Common Fixed Points of Two Nonexpansive Mappings by a Modified Faster Iteration Scheme

Abstract

Abstract. We introduce an iteration scheme for approximating common fixed points of two mappings. On one hand, it extends a scheme due to Agarwal et al.[2] to the case of two mappings while on the other hand, it is faster than both the Ishikawa type scheme and the one studied by Yao and Chen [18] for the purpose in some sense. Using this scheme, we prove some weak and strong convergence results for approximating common fixed points of two nonexpansive self mappings. We also outline the proofs of these results to the case of nonexpansive nonself mappings.

A new multistep iteration for a finite family of asymptotically quasi-nonexpansive mappings in convex metric spaces

“…The results presented in this paper are extensions and improvements of the corresponding results of Yang [23], Khan and Ahmed [7], Khan et al [9], Suantai [17], Khan and Hussain [10], Gunduz and Akbulut [3,4], Gunduz et al [5], Nilsrakoo and Saejung [13], Khan and Kim [11], Petryshyn and Williamson [14], Khan and Yildirim [24], Shahzad and Udomene [16], Khan et al [8] and Qihou [12]. Remark 3.5.…”

“…And to give strong convergence of the modified three-step iteration (1.4) for three asymptotically quasi-nonexpansive mappings in convex metric spaces. The main result is an extension and improvement of corresponding results in [3,4,5,7,9,10,11,12,13,14,16,17,23,24].…”

Fixed points of multivalued nonexpansive mappings in Banach spaces

“…We then apply this theorem to obtain a result in uniformly convex Banach spaces. We also use the method of direct construction of Cauchy sequence as indicated by Song and Cho [13] (and opposed to [10]) but used also by many other authors including [12,14,15].…”

“…They proved that this scheme converges at a rate faster than both Picard iteration scheme x n+1 = Tx n and Mann iteration scheme for contractions. Following their method, it was observed in [12,Example 3.7] that this scheme also converges faster than Ishikawa iteration scheme.…”

Δ−Convergence Theorems for Multivalued Non-expansive Mappings in Hyperbolic Spaces

“…The study for fixed point problem involve that multivalued contractions and non-expansive mappings used the Hausdorff metric was initiated by Markin [1,2] later, different iterative processes have used to approximate the fixed points of multivalued non-expansive mappings in Banach space, many scholars have made extensive research in [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. An interesting and rich fixed point theory for such mappings was developed which has applications in control theory, convex optimization, differential inclusion, and economics [3].…”