Modified Mann Iteration Process Research Articles

Approximating Common Fixed Points of an Evolution Family on a Metric Space via Mann Iteration

In this article, we present the set of all common fixed points of a subfamily of an evolution family in terms of intersection of all common fixed points of only two operators from the family; that is, for subset M of L , we have F M = F Y ϱ 1 , 0 ∩ F Y ϱ 2 , 0 , where ϱ 1 and ϱ 2 are positive and ϱ 1 / ϱ 2 is an irrational number. Furthermore, we approximate such common fixed points by using the modified Mann iteration process. In fact, we are generalizing the results from a semigroup of operators to evolution families of operators on a metric space.

Strong convergence results for fixed points of nearly weak uniformly L-Lipschitzian mappings of I-Dominated mappings

In this paper, we prove strong convergence results for a modified Mann iterative process for a new class of I- nearly weak uniformly L-Lipschitzian mappings in a real Banach space. The class of I-nearly weak uniformly L-Lipschitzian mappings is an interesting generalization of the class of nearly weak uniformly L-Lipschitzian mappings which inturn is a generalization of the class of nearly uniformly L-Lipschitzian mappings which inturn generalises uniformly L-Lipschitzian mappings. Our theorems include some very recent results in fixed point theory and applications, in the context of nearly uniformly L-Lipschitzian mappings.

Existence and convergence theorem for fixed point problem of various nonlinear mappings and variational inequality problems without some assumptions

The purpose of this article, we give a necessary and sufficient condition for the modified Mann iterative process in order to obtain a strong convergence theorem for finding a common element of the set of fixed point of a finite family of nonexpansive mappings and variational inequality problem in Hilbert space without the conditions ?Ni=1 Fix(Ti)? VI(C,A)??. Moreover, we utilize our main result to fixed point problems of strictly pseudocontractive mappings and the set of solutions of variational inequality problem.

Goebel and Kirk fixed point theorem for multivalued asymptotically nonexpansive mappings

We introduce the concept of a multivalued asymptotically nonexpansive mapping and establish Goebel and Kirk fixed point theorem for these mappings in uniformly hyperbolic metric spaces. We also define a modified Mann iteration process for this class of mappings and obtain an extension of some well-known results for singlevalued mappings defined on linear as well as nonlinear domains.

Approximating common fixed points of semigroups in metric spaces

AbstractIn this paper, we investigate the common fixed points set of nonexpansive semigroups of nonlinear mappings $\{T_{t}\}_{t \geq0}$ { T t } t ≥ 0 , i.e., a family such that $T_{0}(x)=x$ T 0 ( x ) = x , $T_{s+t}=T_{s}(T_{t}(x))$ T s + t = T s ( T t ( x ) ) , where the domain is a metric space $(M,d)$ ( M , d ) . In particular we prove that under suitable conditions, the common fixed points set is the same as the common fixed points set of two mappings from the family. Then we use the modified Mann iteration process to approximate such common fixed points.

Mann Type Iteration Method Involving Three Strictly Hemicontractive Mappings in Banach Spaces

The purpose of this paper is to prove that the modified Mann iteration process can be applied to approximate the common fixed point of three strictly hemicontractive mappings in smooth Banach spaces.

Strong convergence to fixed points of non-Lipschitzian mappings

In this paper, we first show the strong convergence of the modified Moudafi iteration process when E is a real uniformly convex Banach space, S is AQT self-mapping and T is ANI self-mapping satisfying Condition (B). Next, we show the strong convergence of the modified Mann iteration process when T is ANI self-mapping satisfying Condition (A), which generalizes the result due to Kim (J. Nonlinear Convex Anal. 13(3):449–457, 2012). Finally, we show the strong convergence of the Schu iteration process when T is ANI self-mapping satisfying Condition (A), which generalizes the result due to Rhoades (J. Math. Anal. Appl. 183:118–120, 1994). Open image in new window

ON ASYMPTOTICALLY DEMICONTRACTIVE MAPPINGS IN ARBITRARY BANACH SPACES

In this paper, the necessary and sufficient conditions for the strong convergence of a modified Mann iteration process to a fixed point of an asymptotically demicontractive mapping in real Banach spaces are considered. Presented results improve and extend the results of Igbokwe [3], Liu [4], Moore and Nnoli [6] and Osilike [7].

Mann iteration process for asymptotic pointwise nonexpansive mappings in metric spaces

Let (M,d) be a complete 2-uniformly convex metric space. Let C be a nonempty, bounded, closed, and convex subset of M, and let T:C→C be an asymptotic pointwise nonexpansive mapping. In this paper, we prove that the modified Mann iteration process defined by xn+1=tnTn(xn)⊕(1−tn)xn converges in a weaker sense to a fixed point of T.

