<span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.06579971pt" id="M1" height="16.2466pt" version="1.1" viewBox="-0.0657574 -16.1808 22.3349 16.2466" width="22.3349pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-73" d="M828 650H570L564 622C646 614 650 609 634 524L604 366H253L281 524C296 608 308 615 382 622L388 650H132L126 622C211 615 214 609 198 524L121 122C106 42 102 35 23 28L17 0H276L282 28C196 35 191 40 206 122L243 324H597L559 123C544 42 537 35 452 28L446 0H710L716 28C632 35 628 43 643 122L719 524C735 610 741 615 822 622L828 650Z"/></g><g transform="matrix(.012,0,0,-0.012,14.418,-7.578)"><path id="g50-224" d="M563 583C563 664 501 710 433 710C381 710 334 692 285 648C212 582 178 506 149 350L74 -56C49 -190 36 -222 24 -226L30 -257C52 -256 95 -247 121 -229C131 -222 134 -206 165 -21L219 303C254 510 296 665 397 665C447 665 481 626 481 572C481 512 446 457 390 420C367 405 351 400 323 397L302 349C398 338 466 285 466 197C466 117 419 45 306 45C268 45 224 59 198 74L186 50C197 20 228 -14 274 -14C318 -14 359 3 401 26C478 68 554 144 554 225C554 322 493 367 412 396C490 446 563 500 563 583Z"/></g></svg>-</span>Hausdorff Functions and Common Fixed Points of Multivalued Operators in a <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 Space and Their Applications

Alshammari, Fahad Sameer; Alrashedi, Naif R.; George, Reny

doi:https://doi.org/10.1155/2022/1931861

Journal of Function Spaces

On this page

Abstract Introduction Preliminaries Applications Data Availability Conflicts of Interest References Copyright Related Articles

Special Issue

Fixed Point Theory and Applications for Function Spaces 2021

View this Special Issue

Research Article | Open Access

Volume 2022 | Article ID 1931861 | https://doi.org/10.1155/2022/1931861

-Hausdorff Functions and Common Fixed Points of Multivalued Operators in a -Metric Space and Their Applications

Fahad Sameer Alshammari,¹Naif R. Alrashedi,¹and Reny George^1,2

Academic Editor: Alexander Meskhi

Received14 Nov 2021

Revised22 Jan 2022

Accepted13 Jun 2022

Published09 Jul 2022

Abstract

-Hausdorff functions for are introduced, and common fixed-point theorems for a pair of multivalued operators satisfying generalized contraction conditions are proven in a -metric space. Our results are proper extensions and new variants of many contraction conditions existing in literature. In order to demonstrate applications of our result, we have proven an existence theorem for a unique common multivalued fractal of a pair of iterated multifunction systems and also an existence theorem for a common solution of a pair of Volterra-type integral equations.

1. Introduction

In the last few decades, a wide range of extensions, generalizations, and applications of the infamous Banach contraction principle came into existence. In the sequel, Bakhtin [1] initiated the idea of a -metric space followed by Czerwik [2], in which the author by weakening the triangular inequality formally defined a -metric space and proved the Banach contraction principle in a -metric space. Some examples and other details of a -metric space can be found in Kirk and Shahzad [3] whereas a wide range of generalized fixed-point theorems in a -metric space can be found in [4–7]. On the other hand, the study of a metric function on the set of closed and bounded subsets of a metric space was initiated by Pompeiu in [8] and then continued by Hausdorff [9]. Such a metric function is referred to as the Hausdorff-Pompeiu metric. Banach’s contraction principle was extended to a multivalued function in a metric space by Nadler [10] and in a -metric space by Czerwik [2] using the Hausdorff-Pompeiu metric . Further generalized results of multivalued contractions can be found in ([11–14]). Czerwik’s contraction was also generalized in many directions to name a few: -quasi-contraction [15], Hardy-Rogers contraction [16], weak quasi-contraction [17], Ciric contraction [18], etc. More results on multivalued contraction mappings in a -metric space can be found in [19–23]. Very recently, Debnath [24] proved the set-valued Meir–Keeler-type as well as Geraghty- and Edelstein-type fixed-point theorems in a -metric space whereas Altun et al. [25] and Kumar and Luambano [26] proved fixed-point results for multivalued -contraction mappings in complete metric space and partial metric space, respectively. In [27], the authors introduced the concept of -Hausdorff-Pompeiu -metric for some and proved fixed-point theorems for multivalued mappings belonging to various classes of multivalued -contractions in a -metric space. Applications of fixed-point results in dealing with solutions of nonlinear problems arising in engineering and science are an important area in present-day research. Fruitful applications of fixed-point problems in solution of various types of integral equations, fractional differential equations, and optimization problems can be found in [28–32]. Barnsley [33] introduced the idea of data interpolation using the fractal methodology of iterated function systems. Nowadays, fractal functions constitute a method of approximation of nondifferentiable mappings, providing suitable tools for the description of irregular signals (see [34–39]). The aim of this work is to prove common fixed-point theorems for a pair of multivalued mappings in a -metric space using -Hausdorff-Pompeiu -metric and thereby extend and introduce new variants of various fixed-point results for multivalued mappings existing in literature. We have provided two applications of our main results: one to prove the existence of a unique common multivalued fractal of a pair of iterated multifunction system defined on a -metric space and the second to prove the existence of a common solution of a pair of Volterra-type nonlinear integral equations.

