Volume 19, Issue 1 (4-2024)                   IJMSI 2024, 19(1): 135-148 | Back to browse issues page

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Ullah K, Abbas M, Ahmad J, Ahmad F. Approximating Fixed Points of Operators Satisfying the ($B_{gamma,mu}$) Condition. IJMSI 2024; 19 (1) :135-148
URL: http://ijmsi.ir/article-1-1714-en.html

Suppose C is any nonempty subset of a Banach space X. A mapping T : C → C is said to satisfy condition (Bγ,µ) if there exists γ ∈ [0, 1] and µ ∈ [0, 1 /2 ] with 2µ ≤ γ such that for each two elements x, y ∈ C,

γ||x - T x|| ≤ ||x - y|| + µ||y - T y||

implies ||T x - T y|| ≤ (1 - γ)||x - y|| + µ(||x - T y|| + ||y - T x||).

In this research, we suggest some convergence results for these mappings under a up-to-date iterative process in a Banach space setting. Our results are new and improve some known results of the literature.

Type of Study: Research paper | Subject: General

1. M. Abbas, T. Nazir, A New Faster Iteration Process Applied to Constrained Minimization and Feasibility Problems, Mat. Vesnik, 66(2), (2006), 223-234.
2. R. P. Agarwal, D. O'Regan, D. R. Sahu, Fixed Point Theory for Lipschitzian Type Mappings with Applications, Series. Topological Fixed Point Theory and Its Applications, vol. 6. Springer, New York, 2009. [DOI:10.1155/2009/439176]
3. R. P. Agarwal, D. O'Regan, D. R. Sahu, Iterative Construction of Fixed Points of Nearly Asymptotically Nonexpansive Mappings, J. Nonlinear Convex Anal., 8(1), (2007), 61-79.
4. H. H. Bauschke, P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer, New York, 2011. [DOI:10.1007/978-1-4419-9467-7]
5. F. E. Browder, Nonexpansive Nonlinear Operators in a Banach Space, Proc. Nat. Acad. Sci. USA., 54, (1965), 1041-1044. [DOI:10.1073/pnas.54.4.1041]
6. R. Chugh, V. Kumar, S. Kumar, Strong Convergence of a New Three Step Iterative Scheme in Banach Spaces, Am. J. Comp. Math., 2, (2012), 345-357. [DOI:10.4236/ajcm.2012.24048]
7. J. A. Clarkson, Uniformly Convex Spaces, Trans. Am. Math. Soc., 40, (1936), 396-414. [DOI:10.1090/S0002-9947-1936-1501880-4]
8. M. Deepmala, L. N. Jain, V. N. Mishra, A Note on the Paper "Hu et al., Common Coupled Fixed Point Theorems for Weakly Compatible Mappings in Fuzzy Metric Spaces, Fixed Point Theory and Applications 2013, 2013:220", Int. J. Adv. Appl. Math. and Mech., 5(2), (2017), 51-52.
9. D. Gohde, Zum Prinzip Der Kontraktiven Abbildung, Math. Nachr., 30, (1965), 251-258. [DOI:10.1002/mana.19650300312]
10. F. Gursoy, V. Karakaya, A Picard-S Hybrid Type Iteration Method for Solving a Differential Equation with Retarded Argument, arXiv:1403.2546v2, (2014).
11. S. Ishikawa, Fixed Points by a New Iteration Method, Proc. Am. Math. Soc., 44, (1974), 147-150. [DOI:10.1090/S0002-9939-1974-0336469-5]
12. I. Karahan, M. Ozdemir, A General Iterative Method for Approximation of Fixed Points and their Applications, Adv. Fixed Point Theory, 3, (2013), 510-526. [DOI:10.1186/1687-1812-2013-244]
13. S. H. Khan, S.H, A Picard-Mann Hybrid Iterative Process, Fixed Point Theory Appl., (2013). https://doi.org/10.1186/1687-1812-2013-69 [DOI:10.1186/1687-1812-2013-69.]
14. W. A. Kirk, Fixed Point Theorem for Mappings which Do Not Increase Distance, Am. Math. Monthly, 72, (1965), 1004-1006. [DOI:10.2307/2313345]
15. W. R. Mann, Mean Value Methods in Iterations, Proc. Am. Math. Soc., 4, (1953), 506-510. [DOI:10.1090/S0002-9939-1953-0054846-3]
16. L. N. Mishra, V. Dewangan, V. N. Mishra, H. Amrulloh, Coupled Best Proximity Point Theorems for Mixed mathrmg-Monotone Mappings in Partially Ordered Metric Spaces, J. Math. Comput. Sci., 11(5), (2012), 6168-6192.
17. L. N. Mishra, V. Dewangan, V. N. Mishra, S. Karateke, Best Proximity Points of Admissible Almost Generalized Weakly Contractive Mappings with Rational Expressions on b-Metric Spaces, J. Math. Computer Sci., 22(2), (2021), 97-109. [DOI:10.22436/jmcs.022.02.01]
