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*Maldar, Samet ( Aksaray, Yazar ), The aim of this paper is twofold: (a) to revisit the results in Iqbal et al. (Numer Algorithms 1–21, 2019. https:doi.org/10.1007/s11075-019-00854-z) and prove some convergence and stability results for the subclass of multivalued contractive operators under some mild conditions (b) to introduce a multivalued Reich-Suzuki type α-nonexpansive mappings and present some fixed point results for this class of mappings. We considered a more natural notion of stability called weak w2-stability instead of simple stability in Iqbal et al. (Numer Algorithms 1–21, 2019. https:doi.org/10.1007/s11075-019-00854-z). Some illustrative examples to support the results obtained herein are also presented.

We modify an iteration process of Agarwal et al. (J. Nonlinear Convex Anal. 8(1):61–79, 2007) to the case of two mappings and prove some weak and strong convergence theorems for two asymptotically nonexpansive mappings. We also point out that this process cannot be used for three mappings in its existing form even for nonexpansive mappings. We have to impose an extra condition to get convergence. We give an example to show that there do exist two nonexpansive mappings satisfying that condition.

Let E be a reflexive real Banach space with uniformly Gâteaux differentiable norm. Let K be a nonempty closed convex subset of E and \(\{T_{n}\}_{n=1}^{\infty}\) be a sequence of Ln-Lipschitzian mappings of K into itself with Ln≥1, \(\sum_{n=1}^{\infty}(L_{n}-1)\allowbreak

The purpose of this paper is to prove weak and strong convergences of a modified implicit iteration process to common fixed points for a finite family of asymptotically nonexpansive mappings. The results presented in this paper improve and extend some well known results.

Let E be a real uniformly convex and smooth Banach space with P as a sunny nonexpansive retraction, K be a nonempty closed convex subset of E. Let {Si}i=1N, {Ti}i=1N:K→K be two finite families of weakly inward and asymptotically nonexpansive mappings with respect to P. It is proved that the composite implicit iteration process converges weakly and strongly to a common fixed point of {Si}i=1N, {Ti}i=1N under certain conditions. The results of this paper improve and extend some well known corresponding results.

The purpose of this paper is to prove strong convergences of a modified implicit iteration process to a common fixed point for a finite family of generalized asymptotically quasi-nonexpansive mappings. The results presented in this paper improve and extend Shahzad and Zegeye’s corresponding results (Shahzad and Zegeye in Appl. Math. Comput. 189:1058–1065, 2007)