2. Preliminaries

In this section, we provide some preliminary definitions, lemmas, and propositions required in our main results.

Definition 1 (see [1]). Let be a nonempty set and satisfy the following: (1) if and only if for all (2) for all (3)There exists a real number such that for all Then, is a -metric on and is a -metric space with coefficient .

Let be the collection of all nonempty closed and bounded subsets of a -metric space . For , define , , and . Czerwik [2] has shown that is a -metric in the set and is called the Hausdorff-Pompeiu -metric induced by . In [27], the authors introduced the function for some and showed that is a -metric for the set . They called this function the -Hausdorff-Pompeiu -metric induced by the -metric . Note that for or , is the Hausdorff-Pompeiu metric .

Proposition 2 (see [27]). For any , .

Definition 3 (see [18]). The -metric is -continuous if and only if for any and sequence in with , we have .

Proposition 4 (see [19]). For any ,

Lemma 5 (see [18]). Let be a sequence in (). If there exists such that for all , then is a Cauchy sequence.

The following lemma follows immediately from the above lemma.

Lemma 6. If for some , with , for all , then is a Cauchy sequence.

3. Main Results

We introduce pairwise -Hausdorff functions as follows:

Definition 7. Let . For any , and any , if there exist such that then we say that and are pairwise -Hausdorff functions.

For , we get the following.

Definition 8. For any and any if there exist such that then we say that is a -Hausdorff function.

Remark 9. (i)For , is always a -Hausdorff function(ii)If for any , the function is a -Hausdorff function, then for any , the function is a -Hausdorff function

Example 10. Let , and be as follows:

We will show that the functions and satisfy (2). We will consider the values of in as follows: (i). In this case, and are singleton sets and so (2) is obviously true(ii). . If , then we have and , , , and . Thus, (2) is true for all . If , then inequality (2) holds with (iii). , and the result follows in the same way as in (ii) above.(iv). . If , then we have and , , , and . Thus, (2) is true for all . If , then we take and then , , , and . Thus, (2) is true for all . If , inequality (2) holds with

Thus, and are pairwise -Hausdorff functions for . However, and are not pairwise -Hausdorff functions for , as we see that inequality (2) is not satisfied for , , and . In fact, and are not pairwise -Hausdorff functions for .

We now present our main result as follows:

Theorem 11. Let be a complete -metric space with constant , be -continuous, and be multivalued pairwise -Hausdorff functions for some and satisfying the following condition: for all and some , , with , , and . Then, and have a common fixed point.

Proof. Let , , and . By (2), there exist , such that . By (2) again, there exist , such that
Continuing these ways, we construct the sequence such that Now, Therefore, Again, Thus, we have where .
By Lemma 6, the sequence is a Cauchy sequence. Since is complete, there exists such that the Cauchy sequence is convergent to . We will show that . By the definition of , we have

It follows that

Since we have

This implies

Again, we have

It follows that

Since we have

This implies

Now

Using (17) and (22) in the above two inequalities, we get

This gives and . Since and are closed, we have and .

Our next result provides an extension and new variants of Ciric’s quasi-contraction [15] and multivalued weak quasi-contraction [17], for a pair of multivalued mappings in a -metric space.