18. M. N. Noor, New Approximation Schemes for General Variational Inequalities, J. Math. Anal. Appl., 251(1), (2000), 217-229. [DOI:10.1006/jmaa.2000.7042]
19. Z. Opial, Weak and Strong Convergence of the Sequence of Successive Approximations for Nonexpansive Mappings, Bull. Am. Math. Soc., 73, (1967), 591-597. [DOI:10.1090/S0002-9904-1967-11761-0]
20. B. Patir, N. Goswami, V. N. Mishra, Some Results on Fixed Point Theory for a Class of Generalized Nonexpansive Mappings, Fixed Point Theory Appl., (2018). https://doi.org/10.1186/s13663-018-0644-1 [DOI:10.1186/s13663-018-0644-1.]
21. W. Phuengrattana, S. Suantai, On the Rate of Convergence of Mann, Ishikawa, Noor and SP Iterations for Continuous Functions on an Arbitrary Interval, J. Comp. App. Math., 235, (2011), 3006-3014. [DOI:10.1016/j.cam.2010.12.022]
22. D. R. Sahu, A. Petrusel, Strong Convergence of Iterative Methods by Strictly Pseudocontractive Mappings in Banach spaces, Nonlinear Anal. Theory, Methods Applications, 74, (2011), 6012-6023. [DOI:10.1016/j.na.2011.05.078]
23. J. Schu, Weak and Strong Convergence to Fixed Points of Asymtotically Nonexpansive Mappings, Bull. Austral. Math. Soc., 43, (1991), 153-159. [DOI:10.1017/S0004972700028884]
24. H. F. Sentor, W. G. Dotson, Approximating Fixed Points of Nonexpansive Mappings, Proc. Am. Math. Soc., 44, (1974), 375-380. [DOI:10.1090/S0002-9939-1974-0346608-8]
25. A. G. Sanatee, L. Rathour, V. N. Mishra, V. Dewangan, Some Fixed Point Theorems in Regular Modular Metric Spaces and Application to Caratheodory's Type Anti-Periodic Boundary Value Problem, The Journal of Analysis, (2022), DOI: https://doi.org/10.1007/s41478-022-00469-z [DOI:10.1007/s41478-022-00469-z.]
26. P. Shahi, L. Rathour, V. N. Mishra, Expansive Fixed Point Theorems for TriSimulation Functions, The Journal of Engineering and Exact Sci., 8(3), (2022), 1-8. DOI: https://doi.org/10.18540/jcecvl8iss3pp14303-01e [DOI:10.18540/jcecvl8iss3pp14303-01e.]
27. N. Sharma, L. N. Mishra, V. N. Mishra, H. Almusawa, Endpoint Approximation of Standard Three Step Multi-Valued Iteration Algorithm for Nonexpansive Mappings, Applied Mathematics and Information Sciences, 15(1), (2021), 73-81. [DOI:10.18576/amis/150109]
28. N. Sharma, L. N. Mishra, V. N. Mishra, S. Pandey, Solution of Delay Differential equation Via Nv 1 Iteration Algorithm, European J. Pure Appl. Math., 13(5), (2020), 1110-1130. [DOI:10.29020/nybg.ejpam.v13i5.3756]
29. N. Sharma, L. N. Mishra, S. N. Mishra, V. N. Mishra, Empirical Study of New Iterative Algorithm for Generalized Nonexpansive Operators, Journal of Mathematics and Computer Science, 25(3), (2022), 284-295. [DOI:10.22436/jmcs.025.03.07]
30. T. Suzuki, Fixed Point Theorems and Convergence Theorems for Some Generalized Nonexpansive Mappings, J. Math. Anal. Appl., 340, (2008), 1088-1095. [DOI:10.1016/j.jmaa.2007.09.023]
31. W. Takahashi, Nonlinear Functional Analysis, Yokohoma Publishers, Yokohoma, 2000.
32. B. S. Thakur, D. Thakur, M. Postolache, A New Iterative Scheme for Numerical Reckoning Fixed Points of Suzuki's Generalized Nonexpansive Mappings, App. Math. Comp., 275, (2016), 147-155. [DOI:10.1016/j.amc.2015.11.065]
33. K. Ullah, J. Ahmad, Iterative Approximation of Fixed Points for Operators Satisfying (Bγ,µ) Condition, Fixed Point Theory, 32(1), (2020), 187-196 (2021)
34. K. Ullah, M. Arshad, M, New Three Step Iteration Process and Fixed point Approximation in Banach Spaces, Journal of Linear and Topological Algebra, 7(2), (2018), 87-100.
35. K. Ullah, M. Arshad, Numerical Reckoning Fixed Points For Suzuki's Generalized Nonexpansive Mappings Via New Iteration Process, Filomat, 32(1), (2018), 187-196. [DOI:10.2298/FIL1801187U]

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