (1)The results in this section are independent of the observation made in [23], and hence, our results are a real generalization of the related results in literature (see [12, 19–22])(2)Theorem 21 remains true if is replaced by The following example explains the main results.

Example 5. Let Define the partial -metric function by

Then (, ) is a complete partial -metric space. Define by for each , then . Let the function be defined by for all Then is continuous comparison. Define the binary relation on by

Define the mappings by

We observe that there exists such that by definition of , so assumption (a) is satisfied in Theorem 17. Let , then we have for each , so . Thus, is -closed (this verifies assumption (b) of Theorem 17. Also, are continuous (assumption (c) is satisfied). Now, we show that form -contraction. It is noted that the mappings , do not form Banach contraction in the partial -metric sense. Indeed,

We noticed that for each Thus, Consider

This implies

For and , the inequality (41) reduces to . Thus, (41) holds for this case. For and , the inequality (41) gets the form . Similarly, for each (41) holds true. Thus, we have

We note that .

3.2. Discussions

In this part of the current section, we state some corollaries which are themselves prominent fixed-point theorems in the literature.

The following corollary generalizes the results presented by Jleli and Samet [6] and al-Sulami et al. [20].

Corollary 23. Let be a complete partial -metric space and be a transitive and antisymmetric binary relation on . If the self-mappings and defined on satisfy the following conditions: (a) is nonempty for all (b)There exists such that and -closed(c)Either are continuous or is (d)There exists a function such that for all ,Then the mappings and admit a unique common fixed point.

Proof. Setting and in Theorem 17 and following the proofs of Theorems 17, 18, and 20 respectively, we obtain the required result.☐

The following corollary generalizes and improves the results presented by Zada and Sarwar [21] and Wardowski [25].

Corollary 24. Suppose that the self-mappings and defined on the complete partial -metric space satisfy the following conditions: (a) is nonempty for all (b)There exists such that and -closed(c)Either are continuous or is (d)There exists such that for all ,If is a transitive and antisymmetric binary relation on , then the mappings admit a unique common fixed point.

Proof. Setting and in Theorem 17 and following the proofs of Theorems 17, 18, and 20, respectively, we obtain the required result.☐

Corollary 25 (see [21]). Let be a complete partial -metric space and be a transitive and antisymmetric binary relation on . If the self-mappings and defined on satisfy the following conditions: (a) is nonempty for all (b)There exists such that and -closed(c)Either are continuous or is (d)There exists , such that for all ,Then and have a unique common fixed point in

Proof. This proof follows the proof of Corollary 24.
The following corollary improves the fixed-point results presented by Geraghty [24].☐

Corollary 26. Let be a complete partial -metric space and be a transitive and antisymmetric binary relation on . If the self-mappings and defined on satisfy the following conditions: (a) is nonempty for all (b)There exists such that and -closed(c)Either are continuous or is (d)For all and where such that , for each

Proof. By defining and in Theorem 17 and following the proofs of Theorems 17, 18, and 20, respectively, we obtain the required result.☐

Remark 27. (1)For Theorems 17, 18, and 20 establish criteria for the existence of common fixed points of -contractions in the partial metric spaces [12] and correspondingly for Corollaries 23, 24, 25, and 26(2)For the zero self-distance () and for the zero self-distance with , the results stated in Remark 27 (1) hold in the -metric spaces and metric spaces, respectively

4. Application to the System of Boundary Value Problems

We will apply Theorem 17 to achieve the existence of a common solution to the following system of boundary value problems: where represents the set of continuous functions defined on The functions are continuous and nondecreasing according to ordinates. We define the binary relation on as follows:

The associated Green function to (66) and (67) can be defined as follows:

Let the mapping be defined by

It is claimed that is a complete -metric space. By integration, we see that (66) and (67) can be written as and , where are defined by

It is remarked that the common solution to (66) and (67) is the common fixed point of the operators Suppose the following conditions: (a) such that for , we have(b) such that

The following theorem states the conditions under which equations (66) and (67) have a common solution.

Theorem 28. Let the functions satisfy conditions (a) and (b) Then equations (66) and (67) have a common solution.

Proof. We will apply Theorem 17 to show the existence of the common solution to (66) and (67). By condition (b), there exists such that . Since the functions are continuous, so defined above are continuous. Since it is given that are nondecreasing, thus, is closed. To show that the mappings form DC-contraction, we proceed as follows: ☐

Since , for all , thus, taking supremum on both sides of the above inequality, we have

Define the -metric on by

Inequality (75) can be written as

Defining the functions , F, and by and , respectively, for all we have

Hence, applying Theorem 17, we say that the boundary value problems (66) and (67) have a common solution in

Data Availability

No data were used to support this study.

Conflicts of Interest

All authors declare that they have no competing interests.

Authors’ Contributions

All authors contributed equally to this work.

Acknowledgments

The authors thank their universities for recommending this research work.

