Analyzing Stability and Data Dependence Notions by a Novel Jungck-Type Iteration Method

Abstract

Finding the ideal circumstances for a mapping to have a fixed point is the fundamental goal of fixed point theory. These criteria can also be used for the structure under investigation. One of this theory’s most well-known theorems, Banach’s fixed point theorem, has been expanded adopting various methods, making it possible to conduct numerous research studies. Thanks to the Jungck-Contraction Theorem, which has been proven through commutative mappings, many fixed point theorems have been obtained using classical fixed point iteration methods and newly defined methods. This study aims to investigate the convergence, stability, convergence rate, and data dependency of the new four-step fixed-point iteration method. Nontrivial examples are also included to support some of the results herein.

Keywords

• Jungck-contraction principle
• fixed point
• iteration method
• stability
• data dependence

References

1. A. Amini-Harandi, H. Emami, A Fixed Point Theorem for Contraction Type Maps in Partially Ordered Metric Spaces and Application to Ordinary Differential Equations, Nonlinear Analysis: Theory, Methods and Applications 72 (5) (2010) 2238--2242.
2. A. Wieczorek, Applications of Fixed-Point Theorems in Game Theory and Mathematical Economics, Wisdom Mathematics (28) (1988) 25--34.
3. 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.
4. J. Borwein, B. Sims, Fixed-Point Algorithms for Inverse Problems in Science and Engineering, Vol. 49 of The Douglas–Rachford Algorithm in the Absence of Convexity, Springer, New York, 2011, Ch. 6, pp. 93-109.
5. K. C. Border, Fixed Point Theorems with Applications to Economics and Game Theory, Cambridge University Press, Cambridge, 1989.
6. M. Chen, W. Lu, Q. Chen, K. J. Ruchala, G. H. Olivera, A Simple Fixed-Point Approach to Invert a Deformation Field, Medical Physics 35 (1) (2008) 81--88.
7. S. Banach, Sur Les Opérations Dans Les Ensembles Abstraits Et Leur Application Aux Equations Intégrales, Fundamenta Mathematicae 3 (1) (1922) 133--181.
8. 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 (2013) Article ID 560258 9 pages.

