On the Fractal interpolation functions associated with Matkowski contractions

Najmeddine Attia; Mohamed balegh; Rim Amami; Rimah Amami; Najmeddine Attia; Mohamed balegh; Rim Amami; Rimah Amami

doi:10.3934/era.2023238

Electronic Research Archive

2023, Volume 31, Issue 8: 4652-4668. doi: 10.3934/era.2023238

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Research article

On the Fractal interpolation functions associated with Matkowski contractions

1.
Department of Mathematics and Statistics, College of Science, King Faisal University, Al-Ahsa 31982, Saudi Arabia
2.
Department of Mathematics, College of Science and Arts, Muhayil, King Khalid University, Abha 61413, Saudi Arabia
3.
Department of Basic Sciences, Deanship of Preparatory Year and Supporting Studies, Imam Abdulrahman Bin Faisal University, Dammam 34212, Saudi Arabia
4.
Computer Department, Deanship of Preparatory Year and Supporting Studies, Imam Abdulrahman Bin Faisal University, Dammam 34212, Saudi Arabia

Received: 04 April 2023 Revised: 01 June 2023 Accepted: 15 June 2023 Published: 29 June 2023

In this paper we investigate an iterated function system that defines a fractal interpolation function, where ordinate scaling, that is Lipschitz constant in Banach contraction principle is substituted by real-valued control function. In such a manner, fractal interpolation functions associated with Matkowski contractions are obtained and provide a new framework of approximating experimental data. Furthermore, given a data generating function $ f $, we study a new class of fractal interpolation functions which converge to $ f $.
- fractal interpolation function,
- iterated function system,
- generalized iterated function system,
- Matkowski contraction,
- Banach contraction
Citation: Najmeddine Attia, Mohamed balegh, Rim Amami, Rimah Amami. On the Fractal interpolation functions associated with Matkowski contractions[J]. Electronic Research Archive, 2023, 31(8): 4652-4668. doi: 10.3934/era.2023238

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Abstract

In this paper we investigate an iterated function system that defines a fractal interpolation function, where ordinate scaling, that is Lipschitz constant in Banach contraction principle is substituted by real-valued control function. In such a manner, fractal interpolation functions associated with Matkowski contractions are obtained and provide a new framework of approximating experimental data. Furthermore, given a data generating function $ f $, we study a new class of fractal interpolation functions which converge to $ f $.