Theorem 12. Let be a complete -metric space with constant , be -continuous, and be multivalued pairwise -Hausdorff functions for some and satisfying the following condition: for all , some with and . Then, and have a common fixed point.

Proof. Proceeding as in the proof of Theorem 11, for some , , and , we construct the sequence satisfying (7). Then, we have Therefore, Again, and we get Thus, we have where

By Lemma 6, the sequence is a Cauchy sequence. Since is complete, there exists such that the Cauchy sequence is convergent to . We will show that . By the definition of , we have

It follows that

Since we have

This implies

Again, we have

It follows that

Since we have

This implies

Now,

Using (35) and (40) in the above two inequalities, we get

Since and , we get and . As and are closed, we have and .

Applying the same technique as in the proof of Theorem 12, we can prove the following extension and new variant of Ciric’s contraction for a pair of multivalued mappings in a -metric space.

Theorem 13. Let be a complete -metric space with constant , be -continuous, and be multivalued pairwise -Hausdorff functions for some and satisfying the following condition: for all and some with . Then, and have a common fixed point.

For in Theorem 11, we get the following result:

Corollary 14. Let be a complete -metric space with constant , be -continuous, and be a multivalued -Hausdorff function for some and satisfying the following condition: for all and some , with , , and . Then, has a fixed point.

Example 15. Let , , and be as follows: We will show that the functions and satisfy contraction condition (6) for .

Case 1. . By Proposition 2, we have

Case 2. . We have . The minimum value of for is . The maximum value of for is 2 (at ). Thus, for any .

Case 3. . We have . The minimum value of for is . . The maximum value of for is 2 (at ). Thus, for any .

Thus, and satisfy contraction condition (6) for , and . Simple calculations show that and are pairwise -Hausdorff functions. All conditions of Theorem 11 are satisfied, and is a common fixed point of and . However, we see that at , and do not satisfy contraction condition (6) for and so do not satisfy Nadler’s contraction and Czerwik’s contraction.

Remark 16. In Example 15, simple calculations show that and do not satisfy contraction condition (6) for . However, in view of Remark 9 (i), there may exist functions and which satisfy contraction condition (6) for but may not satisfy for . Thus, for , Theorem 11 is an extension of Nadler’s contraction [10], Czerwik’s contraction [2], and many of their generalizations. For , Theorem 11 provides new variants of Nadler’s contraction [10], Czerwik’s contraction [2], and many of their generalizations.

Example 17. Let , and be as follows: We will show that satisfies (44) with .
For if , then the result is clear. Suppose and . Then, and so that . Also, we have or .

If , then . Now . So and , that is, . Thus, we have , where .

Similarly, if , we get , where .

However, for and , we have

We see that does not satisfy condition (2.2) of [24] and condition (2.1) of [26]. Thus, Theorem 2.2 of Debnath [24] and Theorem 2.3 of Kumar and Luambano [26] are not applicable.

Remark 18 (an open question). Obtain the version of results in fixed points in the sense of Debnath [24], Kumar and Luambano [26], and Altun et al. [25] for two or more mappings using -Hausdorff-Pompeiu -metric, which will give extension and new variants of the respective results and will also generalize Corollary 19.

By taking different values of in Theorem 11, we get the following extension and new variants of well-known contraction principles:

For , , we have the following.

Corollary 19 (Nadler’s and Czerwik’s contraction). Let be a complete -metric space with constant and be multivalued pairwise -Hausdorff functions for some and satisfying the following condition: for all and . Then, and have a common fixed point.

For , , we have the following.

Corollary 20 (Kannan’s contraction). Let be a complete -metric space with constant and be multivalued pairwise -Hausdorff functions for some and satisfying the following condition: for all and . Then, and have a common fixed point.

For , , we have the following.

Corollary 21 (Chattarjee contraction). Let be a complete -metric space with constant and be multivalued pairwise -Hausdorff functions for some and satisfying the following condition: for all and . Then, and have a common fixed point.

For , , we have the following.

Corollary 22 (Hardy-Rogers contraction). Let be a complete -metric space with constant and be multivalued pairwise -Hausdorff functions for some and satisfying the following condition: for all and , , and . Then, and have a common fixed point.

