Results of Convergence, Stability, and Data Dependency for an Iterative Algorithm

Abstract

In this study, we reconstruct an existing result related to the strong convergence of a recently introduced iterative algorithm by removing certain restrictions on the coefficient sequences. We then obtain some new results on the stability and data dependency of this algorithm. To validate our results, we provide a series of nontrivial complex examples, demonstrating the significance and accuracy of our theoretical contributions.

Keywords

• Data dependency
• fixed point
• iterative algorithm
• stability
• strong convergence

References

1. K. C. Border, Fixed point theorems with applications to economics and game theory, Cambridge University Press, Cambridge, 1989.
2. L. C. Ceng, Q. Ansari, J. C. Yao, Some iterative methods for finding fixed points and for solving constrained convex minimization problems, Nonlinear Analysis: Theory, Methods and Applications 74 (2011) 5286-5302.
3. W. R. Mann, Mean value methods in iteration, Proceedings of the American Mathematical Society 4 (3) (1953) 506-510.
4. S. Ishikawa, Fixed points by a new iteration method, Proceedings of the American Mathematical Society 44 (1) (1974) 147-150.
5. S. Thianwan, Common fixed points of new iterations for two asymptotically nonexpansive nonself-mappings in a Banach space, Journal of Computational and Applied Mathematics 224 (2) (2009) 688-695.
6. W. Phuengrattana, S. Suantai, On the rate of convergence of Mann, Ishikawa, Noor and SP-iterations for continuous functions on an arbitrary interval, Journal of Computational and Applied Mathematics 235 (9) (2011) 3006-3014.
7. S. Maldar, Y. Atalan, K. Doğan Comparison rate of convergence and data dependence for a new iteration method, Tbilisi Centre for Mathematical Sciences 13 (4) (2020) 65-79.
8. E. Hacıoğlu, A comparative study on iterative algorithms of almost contractions in the context of convergence, stability and data dependency, Computational and Applied Mathematics 40 (8) (2021) 282 25 pages.

9. J. Ali, M. Jubair, Fixed points theorems for enriched non-expansive mappings in geodesic spaces, Filomat 37 (11) (2023) 3403-3409.
10. K. Ullah, J. Ahmad, A. B. Khan, On multi-valued version of M-iteration process, Asian-European Journal of Mathematics 16 (02) (2023) 2350017 13 pages.
11. H. Fan, C. Wang, Stability and convergence rate of Jungck-type iterations for a pair of strongly demicontractive mappings in Hilbert spaces, Computational and Applied Mathematics 42 (1) (2023) 33 17 pages.
12. A. Keten Çopur, E. Hacıoğlu, F. Gürsoy, New insights on a pair of quasi-contractive operators in Banach spaces: Results on Jungck type iteration algorithms and proposed open problems, Mathematics and Computers in Simulation 215 (2024) 476-497.
13. S. S. Chauhan, N. Kumar, M. Imdad, M. Asim, New fixed point iteration and its rate of convergence, Optimization 72 (9) (2023) 2415-2432.
14. V. Karakaya, K. Doğan, F. Gürsoy, M. Ertürk, Fixed point of a new three-step iteration algorithm under contractive-like operators over normed spaces, Abstract and Applied Analysis 2013 (1) (2013) 560258 9 pages.
15. M. O. Osilike, Stability results for fixed point iteration procedures, Journal of the Nigerian Mathematical Society 14 (15) (1995/1996) 17-29.
16. C. O. Imoru, M. O. Olatinwo, On the stability of Picard and Mann iteration processes, Carpathian Journal of Mathematics 19 (2) (2003) 155-160.
17. A. O. Bosede, B. E. Rhoades, Stability of Picard and Mann iteration for a general class of functions, Journal of Advanced Mathematical Studies 3 (2) (2010) 23-26.
18. V. Berinde, Iterative approximation of fixed points, Springer, Berlin, 2007.
19. L. Qihou, A convergence theorem of the sequence of Ishikawa iterates for quasi-contractive mappings, Journal of Mathematical Analysis and Applications 146 (2) (1990) 301-305.
20. J. Lindenstrauss, L. Tzafriri, Classical Banach spaces I: sequence spaces, Springer, Berlin, 1977.
21. Y. S. Choi, J. M. Kim, The dual space of $(L(X,Y),\tau_p)$ and the $p$-approximation property, Journal of Functional Analysis 259 (2010) 2437-2454.
22. A. Grothendieck, Produits tensoriels topologiques et espaces nucleaires (in French), No. 16 of Memoirs of the American Mathematical Society, Providence, 1955.