Abstract
In this paper, a new pair of nonself mappings is first introduced. And then, we study the stability and convergence rate of Jungck-type iterations for the mappings. Some numerical examples are also given to support the results. Our results generalize and improve some known results in the literature.
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References
Abbas M, Nazir T (2014) A new faster iteration process applied to constrained minimization and feasibility problems. Mat Vesn 66(2):223–234
Babu GVR, Vara Prasad KNVV (2006) Mann iteration converges faster than Ishikawa iteration for the class of Zamfirescu operators. Fixed Point Theory Appl 1–6. https://doi.org/10.1155/FPTA/2006/49615
Berinde V (2004) Picard iteration converges faster than Mann iteration for a class of quasi-contractive operators. Fixed Point Theory Appl 2:97–105. https://doi.org/10.1155/S1687182004311058
Berinde V (2007) Iterative approximation of fixed points. Springer Verlag, Berlin
Berinde V (2016) On a notion of rapidity of convergence used in the study of fixed point iterative methods. Creat Math Inf 25(1):29–40
Chugh R, Kumar V (2011) Strong convergence and stability results for Jungck-SP iterative scheme. Int J Comput Appl 36(12):40–46
Fathollahi S, Ghiura A, Postolache M, Rezapour S (2015) A comparative study on the convergence rate of some iteration methods involving contractive mappings. Fixed Point Theory Appl 2015:1–24. https://doi.org/10.1186/s13663-015-0490-3
Gürsoy F, Khan AR, Ertürk M, Karakaya V (2019) Weak \(w^2\)-stability and data dependence of Mann iteration method in Hilbert spaces. Rev Real Acad Cienc Exactas Fis Nat Ser Mat 113:11–20
Hacioğlu E (2021) A comparative study on iterative algorithms of almost contractions in the context of convergence, stability and data dependency. Comput Appl Math 40:1–25
Hussain N, Kumar V, Kutbi MA (2013) On rate of convergence of Jungck-Type iterative schemes. Abstr Appl Anal 2013:1–15. https://doi.org/10.1155/2013/132626
Ishikawa S (1974) Fixed points by a new iteration method. Proc Am Math Soc 44(1):147–150
Jungck G (1976) Commuting mappings and fixed points. Am Math Mon 83(4):261–263
Khan AR, Kumar V, Hussain N (2014) Analytical and numerical treatment of Jungck-type iterative schemes. Appl Math Comput 231:521–525
Khan AR, Gürsoy F, Kumar V (2016) Stability and data dependence results for the Jungck-Khan iterative scheme. Turk J Math 40:631–640
Kumar N, Chauhan SS (2018) Analysis of Jungck-Mann and Jungck-Ishikawa iteration schemes for their speed of convergence. AIP conference Proceedings 2050(1):020011
Mann WR (1953) Mean value methods in iteration. Proc Am Math Soc 4(6):506–510
Maruster L, St Maruster (2015) On the error estimation and T-stability of the Mann iteration. J Comput Appl Math 276:110–116
Noor MA (2000) New approximation schemes for general variational inequalities. J Math Anal Appl 251(1):217–229
Olatinwo MO (2008) Some stability and strong convergence results for the Jungck-Ishikawa iteration process. Creative Math Inf 17:33–42
Olatinwo MO (2008) A generalization of some convergence results using a Jungck-Noor three-step iteration process in arbitrary Banach space. Polytech Posnan 40(40):37–43
Phuengrattana W, Suantai S (2011) On the rate of convergence of Mann, Ishikawa, Noor and SP-iterations for continuous functions on an arbitrary interval. J Comput Appl Math 235:3006–3014
Rhoades BE (1976) Comments on two fixed point iteration methods. J Math Anal Appl 56(3):741–750
St Maruster (1977) The solution by iteration of nonlinear equations in Hilbert spaces. Proc Am Math Soc 63:69–73
Singh SL, Bhatnagar C, Mishra SN (2005) Stability of Jungck type iterative procedures. Int J Math Math Sci 19:3035–3043
Wang C (2015) A note on the error estimation of the Mann iteration. J Comput Appl Math 285:226–229
Wang C, Li XL (2022) Fixed point theorems in generalized convex metric space and an application to the solution of Volterra integral equations. J Integr Eq Appl 34(2):257–265
Wang C, Li XL, Huang PK (2019) On the error estimation and T-stability of the Ishikawa iteration for strongly demicontractive mappings. J Inequal Appl 75:1–12
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This paper is supported by the National Natural Science Foundation of China(10671182).
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Fan, H., Wang, C. Stability and convergence rate of Jungck-type iterations for a pair of strongly demicontractive mappings in Hilbert spaces. Comp. Appl. Math. 42, 33 (2023). https://doi.org/10.1007/s40314-022-02168-8
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DOI: https://doi.org/10.1007/s40314-022-02168-8