References

[1]	M. F. Barnsley, Fractals Everywhere, 2nd edition, Academic Press, 1988.
[2]	M. F. Barnsley, Fractal functions and interpolation, Constr. Approx, 2 (1986), 303–329. https://doi.org/10.1007/BF01893434 doi: 10.1007/BF01893434
[3]	J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J., 30 (1981), 713–747.
[4]	M. F. Barnsley, A. N. Harrington The Calculus of fractal interpolation functions, J. Approx. Theory, 57 (1989), 14–34. https://doi.org/10.1016/0021-9045(89)90080-4 doi: 10.1016/0021-9045(89)90080-4
[5]	N. A. Secelean, Countable iterated function systems, Far East J. Dyn. Syst., 3 (2001), 149–167.
[6]	K. Leśniak, Infinite iterated function systems: A multivalued approach, Bull. Pol. Acad. Sci. Math., 52 (2004), 1–8.
[7]	A. Mihail, R. Miculescu, Generalized IFSs on non-compact spaces, Fixed Point Theory Appl., 2010 (2010), 584215. https://doi.org/10.1155/2010/584215 doi: 10.1155/2010/584215
[8]	F. Strobin, J. Swaczyna, On a certain generalization of the iterated function system, Bull. Aust. Math. Soc., 87 (2013), 37–54. https://doi.org/10.1017/S0004972712000500 doi: 10.1017/S0004972712000500
[9]	K. R. Wicks, Fractals and Hyperspaces, Springer-Verlag, Berlin, 2006.
[10]	A. K. B. Chand, G. P. Kapoor, Generalized cubic spline fractal interpolation functions, SIAM J. Numer. Anal., 44 (2006), 655–676. https://doi.org/10.1137/0406110 doi: 10.1137/0406110
[11]	Y. Chen, G. A. Kopp, D. Surry, Interpolation of wind-induced pressure time series with an artificial network, J. Wind Eng. Ind. Aerodyn, 90 (2002), 589–615. https://doi.org/10.1016/S0167-6105(02)00155-1 doi: 10.1016/S0167-6105(02)00155-1
[12]	N. Vijender, Bernstein fractal trigonometric approximation, Acta Appl. Math., 159 (2018), 11–27. https://doi.org/10.1007/s10440-018-0182-1 doi: 10.1007/s10440-018-0182-1
[13]	P. Viswanathan, A. K. B. Chand, M. A. Navascuès, Fractal perturbation preserving fundamental shapes: Bounds on the scale factors, J. Math. Anal. Appl., 419 (2014), 804–817. https://doi.org/10.1016/j.jmaa.2014.05.019 doi: 10.1016/j.jmaa.2014.05.019
[14]	S. Ri, A new nonlinear fractal interpolation function, Fractals, 25 (2017). https://doi.org/10.1142/S0218348X17500633 doi: 10.1142/S0218348X17500633
[15]	S. Ri, New types of fractal interpolation surfaces, Chaos Solitons Fractals, 119 (2019), 291–297.
[16]	M. A. Navascués, C. Pacurar, V. Drakopoulos, Scale-free fractal interpolation, Fractal Fract, 6 (2022), 602. https://doi.org/10.3390/fractalfract6100602 doi: 10.3390/fractalfract6100602
[17]	J. Kim, H. Kim, H. Mun, Nonlinear fractal interpolation curves with function vertical scaling factors, Indian J. Pure Appl. Math., 51 (2020), 483–499. https://doi.org/10.1007/s13226-020-0412-x doi: 10.1007/s13226-020-0412-x
[18]	N. Attia, H. Jebali, Fractal interpolation functions with contraction condition of integral type, Chaos Solitons Fractal.
[19]	J. Matkowski, Integrable Solutions of Functional Equations, Instytut Matematyczny Polskiej Akademi Nauk(Warszawa), 1975.
[20]	S. Banach, Sur les opérations dans les ensembles abstraits et leur applications aux équations intégrales, Fund. Math., 3 (1922), 133–181.
[21]	A. Mihail, R. Miculescu, Applications of fixed point theorems in the theory of generalized IFS, Fixed Point Theory Appl., (2008), 312876. https://doi.org/10.1155/2008/312876 doi: 10.1155/2008/312876
[22]	N. Secelean, Generalized iterated function systems on the space $l^\infty(X)$, J. Math. Anal. Appl., 410 (2014), 847–858. https://doi.org/10.1016/j.jmaa.2013.09.007 doi: 10.1016/j.jmaa.2013.09.007
[23]	F. Strobin, J. Swaczyna, A code space for a generalized IFS, Fixed Point Theory, preprint, arXiv: 1310.3097v2. https://doi.org/10.48550/arXiv.1310.3097
