Fixed point approximation of contractive-like mappings using a stable iterative family and its dynamics via quadratic polynomials

It is known that the Mann iteration is for approximating fixed points of nonexpans-ive single-valued mappings. However, in general the Mann iteration process has only weak convergence. In recent years, Sastry, Babu and Panyanak introduced the Mann and Ishikawa iteration scheme for non-expansive multi-valued mapping and obtained the strong convergen-ce theorems. In this paper, we introduce a new iterative method for quasi-nonexpansive multi-valued maps in Banach spaces, and obtain the strong convergence theorems.

The hybrid algorithms for constructing fixed points of nonlinear mappings have been studied extensively in recent years. The advantage of this methods is that one can prove strong convergence theorems while the traditional iteration methods just have weak convergence. In this paper, we propose two types of hybrid algorithm to find a common fixed point of a finite family of asymptotically nonexpansive mappings in Hilbert spaces. One is cyclic Mann′s iteration scheme, and the other is cyclic Halpern′s iteration scheme. We prove the strong convergence theorems for both iteration schemes.

A class of demicontractive mappings was first introduced in [Hicks, T. L. and Kubicek, J. D.,On the Mann ite-ration process in a Hilbert space, J. Math. Anal. Appl.,59(1977) 498–504 and M ̆arus ̧ter, S ̧ .,The solution by iterationof nonlinear equations in Hilbert spaces, Proc. Amer. Math. Soc.,63(1977), 69–73] and was first mentioned in thecase of multi-valued mappings in [Chidume, C. E., Bello, A. U. and Ndambomve, P.,Strong and∆-convergencetheorems for common fixed points of a finite family of multivalued demicontractive mappings in CAT(0) spaces, Abstr.Appl. Anal.,2014(2014), https://doi.org/10.1155/2014/805168 and Isiogugu, F. O. and Osilike, M. O.,Conver-gence theorems for new classes of multivalued hemicontractive-type mappings, Fixed Point Theory Appl.,2014(2014),https://doi.org/10.1186/1687-1812-2014-93]. The demicontractivity with some weak smoothness conditionsensures only weak convergence of Mann iteration. In 2015, M ̆arus ̧ter and Rus [Kannan contractions and stronglydemicontractive mappings, Creat. Math. Inform.,24(2015), No. 2, 173–182], introduced a class of strongly de-micontractive mappings, and also discussed some relationships between strongly demicontractive mappingsand Kannan contractions. In this paper, we introduce a new class of strongly demicontractive multi-valuedmappings in Hilbert spaces. Strong convergence theorems of Picard and Mann iterative methods for stronglydemicontractive multi-valued mappings are established under some suitable coefficients and control sequences.

Self-similarity is a common feature among mathematical fractals and various objects of our natural environment. Therefore, escape criteria are used to determine the dynamics of fractal patterns through various iterative techniques. Taking motivation from this fact, we generate and analyze fractals as an application of the proposed Mann iterative technique with h-convexity. By doing so, we develop an escape criterion for it. Using this established criterion, we set the algorithm for fractal generation. We use the complex function f(x)=xn+ct, with n≥2,c∈C and t∈R to generate and compare fractals using both the Mann iteration and Mann iteration with h-convexity. We generalize the Mann iterative scheme using the convexity parameter h(α)=α2 and provide the detailed representations of quadratic and cubic fractals. Our comparative analysis consistently proved that the Mann iteration with h-convexity significantly outperforms the standard Mann iteration scheme regarding speed and efficiency. It is noticeable that the average number of iterations required to perform the task using Mann iteration with h-convexity is significantly less than the classical Mann iteration scheme. Moreover, the relationship between the fractal patterns and the input parameters of the proposed iteration is extremely intricate.

Construction of fixed points for asymptotically quasi-nonexpansive mappings is one of the important subjects in theory of nonexpansive mappings. Asymptotically quasi-nonexpansive mappings are widely used in a number of applied areas, such as image recovery and signal processing, etc. Consequently, considerable research efforts have been devoted to the study of iterative algorithms for finding fixed points for nonexpansive mappings. Mann iteration method and Ishikawa iteration method are among the most basic and famous iterative methods. In this paper, by combining the idea of Mann iteration and Ishikawa iteration, an iterative algorithm with errors involving three different asymptotically nonexpansive mappings is presented. And, under some conditions, the results that the algorithm converges weakly or strongly to the common fixed points of these three mappings in a uniformly convex Banach space are obtained.