9. M. Özdemir, S. Akbulut, On the Equivalance of Some Fixed Point Iterations, Kyungpook Mathematical Journal 46 (2) (2006) 211--217.
10. V. Karakaya, Y. Atalan, K. Doğan, N. Bouzara, Some Fixed Point Results for a New Three Steps Iteration Process in Banach Spaces, Fixed Point Theory 18 (2) (2017) 625--640.
11. Y. Atalan, V. Karakaya, Investigation of Some Fixed Point Theorems in Hyperbolic Spaces for a Three Step Iteration Process, Korean Journal of Mathematics 27 (4) (2019) 929--947.
12. V. Karakaya, F. Gürsoy, K. Doğan, M. Ertürk, Data Dependence Results for Multistep and CR Iterative Schemes in the Class of Contractive-like Operators, Abstract and Applied Analysis 2013 (2013) Article ID 381980 7 pages.
13. S. Maldar, Y. Atalan, K. Doğan, Comparison Rate of Convergence and Data Dependence for a New Iteration Method, Tbilisi Mathematical Journal 13 (4) (2020) 65--79.
14. Y. Atalan, On Numerical Approach to the Rate of Convergence and Data Dependence Results for a New Iterative Scheme, Konuralp Journal of Mathematics 7 (1) (2019) 97--106.
15. S. Maldar, Y. Atalan, Common Fixed Point Theorems for Complex-Valued Mappings with Applications, Korean Journal of Mathematics 30 (2) (2022) 205--229.
16. Y. Atalan, V. Karakaya, Obtaining New Fixed Point Theorems Using Generalized Banach-Contraction Principle, Erciyes University Journal of the Institute of Science and Technology 35 (3) (2019) 34--45.
17. K. Doğan, F. Gürsoy, V. Karakaya, S. H. Khan, Some New Results on Convergence, Stability and Data Dependence in N-normed Spaces, Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 69 (1) (2020) 112--122.
18. L. J. Ciric, A Generalization of Banach's Contraction Principle, Proceedings of American Mathematical Society 45 (2) (1974) 267--273.
19. M. Edelstein, An Extension of Banach's Contraction Principle, Proceedings of the American Mathematical Society 12 (1) (1961) 7--10.
20. S. B. Presic, Sur Une Classe D' Inequations Aux Differences Finite Et. Sur La Convergence De Certaines Suites, Publications De I'institut Mathématique 5 (25) (1965) 75--78.
21. G. Jungck, Commuting Mappings and Fixed Points, American Mathematical Monthly 83 (4) (1976) 261--263.
22. E. Picard, Memoire Sur La Theorie Des Equations Aux Derivees Partielles Et La Methode Des Approximations Successives, Journal de Mathématiques Pures et Appliquées 6 (1890) 145--210.
23. R. Chugh, V. Kumar, Strong Convergence and Stability Results for Jungck-SP Iterative Scheme, International Journal of Computer Applications 36 (12) (2011) 40--46.
24. N. Hussain, V. Kumar, M. A. Kutbi, On Rate of Convergence of Jungck-type Iterative Schemes, Abstract and Applied Analysis 2013 (2013) Article ID 132626 15 pages.
25. R. Chugh, S. Kumar, On the Stability and Strong Convergence for Jungck-Agarwal et al. Iteration Procedure, International Journal of Computer Applications 64 (7) (2013) 39--44.
26. A. R. Khan, V. Kumar, N. Hussain, Analytical and Numerical Treatment of Jungck-Type Iterative Schemes, Applied Mathematics and Computation 231 (2014) 521--535.
27. W. Pheungrattana, S. Suantai, On the Rate of Convergence of Mann, Ishikawa, Noor and SP Iterations for Continuous on an Arbitrary Interval, Journal of Computational and Applied Mathematics 235 (9) (2011) 3006--3014.
28. R. Chugh, V. Kumar, S. Kumar, Strong Convergence of a New Three Step Iterative Scheme in Banach Spaces, American Journal of Computational Mathematics 2 (4) (2012) 345--357.
29. R. P. Agarwal, D. O. Regan, D. R. Sahu, Iterative Construction of Fixed Points of Nearly Asymptotically Nonexpansive Mappings, Journal of Nonlinear and Convex Analysis 8 (1) (2007) 61--79.
30. D. R. Sahu, A. Petruşel, Strong Convergence of Iterative Methods by Strictly Pseudocontractive Mappings in Banach Spaces, Nonlinear Analysis: Theory, Methods and Applications 74 (17) (2011) 6012--6023.
31. V. Berinde, Picard Iteration Converges Faster Than Mann Iteration for a Class of Quasicontractive Operators, Fixed Point Theory and Applications 2004 (2004) Article Number 716359 9 pages.
32. S. L. Singh, C. Bhatnagar, S. N. Mishra, Stability of Jungck-Type Iterative Procedures, International Journal of Mathematics and Mathematical Sciences 2005 (2005) Article ID 386375 9 pages.
33. M. Kumar, P. Kumar, S. Kumar, Common Fixed Point Theorems in Complex Valued Metric Spaces, Journal of Analysis and Number Theory 2014 (2014) Article ID 587825 7 pages.
34. V. Berinde, On a Family of First Order Difference Inequalities Used in the Iterative Approximation of Fixed Points, Creative Mathematics and Informatics 18 (2) (2009) 110--122.
35. S. M. Şoltuz, T. Grosan, Data Dependence for Ishikawa Iteration when Dealing with Contractive Like Operators, Fixed Point Theory and Applications 2008 (2008) Article Number 242916 7 pages.
36. V. Berinde, Iterative Approximation of Fixed Points, Springer-Verlag, Berlin, 2007.