[24]	F. Strobin, Attractors of generalized IFSs that are not attractors of IFSs, J. Math. Anal. Appl., 422 (2015), 99–108. https://doi.org/10.1016/j.jmaa.2014.08.029 doi: 10.1016/j.jmaa.2014.08.029
[25]	R. Pasupathi, A. K. B. Chand, M. A. Navascuès, M. V. Sebastian, Cyclic generalized iterated function systems, Comput. Math. Methods, 3 (2021). https://doi.org/10.1002/cmm4.1202 doi: 10.1002/cmm4.1202
[26]	J. Jachymski, I. Jóź wik, Nonlinear contractive conditions: A comparison and related problems, Banach Center Publ. Polish Acad. Sci., 77 (2007), 123–146. https://doi.org/10.4064/bc77-0-10 doi: 10.4064/bc77-0-10
[27]	P. Viswanathan, A. K. B. Chand, M. A. Navascués, Fractal perturbation preserving fundamental shapes: Bounds on the scale factors, J. Math. Anal. Appl., 419 (2014), 804–817. https://doi.org/10.1016/j.jmaa.2014.05.019 doi: 10.1016/j.jmaa.2014.05.019
[28]	M. A. Navascués, Non-smooth polynomials, Int. J. Math. Anal., 1 (2007), 159–174.
[29]	E. Rakotch, A note on contractive mappings, Proc. Amer. Math. Soc., 13 (1962), 459–465. http://dx.doi.org/10.1090/S0002-9939-1962-0148046-1 doi: 10.1090/S0002-9939-1962-0148046-1
[30]	F. E. Browder, On the convergence of successive approximations for nonlinear functional equations, Nederl. Akad. Wetensch. Proc. Ser. Indag. Math., 71 (1968), 27–35. https://doi.org/10.1016/S1385-7258(68)50004-0 doi: 10.1016/S1385-7258(68)50004-0
[31]	M. F. Barnsley, J. Elton, D. P. Hardin, P. R. Massopust, Hidden variable fractal interpolation functions, SIAM J. Math. Anal., 20 (1989), 1218–1242. https://doi.org/10.1137/0520080 doi: 10.1137/0520080
[32]	S. G. Gal, Shape Preserving Approximation by Real and Complex Polynomials, Springer Science Business Media, 2010.
[33]	M. A. Navascuès, Fractal polynomial interpolation, Z. Anal. Anwend., 24 (2005), 401–418. https://doi.org/10.4171/ZAA/1248 doi: 10.4171/ZAA/1248
[34]	P. R. Massopust, Fractal surfaces, J. Math. Anal. Appl., 151 (1990), 275–290. https://doi.org/10.1016/0022-247X(90)90257-G doi: 10.1016/0022-247X(90)90257-G
[35]	P. Wong, J. Howard, J. Lin, Surfaces roughening and the fractal nature of rocks, Phys. Rev. Lett., 57 (1986), 637–640. https://doi.org/10.1103/PhysRevLett.57.637 doi: 10.1103/PhysRevLett.57.637
[36]	B. B. Nakos, C. Mitsakaki, On the fractal character of rock surfaces, Int. J. Rock Mech. Min. Sci. Geomech. Abstr., 28 (1991), 527–533. https://doi.org/10.1016/0148-9062(91)91129-F doi: 10.1016/0148-9062(91)91129-F
[37]	C. S. Pande, L. R. Richards, S. Smith, Fractal charcteristics of fractured surfaces, J. Met. Sci. Lett., 6 (1987), 295–297. https://doi.org/10.1007/BF01729330 doi: 10.1007/BF01729330
[38]	H. Xie, J. Wang, E. Stein, Direct fractal measurement and multifractal properties of fracture surfaces, Phys. Lett. A, 242 (1998), 41–50. https://doi.org/10.1016/S0375-9601(98)00098-X doi: 10.1016/S0375-9601(98)00098-X
[39]	X. C. Jin, S. H. Ong, Jayasooriah, Fractal characterization of Kidney tissue sections, IEEE Int. Conf. Eng. Med. Biol. Baltimore, 2 (1994), 1136–1137. https://doi.org/10.1109/IEMBS.1994.415361 doi: 10.1109/IEMBS.1994.415361
[40]	M. Samreen, T. Kamran, M. Postolache, Extended $b$- Metric space, extended b-comparison function and nonlinear contractions, U.P.B. Sci. Bull., Series A, 80 (2018).
[41]	C. Wolf, A mathematical model for the propagation of a hantavirus in structured populations, Discrete Contin. Dyn. Syst. B, 4 (2004), 1065–1089. https://doi.org/10.3934/dcdsb.2004.4.1065 doi: 10.3934/dcdsb.2004.4.1065
[42]	S. S. Al-Bundi, Iterated function system in $\emptyset$-Metric spaces, Bol. Soc. Paran. Mat., 40 (2022), 1–10. https://doi.org/10.5269/bspm.52556 doi: 10.5269/bspm.52556

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