Common Attractive Point Theorems for a Finite Family of Multivalued Nonexpansive Mappings in Banach Spaces

Year 2022, Volume: 5 Issue: 3, 372 - 386, 30.09.2022

Atit Wiriyapongsanon Warunun Inthakon Narawadee Phudolsitthiphat

https://doi.org/10.53006/rna.1128729

Abstract

OurmainpurposeofthispaperistointroducethemodifiedMannandIshikawaiteratesforfindingacommonattractive
point of a finite family of multivalued nonexpansive mappings in the setting of uniformly convex Banach spaces. We
obtain necessary and sufficient conditions to guarantee the strong convergence of the proposed algorithms without
closedness of the domain of such mappings. Moreover, we derive some consequences from our main result to fixed
point result of such mappings. Finally, the numerical results are provided to support our main theorem.

Keywords

common attractive point , multivalued nonexpansive mapping , uniformly convex Banach space , demicompact

References

• 1] W. Inthakon, Strong convergence theorems for generalized nonexpansive mappings with the system of equilibrium problems in Banach spaces, Journal of Nonlinear and Convex Analysis, 2014, 15(4), 753–763.
• [2] E. Karapınar, A. Fulga, M. Rashid, L. Shahid, H. Aydi, Large contractions on quasi-metric spaces with an application to nonlinear fractional differential equations, Mathematics 2019, 7, 444;https://doi:10.3390/math7050444.
• [3] W. Inthakon, S. Suantai, P. Sarnmeta, D. Chumpungam, A new machine learning algorithm based on optimization method for regression and classification problems, Mathematics, 2020, 8(6), 1007; https://doi.org/10.3390/math8061007.
• [4] S.A.R. Hosseiniun, M. Nabiei, Some applications of fixed point theorem in economics and nonlinear functional analysis, International Mathematical Forum, 2010, 5(49), 2407–2414.
• [5] R.S. Adiguzel, U. Aksoy, E. Karapınar, I.M. Erhan, On the solution of a boundary value problem associated with a fractional differential equation, Mathematical Methods in the Applied Sciences, 2020, https://doi.org/10.1002/mma.6652.
• [6] N. Nanan, S. Dhompongsa, A common fixed point theorem for a commuting family of nonexpansive mappings one of which is multival- ued, Fixed Point Theory and Applications, 2011, 54, 1–10.
• [7] S. Dhompongsa, N. Nanan, Fixed point theorems by ways of ultra-asymptotic centers, Abstract and Applied Analysis, 2011, https://doi.org/10.1155/2011/826851.
• [8] C. Jinakul, A. Wiwatwanich, A. Kaewkhao, Common fixed point theorem for multi-valued mappings on b-metric spaces, International Journal of Pure and Applied Mathematics, 2017, 113(1), 167–179.
• [9] N. Phudolsitthiphat, P. Charoensawan, Common fixed point results for three maps one of which is multivalued in G-metric spaces, Thai Journal of Mathematics, 2018, 16(2), 455–465.
• [10] W. Takahashi, N.C. Wong, J.C. Yao, Attractive point and mean convergence theorems for new generalized nonspreading mappings in banach spaces, Infinite Products of Operators and their Applications, 2015, 636, 225–248.
• [11] S. Dhompongsa, A. Kaewkhao, A note on properties that imply the fixed point property, Abstract and Applied Analysis, 2006, https://doi.org/10.1155/AAA/2006/34959.
• [12] S. Dhompongsa, A. Kaewcharoen, A. Kaewkhao, Fixed point property of direct sums, Nonlinear Analysis, 2005, 63, 2177–2188.
• [13] H. Aydi, M.F. Bota, E. Karapınar, S. Mitrovic, A fixed point theorem for set-valued quasi-contractions in b-metric spaces, Fixed Point Theory and Applications, 2012, 2012(1), 1–8.
• [14] Panyanak, B. Mann and Ishikawa iterative processes for multivalued mappings in Banach spaces. Computers & Mathematics with Appli- cations 2007, 54(6), 872–877.
• [15] S.H. Khan, I. Yildirim, Fixed points of multivalued nonexpansive mappings in Banach spaces, Fixed Point Theory and Applications, 2012, 73, 1–9.
• [16] P. Sunthrayuth, P. Kumam, A new composite general iterative scheme for nonexpansive semigroups in banach spaces, International Journal of Mathematics and Mathematical Sciences, 2011, 2011, 1–18.
• [17] W. Takahashi, N.C. Wong, J.C. Yao, Attractive point and weak convergence theorems for new generalized hybrid mappings in Hilbert spaces, Journal of Nonlinear and Convex Analysis, 2012, 13, 745–757.
• [18] P. Sunthrayuth, P. Kumam, Fixed point solutions of variational inequalities for a semigroup of asymptotically nonexpansive mappings in Banach spaces, Fixed Point Theory and Applications, 2012, 177, 1–18.
• [19] Y. Song, H. Wang, Convergence of iterative algorithms for multivalued mappings in Banach spaces, Nonlinear Analysis, 2009, 70(4), 1547–1556.
• [20] Chen, L.; Yang, N.; Zhou, J. Common attractive points of generalized hybrid multi-valued mappings and applications. Mathematics 2020, 8(8), 1307; https://doi.org/10.3390/math8081307.
• [21] N. Shahzad, H. Zegeye, On Mann and Ishikawa iteration schemes for multi-valued maps in Banach spaces, Nonlinear Analysis, 2009, 71, 838–844.
• [22] K.P.R. Sastry, G.V.R. Babu, Convergence of Ishikawa iterates for a multivalued mapping with a fixed point, Czechoslovak Mathematical Journal, 2005, 55, 817–826.
• [23] W. Takahashi, Y. Takeuchi, Nonlinear ergodic theorem without convexity for generalized hybrid mappings in a Hilbert space, Journal of Nonlinear and Convex Analysis, 2011, 12(2), 399–406.
• [24] P. Kocourek, W. Takahashi, J.C. Yao, Fixed point theorems and weak convergence theorems for generalized hybrid mappings in Hilbert spaces, Taiwanese Journal of Mathematics, 2010,14(6), 2497–2511.
• [25] M. Farid, Weak convergence to common attractive points of finite families of nonexpansive mappings, JMI International Journal of Mathematical Sciences, 2018, 9, 41–50.
• [26] S.H. Khan, Iterative approximation of common attractive points of further generalized hybrid mappings, Fixed Point Theory and Appli- cations, 2018, 8, 1–10.
• [27] P. Thongphaen, W. Inthakon, Common attractive point theorems of widely more generalized hybrid mappings in Hilbert spaces, Thai Journal of Mathematics, 2020,18(3), 861–869.
• [28] Thongphaen, C.; Inthakon, W.; Suantai, S.; Phudolsitthiphat, N. Common attractive point results for two generalized nonexpansive mappings in uniformly convex Banach spaces. Mathematics 2022, 10, 1275;https://doi.org/10.3390/math10081275.
• [29] S.B. Nadler, Multivalued contraction mappings, Pacific Journal of Math, 1969 30, 475–488.
• [30] J, Schu, Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bulletin of the Australian Mathematical Society, 1991, 43(1), 153–159.
• [31] S. Plubtieng, K. Ungchittrakool, Hybrid iterative methods for convex feasibility problems and fixed point problems of relatively non- expansive mappings in Banach spaces, Fixed Point Theory and Applications, 2008 82, 1–19.
• [32] S.S. Zhang, Generalized mixed equilibrium problem in Banach spaces, Applied Mathematics and Mechanics, 2009 30, 1105–1112.