Remark 23. Corollary 19 is an extension and new variant of the results of Nadler [10] and Czerwik [2], Corollaries 20 and 21 are extended and new variants of the set-valued versions of the Kannan contraction and Chatterjee contraction, respectively, whereas Corollary 22 is an extended and new variant of the result of Mirmostaffae [16].
If are single-valued mappings and then by Proposition 2, for all . So taking in Theorem 11, we get the following results for single-valued mappings.

Corollary 24. Let be a complete -metric space with constant and be single-valued mappings satisfying the following condition: for all and , , and . Then, and have a common fixed point.

Remark 25. Corollary 24 is an extension and -metric version of the result of Wong [40].

4. Applications

In this section, we provide two applications of our results.

4.1. Application to Multivalued Fractals

In this section inspiring from some recent works in [20, 41, 42], we will apply our result to prove the existence of a unique common multivalued fractal for a pair of iterated multifunction systems. Let , , be upper semicontinuous mappings. Then, and form a pair of iterated multifunction systems defined on the -metric space . The extended multifractal operators generated by the iterated multifunction systems and are the operators defined by and , respectively. A common fixed point of and is called the common multivalued fractal of the iterated multifunction systems and .

Theorem 26. Let , , be upper semicontinuous mappings satisfying the following condition:
For , there exist and , , such that for all , Then, (i)For all , (v)The pair of systems and has a unique common multivalued fractal

Proof. Suppose condition (57) holds. Then, for , we have Similarly, we get Then, we have where and . Note that and so Thus, satisfies the conditions of Corollary 24 in the metric space and hence has a common fixed point in , which in turn is the unique common multivalued fractal of the pair of iterated multifunction systems and .

Remark 27. Since , Theorem 26 is a proper improvement and generalization of Theorem 3.4 of [20], Theorem 3.1 of [41], and Theorem 3.8 of [42].

4.2. Application to the Integral Equation

In this section, motivated by the applications given in [28–30] and [31], we establish the sufficient conditions for the existence of a common solution of a pair of nonlinear Volterra-type integral equations.

For some real numbers with and , let be the Banach space of real continuous functions defined on equipped with a norm given by For some , define a -metric on by

Then, is a complete -metric space. Consider the following pair of Volterra-type integral equations: for all , , , and , and are continuous functions and .

Suppose is self-mappings defined by for all , where . It is obvious that is a solution of (64) if and only if it has a common fixed point of and .

Theorem 28. Suppose that the following hypotheses hold:
(H₁) and are closed in
(H₂) There exist nonnegative real numbers with such that where (H₃)
Then, the system (64) of integral equations has unique common solutions in .

Proof. Using and , we have Thus, conditions of Theorem 11 are satisfied. Theorem 11 therefore ensures a common fixed point of and , which in turn is a common solution of the pair of integral equations (64).

Remark 29. Taking and in (64), we get the Volterra-type integral equations considered in Rasham et al. [31] and Alshoraify et al. [30].

Remark 30. Taking and in (64), we get the Fredholm-type integral equations (III.3) considered in Shoaib et al. [29].

Remark 31. Taking and in (64), we get the Fredholm-type integral equations (III.1) considered in Shoaib et al. [29].