On the Convergence of Mann and Ishikawa Iterative Processes for Asymptotically ϕ‐Strongly Pseudocontractive Mappings

We prove the equivalence and the strong convergence of (1) the modified Mann iterative process and (2) the modified Ishikawa iterative process for asymptotically ϕ‐strongly pseudocontractive mappings in a uniformly smooth Banach space.

An extragradient-like approximation method for variational inequalities and fixed point problems

AbstractThe 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 a variational inequality problem for a monotone and Lipschitz continuous mapping. We introduce an extragradient-like iterative algorithm that is based on the extragradient-like approximation method and the modified Mann iteration process. We establish a strong convergence theorem for two sequences generated by this extragradient-like iterative algorithm. 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.1991 MSC: 47H09; 47J20.

Iterative Approximation of Fixed Points for Asymptotically Strict Pseudocontractive Type Mappings in the Intermediate Sense

We introduced the concept of an asymptotically $\kappa$-strict pseudocontractive type mapping in the intermediate sense which is not necessarily Lipschitzian. We proved that the modified Mann iteration process: $x_{n+1} = (1-\alpha_n) x_n + \alpha_n T^n x_n$, $\forall n \geq 1$ where $\{\alpha_n\}$ is a sequence in $(0,1)$ with $\delta \leq \alpha_n \leq 1 - \kappa - \delta$ for $\delta \in (0,1)$ converges weakly to a fixed point of an asymptotically $\kappa$-strict pseudocontractive type mapping $T$ in the intermediate sense. Furthermore, a CQ method which generates a strongly convergent sequence for this class of mappings is proposed and strong convergence result for this CQ method is established.

Strong convergence theorems of modified Mann iterative process for nonexpansive mappings in Hilbert spaces

The purpose of this article is to modify normal Mann's iterative process to have strong convergence for nonexpansive mappings in the formework of Hilbert spaces. Weprove the strong convergence of the proposed iterative algorithm to the fixed point of nonexpansive mappings which is the unique solution of a variational inequality, which is also the optimality condition for a minimization problem.

Convergence theorems from monotone hybrid methods for an infinitely countable family of Lipschitz asymptotically quasi-nonexpansive mappings

In this paper, we prove a weak convergence theorem for the modified Mann iteration process for a uniformly Lipschitzian and asymptotically quasi-nonexpansive mapping in a uniformly convex Banach space. We also introduce two new kinds of monotone hybrid methods and obtain strong convergence theorems for an infinitely countable family of uniformly Lipschitzian and asymptotically quasi-nonexpansive mappings in a Hilbert space. The results of this paper improve on and extend corresponding ones announced by many authors.

Monotone Hybrid Projection Algorithms for an Infinitely Countable Family of Lipschitz Generalized Asymptotically Quasi-Nonexpansive Mappings

We prove a weak convergence theorem of the modified Mann iteration process for a uniformly Lipschitzian and generalized asymptotically quasi-nonexpansive mapping in a uniformly convex Banach space. We also introduce two kinds of new monotone hybrid methods and obtain strong convergence theorems for an infinitely countable family of uniformly Lipschitzian and generalized asymptotically quasi-nonexpansive mappings in a Hilbert space. The results improve and extend the corresponding ones announced by Kim and Xu (2006) and Nakajo and Takahashi (2003).

Strong convergence of the modified Mann iterative method for strict pseudo-contractions

In this paper, we introduce a modified Mann iterative process for approximating a common fixed point of a finite family of strict pseudo-contractions in Hilbert spaces. We establish the strong convergence theorem of the general iteration scheme under some mild conditions. Our results extend and improve the recent ones announced by many others.

Asymptotically strict pseudocontractive mappings in the intermediate sense

It is proved that the modified Mann iteration process: x n + 1 = ( 1 − α n ) x n + α n T n x n , n ∈ N , where { α n } is a sequence in (0, 1) with δ ≤ α n ≤ 1 − κ − δ for some δ ∈ ( 0 , 1 ) , converges weakly to a fixed point of an asymptotically κ -strict pseudocontractive mapping T in the intermediate sense which is not necessarily Lipschitzian. We also develop CQ method for this modified Mann iteration process which generates a strongly convergent sequence.

Strong convergence theorems of modified Mann iterative process for strict pseudo-contractions in Hilbert spaces

In this paper, we modify the normal Mann’s iterative process to have strong convergence for a k -strictly pseudo-contractive non-self mapping in the framework of Hilbert spaces. Our results improve and extend the corresponding results announced by many others.

Strong Cconvergence Theorems for Strictly Pseudocontractive Mappings by Viscosity Approximation Methods

In this paper, we introduce a modified Mann iterative process for strictly pseudocontractive mappings and obtain a strong convergence theorem in the framework of Hilbert spaces. Our results improve and extend the recent onesannounced by many others.