Development of a robust iterative algorithm for contractive-like mapping in hyperbolic space: application to computational fractal imaging of polynomiographs

• Grace Margrate Mary, J. ^[1] ; Uthayakumar, R. ^[1]
  1. [1] Gandhigram Rural Institute
• Localización: Applied general topology, ISSN-e 1989-4147, ISSN 1576-9402, Vol. 27, Nº. 2, 2026
• Idioma: inglés
• DOI: 10.4995/agt.24652
• Enlaces
  • Texto completo
• Resumen

  • The primary objective is to establish the fixed-point convergence analysis of our newly proposed iteration process, within the framework of hyperbolic spaces, particularly in the presence of contractive-like mapping and further investigate its stability characteristics. To demonstrate its effectiveness, we present a numerical example that highlights the superiority of our iteration process in terms of dominance and faster convergence compared to existing iterative methods. Furthermore, we explore a practical application of our iteration process by employing polynomiographs, which are generated using traditional iteration schemes from the literature and visualized through fractal images. The scheme is implemented computationally using MATLAB, where a variety of visual outputs are produced to illustrate the structure and behavior of the convergence zones. We then compare these results with polynomiographs produced using our novel Picard-New iteration process, demonstrating its advantages in generating fractal images more effectively.

• Referencias bibliográficas
  • A. Bera, A. Chanda, L. K. Dey, and J. Ali, Iterative approximation of fixed points of a general class of non-expansive mappings in hyperbolic...
  • V. Berinde, On the approximation of fixed points of weak contractive mappings Carpathian Journal of Mathematics 19 (2003), 7-22.
  • V. Berinde, Picard iteration converges faster than Mann iteration for a class of quasi-contractive operators, Fixed Point Theory and Applications...
  • V. Berinde, Approximating fixed points of weak contractions using the Picard iteration, Nonlinear Analysis Forum 9 (2004), 43-54. https://doi.org/10.1155/S1687182004311058
  • L. P. Castro and R. C. Guerra, Hyers-Ulam-Rassias stability of Volterra integral equations within weighted spaces, Libertas Mathematica 33,...
  • J. Daengsaen and A. Khemphet, On the rate of convergence of P-iteration, SP-iteration, and D-iteration methods for continuous nondecreasing...
  • K. Eke and H. Akewe, Equivalence of Picard-type hybrid iterative algorithms for contractive mappings, Asian J. Sci. Res. 12, no. 3 (2019),...
  • I. Gosciniak and K. Gdawiec, Control of dynamics of the modified Newton-Raphson algorithm, Communications in Nonlinear Science and Numerical...
  • M. Imdad and S. Dashputre, Fixed point approximation of Picard normal S-iteration process for generalized nonexpansive mappings in hyperbolic...
  • C. O. Imoru and M. O. Olatinwo, On the stability of Picard and Mann iteration, Carpath. J. Math, 19 (2003), 155-160.
  • S. Ishikawa, Fixed points and iteration of a nonexpansive mapping in a Banach space, Proceedings of the American Mathematical Society 59,...
  • B. Kalantari, Polynomiography: from the fundamental theorem of Algebra to art, Leonardo 38, no. 3 (2005), 233-238. https://doi.org/10.1162/0024094054029010
  • B. Kalantari, Polynomial root-finding and polynomiography, World Scientific (2008). https://doi.org/10.1142/6265
  • A. R. Khan, H. Fukhar-ud-din, and M. A. Ahmad Khan, An implicit algorithm for two finite families of nonexpansive maps in hyperbolic spaces,...
  • U. Kohlenbach, Some logical metatheorems with applications in functional analysis, Transactions of the American Mathematical Society 357,...
  • P. Lamba and A. Panwar, A Picard S* iterative algorithm for approximating fixed points of generalized $ α $-nonexpansive mappings, J. Math....
  • B. Nawaz, K. Ullah, and K. Gdawiec, Convergence analysis of Picard-SP iteration process for generalized $ α $-nonexpansive mappings, Numerical...
  • M. A. Noor, New approximation schemes for general variational inequalities, Journal of Mathematical Analysis and applications 251, no. 1 (2000),...
  • A. E. Ofem, J. A. Abuchu, G. C. Ugwunnadi, O. K. Narain, H. Aydi, and C. Park, A mixed-type Picard-S iterative method for estimating common...
  • A. E. Ofem, J. A. Abuchu, G. C. Ugwunnadi, H. Isik, O. K. Narain, On a four-step iterative algorithm and its application to delay integral...
  • A. E. Ofem, H. Isik, G. C. Ugwunnadi, R. George, and O. K. Narain, Approximating the solution of a nonlinear delay integral equation by an...
  • A. E. Ofem, J. A. Abuchu, R. George, G. C. Ugwunnadi, and O. K. Narain, Some new results on convergence, weak w 2-stability and data dependence...
  • A. E. Ofem and D. Igbokwe, New faster four step iterative algorithm for Suzuki generalized nonexpansive mappings with an application, Advances...
  • R. Pant and R. Pandey, Existence and convergence results for a class of nonexpansive type mappings in hyperbolic spaces, Applied General Topology...
  • W. Phuengrattana and S. Suantai, On the rate of convergence of Mann, Ishikawa, Noor and SP-iterations for continuous functions on an arbitrary...
  • S. Reich and I. Shafrir, Nonexpansive iterations in hyperbolic spaces, Nonlinear analysis: theory, methods & applications 15, no. 6 (1990),...
  • M. Saif, B. Almarri, M. Aljuaid, and I. Uddin, Numerical solutions of boundary value problems via fixed point iteration, Computational and...
  • P. Sainuan, Rate of convergence of P-iteration and S-iteration for continuous functions on closed intervals, Thai Journal of Mathematics 13,...
  • G. S. Saluja and H. K. Nashine, Convergence of an implicit iteration process for a finite family of asymptotically quasi-nonexpansive mappings...
  • I. Timis, On the weak stability of Picard iteration for some contractive type mappings, Annals of the University of Craiova-Mathematics and...
  • T. M. Yu, A. A. Shahid, K. Shabbir, N. A. Shah, and Y. M. Li, An iteration process for a general class of contractive-like operators: Convergence,...