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References

I. A. Bakhtin, “The contraction mapping principle in quasimetric spaces,” Functional analysis, vol. 30, pp. 26–37, 1989.
View at: Google Scholar
S. Czerwik, “Nonlinear set-valued contraction mappings in b-metric spaces,” Atti del Seminario Matematico e Fisico dell'Universita di Modena e Reggio Emilia, vol. 46, pp. 263–276, 1998.
View at: Google Scholar
W. A. Kirk and N. Shahzad, Fixed Point Theory in Distance Spaces, Springer, Berlin, 2014.
View at: Publisher Site
M. Younis, D. Singh, and A. A. N. Abdou, “A fixed point approach for tuning circuit problem in dislocated b-metric spaces,” Mathematicsl Methods in the Applied Sciences, 2021.
View at: Publisher Site | Google Scholar
M. Younis, D. Singh, I. Altun, and V. Chauhan, “Graphical structure of extended b-metric spaces: an application to the transverse oscillations of a homogeneous bar,” International Journal of Nonlinear Sciences and Numerical Simulation, 2021.
View at: Publisher Site | Google Scholar
M. Younis, D. Singh, and L. Shi, “Revisiting graphical rectangular b-metric spaces,” Asian-European Journal of Mathematic, vol. 15, no. 4, 2022.
View at: Publisher Site | Google Scholar
M. Younis, D. Singh, and A. Goyal, “A novel approach of graphical rectangular b-metric spaces with an application to the vibrations of a vertical heavy hanging cable,” Journal of Fixed Point Theory and Applications, vol. 21, no. 1, p. 33, 2019.
View at: Publisher Site | Google Scholar
D. Pompieu, Sur la continuite’ des fonctions de variables complexes (These), Annales de la Faculté des sciences de Toulouse: Mathématiques, Gauthier-Villars, Paris, vol. 7, no. 3, 1905.
View at: Publisher Site
F. Hausdorff, Grunclzuege der Mengenlehre, Viet, Leipzig, 1914.
S. B. Nadler, “Multivalued contraction mappings,” Pacific Journal of Mathematics, vol. 30, no. 2, pp. 475–488, 1969.
View at: Publisher Site | Google Scholar
B. Damjanovic, B. Samet, and C. Vetro, “Common fixed point theorems for multi-valued maps,” Acta Mathematica Scientia, vol. 32, no. 2, pp. 818–824, 2012.
View at: Publisher Site | Google Scholar
T. Kamran and Q. Kiran, “Fixed point theorems for multi-valued mappings obtained by altering distances,” Mathematical and Computer Modelling, vol. 54, no. 11-12, pp. 2772–2777, 2011.
View at: Publisher Site | Google Scholar
D. Klim and D. Wardowski, “Fixed point theorems for set-valued contractions in complete metric spaces,” Journal of Mathematical Analysis and Applications, vol. 334, no. 1, pp. 132–139, 2007.
View at: Publisher Site | Google Scholar
Z. Liu, X. Na, Y. C. Kwun, and S. M. Kang, “Fixed points of some set-valued F-contractions,” Journal of Nonlinear Sciences and Applications (JNSA), vol. 9, no. 11, pp. 5790–5805, 2016.
View at: Publisher Site | Google Scholar
H. Aydi, M. F. Bota, E. Karapinar, and S. Mitrovic, “A fixed point theorem for set-valued quasi-contractions in b-metric spaces,” Fixed Point Theory and Applications, vol. 2012, no. 1, 2012.
View at: Publisher Site | Google Scholar
A. K. Mirmostaffae, “Fixed point theorems for set-valued mappings in i-metric spaces,” Fixed Point Theory, vol. 18, no. 1, pp. 305–314, 2017.
View at: Publisher Site | Google Scholar
N. Hussain and Z. D. Mitrovic, “On multivalued weak quasi-contractions in b-metric spaces,” Journal of Nonlinear Sciences and Applications (JNSA), vol. 10, no. 7, pp. 3815–3823, 2017.
View at: Publisher Site | Google Scholar
R. Miculescu and A. Mihail, “New fixed point theorems for set-valued mappings on TVS-cone metric spaces,” Journal of Fixed Point Theory and Applications, vol. 2015, no. 1, 2015.
View at: Publisher Site | Google Scholar
M. Boriceanu, A. Petrusel, and I. A. Rus, “Fixed point theorems for some multivalued generalized contractions in b-metric spaces,” International Journal of Statistics and Mathematics, vol. 6, pp. 65–76, 2010.
View at: Google Scholar
C. Chifu and A. Petrusel, “Fixed points for multivalued contractions in bmetric spaces with applications to fractals,” Taiwanese Journal of Mathematics, vol. 18, no. 5, pp. 1365–1375, 2014.
View at: Publisher Site | Google Scholar
A. A. Mebawondu, C. Izuchukwu, K. O. Aremu, and O. T. Mewomo, “Some fixed point results for a generalized TAC-Suzuki-Berinde type F-contractions in b-metric spaces,” Appl. Math. E-Notes, vol. 19, pp. 629–653, 2019.
View at: Google Scholar
