A New Iterative Scheme for Approximating Fixed Points of Suzuki Generalized Multivalued Nonexpansive Mappings

Abstract

In this paper, we study to approximate fixed points of Suzuki generalized multivalued nonexpansive mappings by using a three-step iterative scheme (1.1) introduced in [17]. We establish some weak and strong convergence results for mappings satisfying condition (C) with the newly proposed iterative scheme in the framework of uniformly convex real Banach spaces.

Keywords

• Multivalued mappings
• generalized multivalued nonexpansive mappings
• strong and weak convergence results
• fixed point

References

1. [1] Nadler, S.B., "Multivalued contraction mappings", Pacific J. Math. 30(2) (1969) : 475-488.
2. [2] Markin, J.T., "A fixed point theorem for set valued mappings", Bull. Amer. Math. Soc. 74(4) (1968) : 545-547.
3. [3] Abkar, A., Eslamian, M., "Fixed point theorems for Suzuki generalized nonexpansive multivalued mappings in Banach spaces", Fixed Point Theory and Applications 2010 (2010) : 1-10.
4. [4] Yildirim, I., "On convergence of an implicit algorithm for multivalued mappings in Banach spaces", Miskolc Mathematical Notes 15(2) (2014) : 771-780.
5. [5] Thakur, B.S., Thakur, D., Postolache, M., "A new iterative scheme for numerical reckoning fixed points of Suzuki's generalized nonexpansive mappings", Applied Mathematics and Computation 275 (2016) : 147-155.
6. [6] Opial,Z., "Weak convergence of the sequence of successive approximations for nonexpansive mappings", Bull Amer. Math. Soc. 73 (1967) : 591-597.
7. [7] Eslamian, M., Abkar, A., "One-step iterative process for a finite family of multivalued mappings", Mathematical and Computer Modelling, 54 (2011) : 105-111.
8. [8] Garcia-Falset, J., Lorens-Fusters, E., Suzuki, T., "Fixed point theory for a class of generalized nonexpansive mappings", J.Math. Anal. Appl. 375 (2011) : 185-195.

9. [9] Kaewchareon, A., Panyanak, B., "Fixed point theorems for some generalized multivalued nonexpansive mappings", Nonlinear Analysis, 74 (2011) : 5578-5584.
10. [10] Ali, J., Ali, F., Kumar, P., "Approximation of fixed points for Suzuki's generalized nonexpansive mappings", Mathematics 7(6) (2019) : 522.
11. [11] Yildirim, I., "Strong and weak convergence of an iterative process for a finite family of multivalued mappings satisfying the condition (C)", Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 66 (2017) : 51-65.
12. [12] Phuengrattana, W., "Approximating fixed points of Suzuki-generalized nonexpansive mappings", Nonlinear Anal. Hybrid Syst. 5(3) (2011) : 583-590.
13. [13] Akkasriworn, N., Sokhuma, K., Chuikamwong, K., "Ishikawa iterative process for a pair of Suzuki generalized nonexpansive single valued and multivalued mappings in Banach spaces , Int. J. Math. Anal. 6(19) (2012) : 923-932.
14. [14] Schu, J., "Weak and strong convergence to fixed points of asymptotically nonexpansive mappings", Bull. Aust. Math. Soc. 43 (1991) : 153-159.
15. [15] Suzuki, T., "Fixed point theorems and convergence theorems for some generalized nonexpansive mappings", J. Math. Anal. Appl. 340(2) (2008) : 1088-1095.
16. [16] Abkar, A., Eslamian, M., "Convergence theorems for a finite family of generalized nonexpansive multivalued mappings in CAT (0) spaces" Nonlinear Analysis: Theory, Methods & Applications 75(4) (2012) : 1895-1903.
17. [17] Kaplan, M., Kopuzlu, A., "Three-step iterative scheme for Approximating fixed points of multivalued nonexpansive mappings", Advances in Fixed Point Theory 3(2) (2013) : 273-285.
18. [18] Sadhu, R., Majee, P., Nahak, C., "Fixed point theorems on generalized -nonexpansive multivalued mappings", The Journal of Analysis 29 (2021) : 1165-1190.
19. [19] Ullah, K., Arshad, M., "Numerical reckoning fixed points for Suzuki’s generalized nonexpansive mappings via new iteration process", Filomat, 32(1) (2018) : 187-196.
20. [20] Khan, S. H., Yildirim, I., "Fixed points of multivalued nonexpansive mappings in Banach spaces", Fixed Point Theory Appl. (2012) : 73.