Some new fixed point results for monotone enriched nonexpansive mappings in ordered Banach spaces

Yıl 2021, , 559 - 567, 30.12.2021

Rahul Shukla , Rajendra Pant

https://doi.org/10.31197/atnaa.954446

Cited By: 4

Öz

We study monotone enriched nonexpansive mappings and present some new existence and convergence theorems for these mappings in the setting of ordered Banach spaces. More precisely, we employ the Krasnosel'skii iterative method to approximate fixed points of enriched nonexpansive under different conditions. This way a number of results from the literature have been extended, generalized and complemented.

Anahtar Kelimeler

Nonexpnasive mapping, Enriched nonexpansive mapping, Banach space

Kaynakça

• [1] M.R. Alfuraidan, M.A. Khamsi, A fixed point theorem for monotone asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 146 (2018), no. 6, 2451-2456.
• [2] M.R. Alfuraidan, M.A. Khamsi, Fixed point theorems and convergence theorems for some monotone generalized nonex- pansive mappings, Carpathian J. Math. 36 (2020), no. 2, 199-204.
• [3] V. Berinde, Approximating fixed points of enriched nonexpansive mappings by Krasnoselskij iteration in Hilbert spaces, Carpathian J. Math. 35 (2019), no. 3, 293-304.
• [4] V. Berinde, Approximating fixed points of enriched nonexpansive mappings in Banach spaces by using a retraction- displacement condition. Carpathian J. Math. 36 (2020), no. 1, 27-34.
• [5] B.A. Bin Dehaish, M.A. Khamsi, Mann iteration process for monotone nonexpansive mappings, Fixed Point Theory Appl. 2015 (2015), 177, 1-7.
• [6] B.A. Bin Dehaish, M.A. Khamsi, Browder and Göhde fixed point theorem for monotone nonexpansive mappings, Fixed Point Theory Appl. 2016 (2016), Paper No. 20, 1-9.
• [7] F.E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. U.S.A. 54 (1965), 1041-1044.
• [8] S. Carl, S. Heikkilä, Fixed point theory in ordered sets and applications: from differential and integral equations to game theory, Springer Science & Business Media (2010).
• [9] J. García-Falset, E. Llorens-Fuster, T. Suzuki, Fixed point theory for a class of generalized nonexpansive mappings, J. Math. Anal. Appl. 375 (2011), 185-195.
• [10] K. Goebel, W.A. Kirk, Topics in metric fixed point theory, volume 28 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, (1990).
• [11] K. Goebel and W.A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 35 (1972), 171-174.
• [12] D. Göhde, Zum Prinzip der kontraktiven Abbildung, Math. Nachr. 30 (1965), 251-258.
• [13] W.A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly, 72 (1965), 1004- 1006.
• [14] M.A. Krasnosel'ski, Two remarks on the method of successive approximations, Uspehi Mat. Nauk (N.S.), 10 (1955), 123-127.
• [15] E. Llorens-Fuster, E. Moreno Gálvez, The fixed point theory for some generalized nonexpansive mappings, Abstr. Appl. Anal. (2011), pages Art. ID 435686, 1-15.
• [16] J.J. Nieto, R. Rodríguez-López, Contractive mapping theorems in partially ordered sets and applications to ordinary di?erential equations, Order, 22 (2006), no. 3, 223-239.
• [17] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967), 591-597.
• [18] R. Pandey, R. Pant, V. Rako£evi¢, R. Shukla, Approximating fixed points of a general class of nonexpansive mappings in Banach spaces with applications, Results Math. 74 (2019), Paper No. 7, 1-24.
• [19] A. Ran, M. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc. 132 (2004), no. 5, 1435-1443.
• [20] H. F. Senter, W. G. Dotson Jr, Approximating fixed points of nonexpansive mappings, Proc. Amer. Math. Soc. 44 (1974), 375-380.
• [21] R. Shukla, R. Pant, M. De la Sen, Generalized α-nonexpansive mappings in Banach spaces, Fixed Point Theory Appl. 2016 (2016), Paper No. 4, 1-16.
• [22] R. Shukla, R. Pant, Z. Kadelburg, H. K. Nashine, Existence and convergence results for monotone nonexpansive type mappings in partially ordered hyperbolic metric spaces, Bull. Iranian Math. Soc., 43 (2017), no. 7, 2547-2565.
• [23] R. Shukla, R. Pant, P. Kumam, On the α-nonexpansive mapping in partially ordered hyperbolic metric spaces, J. Math. Anal. 8 (2017), no. 1, 1-15.
• [24] R. Shukla, A. Wisnicki, Iterative methods for monotone nonexpansive mappings in uniformly convex spaces, Adv. Nonlinear Anal. 10 (2021), no. 1, 1061-1070.
• [25] T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl. 340 (2008), no. 2, 1088-1095.
• [26] H.K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal. 16 (1991), no. 12, 1127-1138.