Fixed points of $\rho$-nonexpansive mappings using MP iterative process

Abstract

This research article introduces a new iterative process called MP iteration and prove some convergence and approximation results for the fixed points of $\rho$-nonexpansive mappings in modular function spaces. To demonstrate that MP iterative process converges faster than some well-known existing iterative processes for $\rho$-nonexpansive mappings, we constructed some numerical examples. In the end, the concept of summably almost T-stability for MP iterative process is discussed.

Keywords

• Fixed point
• ρ-nonexpansive mappings
• MP iteration
• summably almost T-stability
• non-self mappings

References

1. [1] M. Abbas, and T. Nazir, A new faster iteration process applied to constrained minimization and feasibility problems. Mater. Vesn. 66, 223-234, (2014).
2. [2] R.P. Agarwal, D. O'Regan, and D.R. Sahu, Iterative construction of fixed points of nearly asymptotically nonexpansive mappings. J. Nonlinear Convex Anal. 8, 61-79, (2007).
3. [3] V. Berinde, Iterative approximation of fixed points. Editura Efemeride, Baia Mare, (2002).
4. [4] V. Berinde, Summable almost stability of ?xed point iteration procedures. Carpathian J. Math., 19(2), 81-88, (2003).
5. [5] B.A.B. Dehaish, and W.M. Kozlowski, Fixed point iteration processes for asymptotically pointwise nonexpansive mapping in modular function spaces. Fixed Point Theory Appl. 2012, 1-23, (2012).
6. [6] A.M. Harder, and T.L. Hicks, Stability results for fixed point iteration procedures. Math. Japon. 33, 693-706, (1988).
7. [7] S. Ishikawa, Fixed point by new iterative mathod. Proc. Am. Math. Soc. 44, 147-150, (1974).
8. [8] M.A. Khamsi, and W.M. Kozlowski, On asymptotic pointwise nonexpansive mappings in modular function spaces, J. Math. Anal. Appl., 380(2), 697-708, (2011).

9. [9] M.A. Khamsi, W.M. Kozlowski, Fixed Point Theory in Modular Function Spaces. Springer International Publishing, Switzerland, (2015).
10. [10] W.M. Kozlowski, Notes on modular function spaces II. Comment. Math. 28(1), 101-116, (1988).
11. [11] W.R. Mann, Mean value methods in iteration. Proc. Am. Math. Soc. 4, 506-510, (1953).
12. [12] J. Musielak, Orlicz spaces and modular spaces, Lecture notes in Mathematics. 1034, Springer-Verlag, (1983).
13. [13] J. Musielak, and W. Orlicz, On modular spaces. Studia Mathematica, 18(1), 591-597, (1959).
14. [14] Noor, M.A. (2000). New approximation schemes for general variational inequalities. J. Math. Anal. Appl. 251, 217-229.
15. [15] G.A. Okeke, and S.H. Khan, Approximation of fixed point of multivalued ρ-quasi-contractive mappings in modular function spaces, Arab. J. Math. Sci. 26, 75-93, (2019).
16. [16] R. Pant, P. Patel, and R. Shukla, Fixed point results for a class of nonexpansive type mappings in Banach spaces. Adv. Theory Nonlinear Anal. Appl. 5(3), 368-381, (2021).
17. [17] R. Pant, P. Patel, R. Shukla, and M. De la Sen, Fixed point theorems for nonexpansive type mappings in Banach Spaces. Symmetry, 13(4), 1-20, (2021).
18. [18] R. Pant, and R. Shukla, Some new fixed point results for nonexpansive type mappings in Banach and Hilbert spaces. Indian J. Math. 62(1), 1-20, (2020).
19. [19] A. Panwar, and Reena, Approximation of fixed points of multivalued ρ-quasi-nonexpansive mappings for newly defined hybrid iterative scheme. J. Interdisciplinary Math. 22, 593-607, (2019).
20. [20] A. Panwar, and Reena, Approximating the fixed points by a faster iterative process, AIP Conference Proceedings (accepted).
21. [21] V.K. Sahu, H.K. Pathak, and R. Tiwari, Convergence theorems for new iteration scheme and comparison results. Aligarh Bull. Math. 35, 19-42, (2016).
22. [22] Ritika and S.H. Khan, Convergence of RK-iterative process for generalized nonexpansive mappings in CAT(0) spaces, Asian-European Journal of Mathematics, 12(5), 1-13, (2019).
23. [23] H.F. Senter, and W.G. Dotson, Approximating fixed points of nonexpansive mappings, Proc. Am. Math. Sci. 44, 375-380, (1974).
24. [24] D. Thakur, B.S. Thakur, and M. Postolache, New iteration scheme for numerical reckoning fixed points of nonexpansive mapping. J. Inequal. Appl. 328, 1-15, (2014).
25. [25] D. Thakur, B.S. Thakur, B.S. and M. Postolache, New iteration scheme for approximating fixed points of nonexpansive mapping. Filomat. 30, 2711-2720, (2016).