Strong convergence results for a new class of Jungck-strictly pseudo-contractive mappings via inertial and accelerated iterative schemes
DOI:
https://doi.org/10.53704/Keywords:
coincidence point, Halpern iteration, Jungck-type iterative methods, Strong convergence, Pseudo-contractive mappings, Inertial method, Hilbert spaceAbstract
In this paper, we introduce a new class of Jungck-strictly pseudo-contractive mappings in real Hilbert spaces and investigate its relationship with the class of Jungck generalised pseudo-contractive mappings. We prove that every Jungck-strictly pseudo-contractive mapping is Jungck generalised pseudo-contractive, whereas the converse fails, thereby establishing a proper subclass inclusion. Motivated by this operator class, we study three iterative approximation methods: a Jungck–Halpern iteration, an inertial Jungck–Halpern scheme, and an accelerated two-step Jungck iteration. Under suitable assumptions on the mappings and control parameters, we establish strong convergence of the generated sequences to coincidence points. For the representative linear example, the inertial Jungck–Halpern and Jungck–Halpern schemes attained the residual tolerance 10⁻⁶ after 111 and 112 updates, respectively; the accelerated two-step scheme required 195 updates, whereas the Jungck–Mann comparison scheme did not attain the tolerance within 10,000 updates. The numerical experiment was generated directly from the displayed iterative formulas and provides reproducible evidence of convergence without asserting that any scheme is universally fastest for all admissible parameter choices. The results extend existing convergence theory for Jungck-type operators and provide implementable frameworks for coincidence-point approximation in real Hilbert spaces.
References
1. Banach S. Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fund Math. 1922;3:133-81.
2. Browder FE, Petryshyn WV. Construction of fixed points of nonlinear mappings in Hilbert space. J Math Anal Appl. 1967;20(2):197-228.
3. Browder FE, Petryshyn WV. The solution by iteration of nonlinear functional equations in Banach spaces. Bull Amer Math Soc. 1966;72(3):571-5.
4. Goebel K, Kirk WA. Topics in Metric Fixed Point Theory. Cambridge: Cambridge University Press; 1990.
5. Halpern B. Fixed points of nonexpansive maps. Bull Amer Math Soc. 1967;73(6):957-61.
6. Bauschke HH, Combettes PL. Convex Analysis and Monotone Operator Theory in Hilbert Spaces. 2nd ed. New York: Springer; 2017.
7. Mann WR. Mean value methods in iteration. Proc Amer Math Soc. 1953;4(3):506-10.
8. Xu HK. Iterative algorithms for nonlinear operators. J London Math Soc. 2002;66(1):240-56.
9. Opial Z. Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull Amer Math Soc. 1967;73(4):591-7.
10. Marino G, Xu HK. Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces. J Math Anal Appl. 2007;329(1):336-46.
11. Berinde V. Iterative Approximation of Fixed Points. Berlin: Springer; 2007.
12. Berinde V. Weak and strong convergence theorems for enriched strictly pseudocontractive operators in Hilbert spaces. arXiv preprint arXiv:1909.03492. 2019.
13. Berinde V, Khan AR, Fukhar-ud-Din H. Fixed point iterative methods defined as admissible perturbations of generalized pseudocontractive operators. J Nonlinear Convex Anal. 2015;16(3):563-72.
14. Jungck G. Commuting mappings and fixed points. Amer Math Mon. 1976;83(4):261-3.
15. Jungck G. Common fixed points for commuting and compatible maps on compacta. Proc Amer Math Soc. 1988;103(3):977-83.
16. Abbas M, Jungck G. Common fixed point results for noncommuting mappings without continuity in cone metric spaces. J Math Anal Appl. 2008;341(1):416-20.
17. Omidire OJ. Common fixed point theorems for certain generalized contractive conditions in convex metric space settings. Int J Math Sci Optim. 2024;10(3):1-9.
18. Akewe H. The stability of a modified Jungck–Mann hybrid fixed point iteration procedure. Int J Math Sci Optim Theory Appl. 2016;2(1):95-104.
19. Agarwal RP, O'Regan D, Sahu DR. Fixed Point Theory for Lipschitzian-Type Mappings. New York: Springer; 2009.
20. Olatinwo MO, Omidire OJ. Some new convergence and stability results for Jungck generalized pseudo-contractive and Lipschitzian type operators. Rend Circ Mat Palermo (2). 2023;72(2):1067-86.
21. Berinde V, Berinde M. The fastest Krasnoselskij iteration for approximating fixed points of strictly pseudo-contractive mappings. Carpathian J Math. 2005;21(1-2):13-20.
22. Omidire O. J, Ansri A. H., Ariyo R. D., Aduragbemi, M.; Approximating Fixed point of generalized C-class contractivity conditions. Int J Math Sci Optim. 2025;11(1):1-10.
23. Maingé PE. Convergence theorems for inertial KM-type algorithms. J Comput Appl Math. 2008;219(1):223-36.
24. Alvarez F, Attouch H. An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping. Set-Valued Anal. 2001;9(1-2):3-11.
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Copyright (c) 2026 Olaoluwa Omidire, Femi Samuel Adeyinka, Babatunde Ishola, Memudu Olatinwo, Kamiludeen Tijani, Dare Ariyo
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