Strong convergence of modified Mann iterations

The Krasnoselskii---Mann iteration plays an important role in the approximation of fixed points of nonexpansive operators; it is known to be weakly convergent in the infinite dimensional setting. In this present paper, we provide a new inexact Krasnoselskii---Mann iteration and prove weak convergence under certain accuracy criteria on the error resulting from the inexactness. We also show strong convergence for a modified inexact Krasnoselskii---Mann iteration under suitable assumptions. The convergence results generalize existing ones from the literature. Applications are given to the Douglas---Rachford splitting method, the Fermat---Weber location problem as well as the alternating projection method by John von Neumann.

"We prove that the convergence theorems for Mann iteration used for approximation of the fixed points of demicontractive mappings in Hilbert spaces can be derived from the corresponding convergence theorems in the class of quasi-nonexpansive mappings. Our derivation is based on an important auxiliary lemma (Lemma \ref{lem3}), which shows that if $T$ is $k$-demicontractive, then for any $\lambda\in (0,1-k)$, $T_{\lambda}$ is quasi-nonexpansive. In this way we obtain a unifying technique of proof for various well known results in the fixed point theory of demicontractive mappings. We illustrate this reduction technique for the case of two classical convergence results in the class of demicontractive mappings: [M\u aru\c ster, \c St. The solution by iteration of nonlinear equations in Hilbert spaces. {\em Proc. Amer. Math. Soc.} {\bf 63} (1977), no. 1, 69--73] and [Hicks, T. L.; Kubicek, J. D. On the Mann iteration process in a Hilbert space. {\em J. Math. Anal. Appl.} {\bf 59} (1977), no. 3, 498--504]."

The purpose of this article is to investigate the problem of finding a common element of the set of fixed points of a non-expansive mapping and the set of solutions of the variational inequality problem for a monotone, Lipschitz continuous mapping. We introduce a hybrid Mann iterative scheme with perturbed mapping which is based on the well-known Mann iteration method and hybrid (or outer approximation) method. We establish a strong convergence theorem for three sequences generated by this hybrid Mann iterative scheme with perturbed mapping. Utilizing this theorem, we also construct an iterative process for finding a common fixed point of two mappings, one of which is non-expansive and the other taken from the more general class of Lipschitz pseudocontractive mappings.

In this paper, we introduce an iterative scheme by the modification of Mann's iteration process for finding a common element of the set of solutions of a finite family of variational inequality problems and the set of fixed points of an η-strictly pseudo-contractive mapping and a nonexpansive mapping. Moreover, we prove a strong convergence theorem for finding a common element of the set of fixed points of a finite family of ηi-strictly pseudo-contractive mappings for every i =1 , 2, ... , N in uniformly convex and 2-uniformly smooth Banach spaces.

Convergence theorems of fixed points for Lipschitz pseudo-contractions in Hilbert spaces

We use an iteration process due to Rafiq (A. Rafiq, On Mann iteration in Hilbert spaces, Nonlinear Analysis 66 (2007), 2230 – 2236) to approximate fixed points of continuous hemicontractive mappings in Hilbert spaces. We drop the compactness condition on the domain of the operator, imposed in [1] and [26]. Our results extend several well known results in the literature and complement the results in [1] and [26].

Equivalence theorems of the convergence between Ishikawa and Mann iterations with errors for generalized strongly successively Φ-pseudocontractive mappings without Lipschitzian assumptions

Halpern iterative algorithm is one of the most cited in the literature of approximation of fixed points of nonexpansive mappings. Different authors modified this iterative algorithm in Banach spaces to approximate fixed points of nonexpansive mappings. One of which is Hu [8] and Yao et al [21] modification of Halpern iterative algorithm for nonexpansive mappings in Banach spaces. It is the purpose of this paper to thoroughly analyze this modification and its convergence conditions. Unfortunately, Hu [8] and Yao et al [21] control conditions imposed on the modified Halpern iterative algorithm to have strong convergence are found to be not sufficient. In this paper, counterexamples are constructed to prove that the strong convergence conditions of Hu [8] and Yao et al [21] are not sufficient. It is also proved that with some additional conditions on the control parameters, strong convergence of the defined iterative algorithm is obtained in different Banach space settings.

Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups