Convergence theorems for \(SP\)-iteration scheme in a ordered hyperbolic metric space. (English) Zbl 1482.54047
Summary: In this paper, we study the \(\Delta\)-convergence and strong convergence of \(SP\)-iteration scheme involving a nonexpansive mapping in partially ordered hyperbolic metric spaces. Also, we give an example to support our main result and compare SP-iteration scheme with the Mann iteration and Ishikawa iteration scheme. Thus, we generalize many previous results.
References:
| [1] | R.P. Agarwal, D.O. Regan and D.R. Sahu,Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, J. Nonlinear Convex Anal.,8(2007), 61-79. · Zbl 1134.47047 |
| [2] | S. Aggarwal, I. Uddin and J.J. Nieto,A fixed-point theorem for monotone nearly asymptotically nonexpansive mappings, J. Fixed Point Theory Appl., (2019), 21:91. · Zbl 07115533 |
| [3] | S. Aggarwal and I. Uddin,Convergence and stability of Fibonacci-Mann iteration for a monotone non-Lipschitzian mapping, Demonstratio Math.,52(2019), 388-396. · Zbl 1477.47083 |
| [4] | J. Ali and I. Uddin,Convergence ofSP-iteration for generalized nonexpansive mapping in Banach spaces, Ukrainian Math. J.,73(6) (2021), 738-748. · Zbl 07441696 |
| [5] | F.E. Browder,Fixed point theorems for noncompact mappings in Hilbert Space, Proc. Nat. Acad. Sci. USA.,53(1965), 1272-1276. · Zbl 0125.35801 |
| [6] | F.E. Browder,Nonexpansive nonlinear operators in a Banach Space, Proc. Nat. Acad. Sci. USA.,54(1965), 1041-1044. · Zbl 0128.35801 |
| [7] | H. Fukhar-ud-din and M.A. Khamsi,Approximating common fixed pint in hyperbolic spaces, Fixed Point Theory Appl., (2014), 2014:113. · Zbl 1310.47072 |
| [8] | D. G¨ohde,Zum Prinzip der kontraktiven Abbildung, Math. Nachr.,30(1965), 251-258. · Zbl 0127.08005 |
| [9] | C. Garodia and I. Uddin,A new fixed point algorithm for finding the solution of a delay differential equation, AIMS Mathematics,5(4) (2020), 3182-3200. · Zbl 1484.47109 |
| [10] | C. Garodia and I. Uddin,A new iterative method for solving split feasibility problem, J. Appl. Anal. Comput.,10(3) (2020), 986-1004. · Zbl 07331945 |
| [11] | S. Ishikawa,Fixed points by a new iteration method, Proc. Amer. Math. Soc.,44(1974), 147-150. · Zbl 0286.47036 |
| [12] | W.A. Kirk,A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly,72(1965), 1004-1006. · Zbl 0141.32402 |
| [13] | U. Kohlenbach,Some logical metatheorems with application in functional analysis, Trans. Amer. Math. Soc.,357(2005), 89-128. · Zbl 1079.03046 |
| [14] | T.C. Lim,Remarks on some fixed point theorems, Proc. Amer. Math. Soc.,60(1976), 179-182. · Zbl 0346.47046 |
| [15] | L. Leustean,Nonexpansive iterations in uniformly convex W-hyperbolic spaces, in A. Leizarowitz, B. S. Mordukhovich, I. Shafrir and A. Zaslavski, (Eds). Nonlinear Analysis and Optimization I: Nonlinear Analysis, Contemp. Math., Amer. Math. Soc.,513 (2010), 193-209. · Zbl 1217.47117 |
| [16] | W.R. Mann,Mean value methods in iteration, Proc. Amer. Math. Soc.,4(1953), 506510. · Zbl 0050.11603 |
| [17] | J.J. Nieto and R. Rodriguez-L¨opez,Contractive mapping theorems in partially ordered sets and applications to ordinary differential equation, Order,22(2005), 223-239. · Zbl 1095.47013 |
| [18] | M.A. Noor,New approximation schemes for general variational inequalities, J. Math. Anal. Appl.,251(2000), 217-229. · Zbl 0964.49007 |
| [19] | W. Phuengrattana and S. Suantai,On the rate of convergence of Mann, Ishikawa, Noor andSP-iterations for continuous functions on an arbitrary interval, J. Comput. Appl. Math.,235(2011), 3006-3014. · Zbl 1215.65095 |
| [20] | A.C.M. Ran and M.C.B. Reuring,A fixed point theorems in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc., Vol.132(2005), 1435- 1443. · Zbl 1060.47056 |
| [21] | T. Shimizu and W. Takahashi,Fixed points of multivalued mapping in certain convex metric space, Tolol Methds Nonlinear Anal.,8(1996), 197-203 · Zbl 0902.47049 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
