Fractal generation via generalized Fibonacci–Mann iteration with s-convexity

Indian Journal of Pure and Applied Mathematics - Trang 1-15 - 2024
Swati Antal1, Nihal Özgür2, Anita Tomar3, Krzysztof Gdawiec4
1Department of Mathematics, Army Cadet College Wing, Indian Military Academy, Dehradun, India
2Department of Mathematics, İzmir Democracy University, Karabağlar, İzmir, Turkey
3Department of Mathematics, Pt.L.M.S. Campus, Sridev Suman Uttarakhand University, Rishikesh, India
4Institute of Computer Science, University of Silesia in Katowice, Sosnowiec, Poland

Tóm tắt

Recently, the generalized Fibonacci–Mann iteration scheme has been defined and used to develop an escape criterion to study mutants of the classical fractals for a function $$\sin \left( z^{n}\right) +az+c$$ , $$a,c\in \mathbb {C}$$ , $$n\ge 2$$ , and z is a complex variable. In the current work, we use generalized Fibonacci–Mann iteration extended further via the notion of s-convex combination in the exploration of new mutants of celebrated Mandelbrot and Julia sets. Further, we provide a few graphical and numerical examples obtained by the use of the derived criteria.

Tài liệu tham khảo

Alfuraidan, M. R. and Khamsi, M. A. Fibonacci-Mann iteration for monotone asymptotically nonexpansive mappings, Bull. Aust. Math. Soc., 96 (2) (2017), 307–316.

Antal, S., Tomar, A., Prajapati, D. J., Sajid, M., Fractals as Julia sets of complex sine function via fixed point iterations, Fractal Fract. 2021, 5, 272. https://doi.org/10.3390/fractalfract5040272

Antal, S., Tomar, A., Prajapati, D.J., Sajid, M., Variants of Julia and Mandelbrot sets as fractals via Jungck-Ishikawa fixed point iteration system with\( s \)-convexity, AIMS Mathematics, 7 (6) (2022), 10939–10957. https://doi.org/10.3934/math.2022611

Benjamin, A. T., Quinn, J. J., The Fibonacci numbers-exposed more discretely, Math. magazine, 76 (3) (2003), 182-192.

Julia, G., Mémoire sur l’itération des fonctions rationnelles, J. Math. Pures Appl., 8 (1918), 47–745.

Barnsley, M., Fractals everywhere, 2nd ed.; Academic Press: San Diego, CA, USA, 1993.

Mandelbrot, B. B., The fractal geometry of nature, W. H. Freeman, New York, NY, USA, 1982.

Mann, W. R., Mean value methods in iteration, Proc. Amer. Math. Soc., 4 (1953), 506–510.

Devaney, R. L., A first course in chaotic dynamical systems: Theory and Experiment, 2nd ed., Addison-Wesley: Boston, MA, USA, 1992.

Barrallo, J. and Jones, D. M., Coloring algorithms for dynamical systems in the complex plane, in visual mathematics, 1 (4), MISASA, Belgrade, Serbia, 1999.

Gdawiec, K., and Shahid, A. A., Fixed point results for the complex fractal generation in the S-iteration orbit with\( s \)-convexity, Open J. Math. Sci. 2 (1) (2018), 56-72.

Jia, F., Zhang, Y., Application of generalized Julia set graphics in clothing pattern design, Text. Res. J., 36 (7) (2015), 104–109.

Kalman, D., Mena, R., The Fibonacci numbers-exposed, Math. Mag. 76 (3) (2003), 167–181.

Kang, S. M., Nazeer, W., Tanveer, M. and Shahid, A. A., New fixed point results for fractals generation in Jungck-Noor orbit with\( s \)-convexity, J. Funct. Spaces, 2015, Artical ID: 963016, 1–7.

Kumari, S., Gdawiec, K., Nandal, A., Kumar, N., Chugh, R., On the Viscosity Approximation Type Iterative Method and Its Non-linear Behaviour in the Generation of Mandelbrot and Julia Sets, Numerical Algorithms (in press), https://doi.org/10.1007/s11075-023-01644-4

Kumari, S., Gdawiec, K., Nandal, A., Postolache, M., Chugh, R., A Novel Approach to Generate Mandelbrot Sets, Julia Sets and Biomorphs via Viscosity Approximation Method, Chaos, Solitons & Fractals 163, 112540, (2022)

Kumari, S., Kumari, M., and Chugh, R., Generation of new fractals via SP orbit with\( s \)-convexity, Int. J. Eng. Technol, 9 (3), 2491–2504.

Kwun, Y.C., Tanveer, M., Nazeer, W., Gdawiec, K. and Kang, S.M., Mandelbrot and Julia sets via Jungck-CR iteration with\(s\)-convexity, IEEE Access 7 (2019), 12167-12176.

Kwun, Y. C., Shahid, A. A., Nazeer, W., Abbas, M. and Kang, S. M., Fractal generation via\( CR \)-iteration scheme with\( s \)-convexity, IEEE Access 7 (2019), 69986-69997.

Li, D., Tanveer, M., Nazeer, W., Guo, X., Boundaries of filled julia sets in generalized Jungck-Mann orbit, IEEE Access, 7 (2019), 76859-76867.

Ma, Y., Li, S.B., The application of Julia set in the design of textile pattern, Microcomputer Applications, 27 (6) (2006), 739-742.

Nazeer, W., Kang, S.M., Tanveer, M., Shahid A.A., Fixed point results in the generation of Julia and Mandelbrot sets, J. Ineq. Appl., (1) (2015), 1–16.

Özgür, N., Antal, S., Tomar, A., Julia and Mandelbrot sets of transcendental function via Fibonacci-Mann iteration, J. Funct. Spaces, (2022), Article ID 2592573, 13 pages. https://doi.org/10.1155/2022/2592573

Picard, E., Memoire sur la theorie des equations aux derivees partielles et la methode des approximations successives, J. Math. Pures et Appl., 6 (1890), 145210.

Pinheiro, M. R., \(s\)-convexity: foundations for analysis, Differ. Geom. Dyn. Syst. 10 (2008), 257–262.

Shahid, A. A., Nazeer, W., Gdawiec, K., The Picard-Mann iteration with\(s\)-convexity in the generation of Mandelbrot and Julia sets, Monatsh. Math. 195 (4) (2021), 565–584.

Tomar, A., Prajapati, D. J., Antal, S., Rawat, S., Variants of Mandelbrot and Julia fractals for higher-order complex polynomials, Math. Meth. Appl. Sci. (2022), 1-13. https://doi.org/10.1002/mma.8262

Tomar, A., Antal, S., Özgür, N. and Kumar, V., A generalized version of Fibonacci-Mann Iteration scheme in fractal generating process, preprint.

Zhang, H., Tanveer, M., Li, Y. X., Peng, Q., and Shah, N. A., Fixed point results of an implicit iterative scheme for fractal generations, AIMS Math., 6 (12) (2021), 13170-13186.

Thông tin
Thông tin xuất bản

Indian Journal of Pure and Applied Mathematics

1-15

Thông tin tác giả