References

S. Czerwik, “Contraction mapping in -metric space,” Acta Mathematica Et Informatica Universitatis Ostraviensis, vol. 1, pp. 5–11, 1993.
View at: Google Scholar
A. Aghajani, M. Abbas, and J. R. Roshan, “Common fixed point of generalized weak contractive mapping in partially ordered -metric spaces,” Mathematica Slovaka, vol. 64, no. 4, pp. 941–960, 2013.
View at: Google Scholar
V. Berinde, Generalized Contractions and Applications, Editura Cub Press, Baia Mare, Romania, 1997.
M. Boriceanu, M. Bota, and A. Petrusel, “Multivalued fractals in b-metric spaces,” Central European Journal of Mathematics, vol. 8, no. 2, pp. 367–377, 2010.
View at: Publisher Site | Google Scholar
M. Bota, A. Molnar, and V. Csaba, “On Ekeland’s variational principle in -metric spaces,” Fixed Point Theory, vol. 12, pp. 21–28, 2011.
View at: Google Scholar
M. Jleli and B. Samet, “A new generalization of the Banach contraction principle,” Journal of Inequalities and Applications, vol. 2014, no. 1, 2014.
View at: Publisher Site | Google Scholar
M. Jovanović, Z. Kadelburg, and S. Radenović, “Common fixed point results in metric-type spaces,” Fixed Point Theory and Applications, vol. 2010, no. 1, Article ID 978121, 2010.
View at: Publisher Site | Google Scholar
E. Karapinar, A. Fulga, and R. P. Agarwal, “A survey: F-contractions with related fixed point results,” Journal of Fixed Point Theory and Applications, vol. 22, no. 3, 2020.
View at: Publisher Site | Google Scholar
M. A. Khamsi and N. Hussain, “KKM mappings in metric type spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 73, no. 9, pp. 3123–3129, 2010.
View at: Publisher Site | Google Scholar
M. Nazam, N. Hussain, A. Hussain, and M. Arshad, “Fixed point theorems for weakly admissible pair of F-contractions with application,” Nonlinear Analysis: Modelling and Control, vol. 24, no. 6, pp. 898–918, 2019.
View at: Publisher Site | Google Scholar
S. G. Matthews, “Partial metric topology,” Annals of the New York Academy of Sciences, vol. 728, pp. 183–197, 1994.
View at: Publisher Site | Google Scholar
M. Nazam, “On J_c‐contraction and related fixed‐point problem with applications,” Mathematical Methods in the Applied Sciences, vol. 43, no. 17, pp. 10221–10236, 2020.
View at: Publisher Site | Google Scholar
S. Shukla, “Partial b-metric spaces and fixed point theorems,” Mediterranean Journal of Mathematics, vol. 11, no. 2, pp. 703–711, 2014.
View at: Publisher Site | Google Scholar
Z. Mustafa, J. R. Roshan, V. Parvaneh, and Z. Kadelburg, “Some common fixed point results in ordered partial b-metric spaces,” Journal of Inequalities and Applications, vol. 2013, no. 1, 2013.
View at: Publisher Site | Google Scholar
D. Dolicanin-Ðekic, “On some ciric type results in partial b-metric spaces,” Filomat, vol. 31, no. 11, pp. 3473–3481, 2017.
View at: Publisher Site | Google Scholar
D. Singh, V. Chauhan, and I. Altun, “Common fixed point of a power graphic (F; fi)-contraction pair on partial b-metric spaces with application,” Nonlinear Analysis: Modelling and Control, vol. 22, no. 5, pp. 662–678, 2017.
View at: Publisher Site | Google Scholar
H. Aydi, E. Karapinar, Z. D. Mitrovic, and T. Rashid, “A remark on “existence and uniqueness for a neutral differential problem with unbounded delay via fixed point results F-metric spaces”,” Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, vol. 113, no. 4, pp. 3197–3206, 2019.
View at: Publisher Site | Google Scholar
H. Aydi, E. Karapinar, and H. Yazidi, “Modified F-contractions via α-admissible mappings and application to integral equations,” Filomat, vol. 31, no. 5, pp. 1141–1148, 2017.
View at: Publisher Site | Google Scholar
A. Alam and M. Imdad, “Relation-theoretic contraction principle,” Journal of Fixed Point Theory and Applications, vol. 17, no. 4, pp. 693–702, 2015.
View at: Publisher Site | Google Scholar
H. al-Sulami, J. Ahmad, N. Hussain, and A. Latif, “Relation theoretic (Θ,R) contraction results with applications to nonlinear matrix equations,” Symmetry, vol. 10, no. 12, p. 767, 2018.
View at: Publisher Site | Google Scholar
M. B. Zada and M. Sarwar, “Common fixed point theorems for rational -contractive pairs of mappings with applications,” Journal of Inequalities and Applications, vol. 2019, no. 1, 2019.
View at: Publisher Site | Google Scholar
X. D. Liu, S. S. Chang, Y. Xiao, and L. C. Zhao, “Some fixed point theorems concerning (ψ,ϕ )-type contraction in complete metric spaces,” Journal of Nonlinear Sciences and Applications, vol. 9, no. 6, pp. 4127–4136, 2016.
View at: Publisher Site | Google Scholar
N. V. Dung and V. T. Le Hang, “Remarks on partial b-metric spaces and fixed-point theorems,” Matematicki Vesnik, vol. 69, no. 4, pp. 231–240, 2017.
View at: Google Scholar
M. A. Geraghty, “On contractive mappings,” Proceedings of the American Mathematical Society, vol. 40, no. 2, pp. 604–608, 1973.
View at: Publisher Site | Google Scholar
D. Wardowski, “Fixed points of a new type of contractive mappings in complete metric spaces,” Fixed Point Theory and Applications, vol. 2012, no. 1, 2012.
View at: Publisher Site | Google Scholar
M. Nazam, M. Arshad, and M. Abbas, “Existence of common fixed points of improved F-contraction on partial metric spaces,” Applied General Topology, vol. 18, no. 2, pp. 277–287, 2017.
View at: Publisher Site | Google Scholar
A. C. M. Ran and M. C. B. Reurings, “A fixed point theorem in partially ordered sets and some applications to matrix equations,” Proceedings of the American Mathematical Society, vol. 132, no. 5, pp. 1435–1443, 2004.
View at: Publisher Site | Google Scholar

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Copyright © 2021 Muhammad Nazam 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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