H. K. Pathak, R. George, H. A. Nabwey, M. S. El-Paoumy, and K. P. Reshma, “Some generalized fixed point results in a b-metric space and application to matrix equations,” Fixed Point Theory and Applications, vol. 2015, no. 1, 2015.
View at: Publisher Site | Google Scholar
R. George, H. A. Nabwey, R. Ramaswamy, and S. Radenovic, “Some generalized contraction classes and common fixed points in b-metric space endowed with a graph,” Mathematics, vol. 7, no. 8, p. 754, 2019.
View at: Publisher Site | Google Scholar
P. Debnath, “Set-valued Meir–Keeler, Geraghty and Edelstein type fixed point results in b-metric spaces,” Rendiconti del Circolo Matematico di Palermo Series 2, vol. 2, 2021.
View at: Publisher Site | Google Scholar
I. Altun, G. Minak, and H. Dag, “Multivalued F -contractions on complete metric space,” Journal of Nonlinear and Convex Analysis, vol. 16, no. 4, pp. 659–666, 2015.
View at: Google Scholar
S. Kumar and S. Luambano, “On some fixed point theorems for multivalued F-contractions in partial metric spaces,” Demonstratio Mathematica, vol. 54, no. 1, pp. 151–161, 2021.
View at: Publisher Site | Google Scholar
R. George and H. K. Pathak, “Some new extensions of multivalued contractions in a b-metric space and its applications,” Mathematics (MDPI), vol. 9, p. 12, 2021.
View at: Google Scholar
M. Younis and D. Singh, “On the existence of the solution of Hammerstein integral equations and fractional differential equations,” Journal of Applied Mathematics and Computing, vol. 68, no. 2, pp. 1087–1105, 2022.
View at: Publisher Site | Google Scholar
M. Shoaib, T. Abdeljawad, M. Sarwar, and F. Jarad, “Fixed point theorems for multi-valued contractions in b-metric spaces with applications to fractional differential and integral equations,” Access, vol. 7, article 127373, 127383 pages, 2019.
View at: Publisher Site | Google Scholar
S. S. Alshoraify, A. Shoaib, and M. Arshad, “New types of contraction for multivalued mappings and related fixed point results in abstract spaces,” Journal of Function Spaces, vol. 2019, Article ID 1812461, 14 pages, 2019.
View at: Publisher Site | Google Scholar
T. Rasham, A. Shoaib, N. Hussain, M. Arshad, and S. U. Khan, “Common fixed point results for new Ciric-type rational multivalued F-contraction with an application,” Journal of Fixed Point Theory and Applications, vol. 20, no. 1, 2018.
View at: Publisher Site | Google Scholar
H. Zegeye and G. B. Wega, “Approximation of a common f-fixed point of f-pseudocontractive mappings in Banach spaces,” Rendiconti del Circolo Matematico di Palermo Series 2, vol. 70, no. 3, pp. 1139–1162, 2021.
View at: Publisher Site | Google Scholar
M. F. Barnsley, Fractals Everywhere, Dover Publications, Mineola, New York, 2012.
A. K. B. Chand, N. Vijender, and M. A. Navascués, “Shape preservation of scientific data through rational fractal splines,” Calcolo, vol. 51, no. 2, pp. 329–362, 2014.
View at: Publisher Site | Google Scholar
B. B. Mandelbrot, The Fractal Geometry of Nature, Freeman, New York, 1982.
P. Manousopoulos, V. Drakopoulos, and T. Theoharis, “Curve fitting by fractal interpolation,” Transactions on Computational Science, Springer, Berlin, vol. 4750, no. I, pp. 85–103, 2008.
View at: Google Scholar
P. Manousopoulos, V. Drakopoulos, and T. Theoharis, “Parameter identification of 1D recurrent fractal interpolation functions with applications to imaging and signal processing,” Journal of Mathematical Imaging and Vision, vol. 40, no. 2, pp. 162–170, 2011.
View at: Publisher Site | Google Scholar
M. A. Navascués, “Fractal functions of discontinuous approximation,” Journal of Basic & Applied Sciences, vol. 10, pp. 173–176, 2014.
View at: Publisher Site | Google Scholar
M. A. Navascués and A. K. B. Chand, “Fundamental sets of fractal functions,” Acta Applicandae Mathematicae, vol. 100, no. 3, pp. 247–261, 2008.
View at: Publisher Site | Google Scholar
C. S. Wong, “Common fixed points of two mappings,” Pacific Journal of Mathematics, vol. 48, no. 1, pp. 299–312, 1973.
View at: Publisher Site | Google Scholar
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
C. Chifu and A. Petrusel, “Multivalued fractals and generalized multivalued contractions,” Chaos Solitons & Fractals, vol. 36, no. 2, p. 203, 2008.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2022 Fahad Sameer Alshammari 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.

PDF Download Citation

Download other formats

Order printed copies

Views

195

Downloads

356

Citations