Convergence of three-step iterations for asymptotically nonexpansive mappings

Strong convergence theorems for countable families of asymptotically relatively nonexpansive mappings with applications

Viscosity approximation methods for a finite family of nonexpansive mappings in Banach spaces

Modified block iterative algorithm for Quasi- ϕ-asymptotically nonexpansive mappings and equilibrium problem in banach spaces

In this paper, we prove strong convergence theorems of modified Halpern’s iteration for finding a common element of the zero point set of a maximal monotone operator and the fixed point set of a relatively nonexpansive mapping in a Banach space by using two hybrid methods. Using these results, we obtain new convergence results for resolvents of maximal monotone operators and relatively nonexpansive mappings in Banach spaces.

The purpose of this paper is to study the weak and strong convergence of an implicit iteration process to a common fixed point for two finite families of nonexpansive mappings in Banach spaces. The results presented in this paper extend and improve the corresponding results of Xu and Ori, Numer. Funct. Anal. Optim. 2001; 22:767–773, Zhou and Chang, Numer. Fund. Anal. Optim. 2002; 23:911–921, Chidume and Shahzad, Nonlinear. Anal. 2005; 62: 1149–1156.

On Reich’s strong convergence theorem for asymptotically nonexpansive mappings in Banach spaces

In this paper, we prove convergence theorems for a fixed point iterative algorithm of Krasnoselskij-Mann typeassociated to the class of enriched nonexpansive mappings in Banach spaces. The results are direct generaliza-tions of the corresponding ones in [Berinde, V.,Approximating fixed points of enriched nonexpansive mappings byKrasnoselskij iteration in Hilbert spaces, Carpathian J. Math., 35 (2019), No. 3, 293–304.], from the setting of Hilbertspaces to Banach spaces, and also of some results in [Senter, H. F. and Dotson, Jr., W. G.,Approximating fixed pointsof nonexpansive mappings, Proc. Amer. Math. Soc.,44(1974), No. 2, 375–380.], [Browder, F. E., Petryshyn, W. V., Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl., 20 (1967), 197–228.], byconsidering enriched nonexpansive mappings instead of nonexpansive mappings. Many other related resultsin literature can be obtained as particular instances of our results.

In this paper, we introduce a hybrid method, a combination of the steepest-descent method and the Krasnosel'skii-Mann one, for solving a variational inequality over the set of common fixed points of an infinite family of nonexpansive mappings in Banach spaces under two different conditions on the Banach space, either a uniformly smooth Banach space or a reflexive and strictly convex one with a uniformly Gâteaux differentiable norm, without imposing the sequential weak continuity of the normalized duality mapping. The method is an improvement and extension of some other published results. We also give a numerical example to illustrate the convergence analysis of the proposed method.

Convergence of composite iterative methods for finding zeros of accretive operators

A note on “Approximation of fixed points for amenable semigroups of nonexpansive mappings in Banach spaces” [Nonlinear Anal. 67 (2007) 1211–1225

We approximate the fixed point of the Suzuki's generalized nonexpansive mappings via the Picard-Ishikawa hyprid iterative process, recently introduced by Okeke [18].We prove some weak and strong convergence theorems of this type of mappings in the setting of uniformly convex Banach spaces.We apply our results in finding the solution of a mixed type Volterra-Fredholm functional nonlinear integral equation in Banach spaces.Finally, we give several numerical examples to validate our analytical results.Our results extend and improve several known results in literature, including the results of Okeke [18], Ullah and Arshad [30] and Craciun and Serban [7] among others.

This paper investigates fixed points of Reich-Suzuki-type nonexpansive mappings in the context of uniformly convex Banach spaces through an M ∗ iterative method. Under some appropriate situations, some strong and weak convergence theorems are established. To support our results, a new example of Reich-Suzuki-type nonexpansive mappings is presented which exceeds the class of Suzuki-type nonexpansive mappings. The presented results extend some recently announced results of current literature.

In this paper, we prove a strong convergence theorem by the shrinking projection method for a sequence of hemi-relatively nonexpansive mappings in Banach space.

In this paper, we prove a weak convergence theorem and a strong convergence theorem for split common fixed point problem involving a quasi-strict pseudo contractive mapping and an asymptotical nonexpansive mapping in the setting of two Banach spaces. Our results are new and seem to be the first outside Hilbert spaces.

This paper studies the convergence of the sequence defined by \(x_0 \in C,x_{n + 1} = \alpha _n u + (1 - \alpha _n )Tx_n ,n = 0,1,2,...,\) where 0 ≤ αn ≤ 1, limn→∞αn = 0, Σn=0∞αn = ∞, and T is a nonexpansive mapping from a nonempty closed convex subset C of a Banach space X into itself. The iterative sequence {xn} converges strongly to a fixed point of T in the case when X is a uniformly convex Banach space with a uniformly Gateaux differentiable norm or a uniformly smooth Banach space only. The results presented in this paper extend and improve some recent results.