Mann Iterative Scheme Research Articles

On the convergence of Mann iteration scheme for the finite family of \phi-hemicontractive mappings

The purpose of this paper is to characterize conditions for the convergence of the Mann type iterative scheme to the common fixed point of a family of φ-hemicontractive mappings in a nonempty convex subset of an arbitrary Banach space.

Implicit Mann approximation with perturbations for nonexpansive semigroups in CAT(0) spaces

In this paper, we study the convergence of implicit Mann iteration processes with bounded perturbations for approximating a common fixed point of nonexpansive semigroup in CAT(0) spaces. We obtain the △-convergence results of implicit Mann iteration schemes with bounded perturbations for a family of nonexpansive mappings in CAT(0) spaces. Under certain and different conditions, we also get the strong convergence theorems of implicit Mann iteration schemes with bounded perturbations for nonexpansive semigroups in CAT(0) spaces. The results presented in this paper extend and enrich the existing literature.MSC:47H05, 47H10, 47J25.

Strong Convergence of Hybrid Algorithm for Asymptotically Nonexpansive Mappings in Hilbert Spaces

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.

Existence of Bounded Positive Solutions for Partial Difference Equations with Delays

This paper deals with solvability of the third‐order nonlinear partial difference equation with delays . With the help of the Banach fixed‐point theorem, the existence results of uncountably many bounded positive solutions for the partial difference equation are given; some Mann iterative schemes with errors are suggested, and the error estimates between the iterative schemes and the bounded positive solutions are discussed. Three nontrivial examples illustrating the results presented in this paper are also provided.

A modified Mann iterative scheme by generalized f-projection for a countable family of relatively quasi-nonexpansive mappings and a system of generalized mixed equilibrium problems

The purpose of this paper is to introduce a new hybrid projection method based on modified Mann iterative scheme by the generalized f-projection operator for a countable family of relatively quasi-nonexpansive mappings and the solutions of the system of generalized mixed equilibrium problems. Furthermore, we prove the strong convergence theorem for a countable family of relatively quasi-nonexpansive mappings in a uniformly convex and uniform smooth Banach space. Finally, we also apply our results to the problem of finding zeros of B-monotone mappings and maximal monotone operators. The results presented in this paper generalize and improve some well-known results in the literature.2000 Mathematics Subject Classification: 47H05; 47H09; 47H10.

Strong Convergence Theorems for Variational Inequalities and Fixed Point Problems of Asymptotically Strict Pseudocontractive Mappings in the Intermediate Sense

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

Convergence analysis of a hybrid Mann iterative scheme with perturbed mapping for variational inequalities and fixed point problems

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.

Strong Convergence Theorems for Strict Pseudocontractions in Uniformly Convex Banach Spaces

The viscosity approximation methods are employed to establish strong convergence theorems of the modified Mann iteration scheme to -strict pseudocontractions in -uniformly convex Banach spaces with a uniformly Gâteaux differentiable norm. The main result improves and extends many nice results existing in the current literature.

Construction of Fixed Points by Some Iterative Schemes

We obtain strong convergence theorems of two modifications of Mann iteration processes with errors in the doubly sequence setting. Furthermore, we establish some weakly convergence theorems for doubly sequence Mann's iteration scheme with errors in a uniformly convex Banach space by a Frechet differentiable norm.

Strong Convergence of a New Iteration for a Finite Family of Accretive Operators

The viscosity approximation methods are employed to establish strong convergence of the modified Mann iteration scheme to a common zero of a finite family of accretive operators on a strictly convex Banach space with uniformly Gâteaux differentiable norm. Our work improves and extends various results existing in the current literature.

Remarks of Equivalence among Picard, Mann, and Ishikawa Iterations in Normed Spaces

We show that the convergence of Picard iteration is equivalent to the convergence of Mann iteration schemes for various Zamfirescu operators. Our result extends of Soltuz (2005).

Ishikawa iterative process with errors for nonlinear equations of generalized monotone type in Banach spaces

AbstractThe concept of the operators of generalized monotone type is introduced and iterative approximation methods for a fixed point of such operators by the Ishikawa and Mann iteration schemes {xn} and {yn} with errors is studied. Let X be a real Banach space and T : D ⊂ X → 2D be a multi‐valued operator of generalized monotone type with fixed points. A new general lemma on the convergence of real sequences is proved and used to show that {xn} converges strongly to a unique fixed point of T in D. This result is applied to the iterative approximation method for solutions of nonlinear equations with generalized strongly accretive operators. Our results generalize many of know results. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Random Mann iteration scheme and random fixed point theorems

In this paper, some concepts such as random monotone operators, random Mann iteration and so on in a separable real Banach space are introduced. Also the existence and uniqueness theorems of random fixed points for random monotone operators satisfying Condition(H) are proved.

The equivalence of Mann iteration and Ishikawa iteration for a Lipschitzian $ \psi $- uniformly pseudocontractive and $ \psi $- uniformly accretive maps

We show that certain Ishikawa iteration and the corresponding Mann iteration schemes are equivalent when applied to Lipschitzian and $ \psi- $uniformly pseudocontractive or Lipschitzian $ \psi-$ uniformly accretive maps.

A new convergence result for fixed-point iteration in bounded interval of [formula omitted

A fixed-point theorem in bounded interval of R n and a new fixed-point iteration on Mann's iteration scheme are established to compute a fixed-point of real Lipschitz functions. Two numerical examples are presented to show that the method is feasible and that it produces reasonable results.

Convergence results for fixed point iterations in R

In this paper, we consider the extrapolated iteration and Mann's iteration scheme for computing the fixed point of a real Lipschitz function. Convergence of various types of fixed point iterations is well known for a self mapping which is a contraction. Here, we prove convergence of both methods without these assumptions.