The New Iterative Approximating of Endpoints of Multivalued Nonexpansive Mappings in Banach Spaces

Year 2022, Volume: 5 Issue: 2, 64 - 68, 30.06.2022

Makbule Kaplan

https://doi.org/10.32323/ujma.1004517

Abstract

The purpose of this paper is to introduce the new iteration process to approximate endpoints of multivalued nonexpansive mappings in Banach space. We prove weak and strong convergence theorems of proposed iterative scheme under some suitable assumptions in the framework of a uniformly convex Banach space.

Keywords

Multivalued mappings;, endpoint, strong and weak convergence

References

• [1] J. P. Aubin, J. Siegel, Fixed points and stationary points of dissipative multivalued maps, Proc. Am. Math. Soc., 78(3) (1980), 391-398.
• [2] H. W. Corley, Some hybrid fixed point theorems related to optimization, J. Math. Anal. Appl., 120(2) (1986), 528-532.
• [3] B. Panyanak, Approximating endpoints of multi-valued nonexpansive mappings in Banach spaces, J. Fixed Point Theory Appl., 20(2) (2018), Article ID: 77, 8 pages, doi:10.1007/s11784-018-0564-z.
• [4] T. Laokul, Browder’s convergence theorem for multivalued mappings in Banach spaces without the endpoint condition, Abstract and Applied Analysis, 2020 (2020), Article ID: 6150398, 7 pages, doi:10.1155/2020/6150398.
• [5] T. Abdeljawad, K. Ullah, J. Ahmad, N. Mlaiki, Iterative approximation of endpoints for multivalued mappings in Banach spaces, Journal of Function Spaces, 2020 (2020), Article ID: 2179059, doi:10.1155/2020/2179059.
• [6] K. Ullah, J. Ahmad, M. Arshad, M. Sen, M. S. U. Khan, Approximating stationary point of multivalued generalized nonexpansive mappings in Banach spaces, Advances in Mathematical Physics, 2020 (2020), Article ID: 9086078, 6 pages, doi:10.1155/2020/9086078.
• [7] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc., 73(4)(1967), 591-597.
• [8] H. K. Xu, Inequalities in Banach spaces with applications, Nonlinear Analysis: Theory, Methods & Applications, 16(12) (1991), 1127-1138, doi:10.1016/0362-546X(91)90200-K.
• [9] B. Panyanak, Mann and Ishikawa iterative processes for multivalued mappings in Banach spaces, Comp. Math. Appl., 54(6) (2007), 872-877 doi:10.1016/j.camwa.2007.03.012.
• [10] B. Panyanak, Endpoints of multivalued nonexpansive mappings in geodesic spaces, Fixed Point Theory Appl., 2015, Article ID: 147, 11 pages, doi:10.1186/s13663-015-0398-y.
• [11] B. Panyanak, The demiclosed principle for multi-valued nonexpansive mappings in Banach spaces. J. Nonlinear Convex Anal., 17(10) (2016), 2063-2070.
• [12] P. Chuadchawna, A Farajzadeh, A. Kaewcharoen, Convergence theorems and approximating endpoints for multivalued Suzuki mappings in hyperbolics spaces, J. Comp. Anal. Appl., 28(5) (2020), 903-916.