Approximating Common Fixed Point of Three $C$-$\alpha$ Nonexpansive Mappings

Abstract

In this paper, we consider a new class of nonlinear mappings presented in \cite{Shukla} that generalizes two well-known classes of nonexpansive type mappings and extends some other classes of mappings. We introduce approximating common fixed point of three C-$\alpha$ nonexpansive mappings through weak and strong convergence of an iterative sequence in a uniformly convex Banach space. We also numerically illustrate the common fixed point approximations of the presented iteration for the three $C$-$\alpha$ nonexpansive mappings.

Keywords

• Suzuki’s generalized nonexpansive mappings
• α−nonexpansive mappings
• Opial property
• uniformly convex Banach space
• common fixed point
• convergence theorems.

References

1. [1] J.Ali, F. Ali, F. P.Kumar, Approximation of Fixed Points for Suzuki’s Generalized Non-Expansive Mappings, Mathematics 7(6), 522 (2019), 1-11.
2. [2] D. Ariza-Ruiz, C. Hernandez Linares, E. Llorens-Fuster and E. Moreno-Galvez, On a􀀀nonexpansive mappings in Banach spaces, Carpathian J. Math. 32 (2016), 13-28.
3. [3] K. Aoyama and F. Kohsaka, Fixed point theorem for a􀀀nonexpansive mappings in Banach spaces, Nonlinear Analy. 74 (13) (2011), 4378-4391.
4. [4] A. Ekinci and S. Temir, Convergence theorems for Suzuki generalized nonexpansive mapping in Banach spaces, Tamkang Journal of Mathematics 54 (1) (2023), 57-67.
5. [5] J. Garcia-Falset, E. Llorens-Fuster, T. Suzuki, Fixed Point Theory for A Class of Generalized Nonexpansive Mappings, Journal of Mathematical Analysis and Applications 375(1), (2011), 185-195.
6. [6] G. Maniu, On a three-step iteration process for Suzuki mappings with qualitative study, Numerical Functional Analysis and Optimization, 41:8 (2020), 929-949.
7. [7] E. Naraghirad, N.-C. Wong and J.-C. Yao, Approximating fixed points of a􀀀nonexpansive mappings in uniformly convex Banach spaces and CAT(0) spaces, Fixed Point Theory and Applications 2013/1/57, (2013), 20 pages.
8. [8] M.A. Noor, New approximation schemes for general variational inequalities, Journal of Mathematical Analysis and Applications, 251 (2000), 217-229.

9. [9] Z. Opial, Weak convergence of successive approximations for nonexpansive mappings, Bull. Ame. Math.Soc. 73 (1967), 591-597.
10. [10] R. Pandey, R. Pant, W. Rakocevic, R. Shukla, Approximating Fixed Points of A General Class of Nonexpansive Mappings in Banach Spaces with Applications, Results in Mathematics, 74(7) (2019), 24 pages.
11. [11] R. Pant and R. Shukla, Approximating fixed points of generalized a-nonexpansive mappings in Banach spaces, Numer. Funct. Anal. Optim. 38(2) (2017), 248-266.
12. [12] R. Pant and R. Shukla, Fixed point theorems for a new class of nonexpansive mappings, Appl. Gen. Topol. 23(2) (2022), 377-390.
13. [13] H. Piri, B. Daraby, S. Rahrovi, M. Ghasemi, Approximating fixed points of generalized ?-nonexpansive mappings in Banach spaces by new faster iteration process, Numerical Algorithms 81 (2019),1129–1148, DOI:10.1007/s11075-018-0588-x.
14. [14] T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, Journal of Mathematical Analysis and Applications, 340(2) (2008), 1088-1095.
15. [15] S. Temir, Weak and strong convergence theorems for three Suzuki’s generalized nonexpansive mappings, Publications de l’Institut Mathematique 110 (124) (2021), 121-129.
16. [16] S. Temir and O. Korkut, Approximating fixed points of generalized a􀀀nonexpansive mapping by the new iteration process, Journal of Mathematical Sciences and Modelling 5(1) (2022), 35-39.
17. [17] S. Temir and O. Korkut, Some Convergence Results Using A New Iteration Process for Generalized Nonexpansive Mappings in Banach Spaces, Asian-European Journal of Mathematics, 16(05) 2350077 (2023).
18. [18] S. Temir, Convergence theorems for a general class of nonexpansive mappings in Banach spaces, International Journal of Nonlinear Analysis and Applications (in press).
19. [19] B.S.Thakur, D.Thakur, M.Postolache, A new iterative scheme for numerical reckoning fixed points of Suzuki’s generalized nonexpansive mappings, Applied Mathematics and Computation, 275 (2016), 147-155.
20. [20] K.Ullah and M.Arshad, Numerical reckoning fixed points for Suzuki’s generalized nonexpansive mappings via new iteration process, Filomat 32(1) (2018), 187-196.
21. [21] H. K. Xu, Inequalities in Banach spaces with applications, Nonlinear Analysis 16 (1991), 1127-1138.
22. [22] I. Yildirim, On fixed point results for mixed nonexpansive mappings, Mathematical Methods for Engineering Applications, ICMASE 2021, Salamanca, Spain, July 1–2, 2022/4/16.
23. [23] I. Yildirim, N. Karaca, Generalized (a;b)􀀀nonexpansive multivalued mappings and their properties, 1st Int. Cong. Natural Sci., (2021), 672–679.