Ishikawa Iterative Scheme Research Articles

A modified Ishikawa iteration scheme for b‐enriched nonexpansive mapping to solve split variational inclusion problem and fixed point problem in Hilbert spaces

In this article, an Ishikawa iteration scheme is modified for ‐enriched nonexpansive mapping to solve a fixed point problem and a split variational inclusion problem in real Hilbert spaces. Under some suitable conditions, we obtain convergence theorem. Moreover, to demonstrate the effectiveness and performance of proposed scheme, we apply the scheme to solve a split feasibility problem and compare it with some existing iterative schemes.

Some fixed point theorems for multi-valued mappings via Ishikawa-type iterations in CAT(0) spaces

In this manuscript, we establish strong convergence theorems for the Ishikawa iteration scheme involving quasi-nonexpansive multi-valued maps in the setting of CAT(0) spaces. Our results extend and improve some related results in the literature.

Strong Convergence Theorems for a Finite Family of Enriched Strictly Pseudocontractive Mappings and ΦT-Enriched Lipschitizian Mappings Using a New Modified Mixed-Type Ishikawa Iteration Scheme with Error

In this paper, we introduce two new classes of mappings known as λ-enriched strictly pseudocontractive mappings and ΦT-enriched Lipshitizian mappings in the setup of a real Banach space. In addition, a new modified mixed-type Ishikawa iteration scheme was constructed, and it was proved that our iteration method converges strongly to the common fixed points of finite families of the above mappings in the framework of a real uniformly convex Banach space. Moreover, we provided a non-trivial example to support our main result. Our results extend and generalize several results existing in the literature.

Fuzzy Fixed Point Results in Convex C ∗ -Algebra-Valued Metric Spaces

The purpose of this note is to come up with some new directions in fuzzy fixed point theory. To this effect, notions of a C ∗ -algebra-valued fuzzy λ -contraction and related concepts in a convex C ∗ -algebra-valued metric space ( C ∗ -AVMS) are set-up. In line with the view of a Hausdorff distance function, an idea of a distance between two approximate quantities is proposed. Consequently, two fixed point results of a C ∗ -algebra-valued fuzzy mapping ( C ∗ -AVFM) for the new type of contractions are established using Mann and Ishikawa iterative schemes. For some future investigations of our results, two open problems are noted concerning sufficient criteria guaranteeing the existence of fixed points of a C ∗ -algebra-valued fuzzy λ -contraction and whether or not the Picard iteration for a C ∗ -algebra-valued fuzzy λ -contraction converges.

Weak and Strong Convergence Theorems of Modified Projection-Type Ishikawa Iteration Scheme for Lipschitz α-Hemicontractive Mappings

In this paper, we establish weak and strong convergence theorems of a two-step modified projection-type Ishikawa iterative scheme to the fixed point of α-hemicontractive mappings without any compactness assumption on the operator or the space. Our results extend, improve and generalize several previously known results of the existing literature.

Iterative approximation to common best proximity points of proximally mean nonexpansive mappings in Banach spaces

In this paper, we introduce the new class of mappings called proximally mean nonexpansive mappings. Further, we construct the Ishikawa iteration scheme associated with two non-self mappings, and we approximate a common best proximity point for a pair of proximally mean nonexpansive mappings via this Ishikawa iteration. As a consequence of our main result, we approximate a best proximity point for proximal mean nonexpansive mappings. Our new results complement and extend recent related results in the literature.

Strong and weak convergence of Ishikawa iterations for best proximity pairs

Abstract Let A and B be nonempty subsets of a normed linear space X. A mapping T : A ∪ B → A ∪ B is said to be a noncyclic relatively nonexpansive mapping if T(A) ⊆ A, T(B) ⊆ B and ∥Tx − Ty∥ ≤ ∥x − y∥ for all (x, y) ∈ A × B. A best proximity pair for such a mapping T is a point (p, q) ∈ A × B such that p = Tp, q = Tq and d(p, q) = dist(A, B). In this work, we introduce a geometric notion of proximal Opiaľs condition on a nonempty, closed and convex pair of subsets of strictly convex Banach spaces. By using this geometric notion, we study the strong and weak convergence of the Ishikawa iterative scheme for noncyclic relatively nonexpansive mappings in uniformly convex Banach spaces. We also establish a best proximity pair theorem for noncyclic contraction type mappings in the setting of strictly convex Banach spaces.

Strong convergence of modified Ishikawa iterative-type algorithm in CAT(0) spaces

We introduce a modified Ishikawa iterative type scheme in CAT(0) space and prove strong convergence theorem for total asymptotically demicontractive mappings. Our result improve and extend many results in the literature.

Approximating Fixed Points of Bregman Generalized α-Nonexpansive Mappings

In this paper, we introduce a new class of Bregman generalized α -nonexpansive mappings in terms of the Bregman distance. We establish several weak and strong convergence theorems of the Ishikawa and Noor iterative schemes for Bregman generalized α -nonexpansive mappings in Banach spaces. A numerical example is given to illustrate the main results of fixed point approximation using Halpern’s algorithm.

New hybrid method for equilibrium problems and relatively nonexpansive mappings in Banach spaces

In this paper, applying hybrid projection method, a new modified Ishikawa iteration scheme is presented for finding a common element of the solution set of an equilibrium problem and the set of fixed points of relatively nonexpansive mappings in Banach spaces. A numerical example is given and the numerical behaviour of the sequences generated by this algorithm is compared with several existence results in literature to illustrate the usability of obtained results.

On Some Convergence Properties of the Modified Ishikawa Scheme for Asymptotic Demicontractive Self-Mappings with Matricial Parameterizing Sequences

This paper is focused on the modified Ishikawa iterative scheme by admitting that the parameterizing sequences might be vectors of distinct components. It is also assumed that the auxiliary self-mapping which supports the iterative scheme is asymptotically demicontractive.

Strong convergence theorems of an iterative scheme for strictly pseudocontractive mappings in Banach spaces

The article establishes the sufficient and necessary conditions for the strong convergence of the Ishikawa iterative scheme with random errors in the sense of Xu for a strictly pseudocontractive mapping in an arbitrary real Banach space under certain assumptions. It affirmatively answers the open question that put forth by Zeng. The results in this paper extend and improve the corresponding results of Browder, Petryshyn, Rhoades, Osilike and Zeng.

Ishikawa iterative process for hemicontractive multi-valued non-self mappings in CAT(0) spaces

Our purpose in this paper is to construct Ishikawa iterative scheme formulti-valued non-self mappings in CAT(0) spaces. Then we obtain strong convergence of the scheme to a fixed point of multi-valued hemicontractive non-self mapping in a complete CAT(0) space. In addition, we define pseudocontractive mapping in CAT(0) spaces and show that a pseudocontractive mapping T with \(F(T)\not =\emptyset \) and \(Tp=\{p\}, \forall p\in F(T)\) is hemicontractive mapping. Furthermore, we give an example of hemicontractive mapping which is not pseudocontarctive to show that the converse is not necessarily true. Our theorems improve and unify most of the results in the literature.

Fixed points of a finite family of I-asymptotically quasi-nonexpansive mappings in a convex metric space

In this paper, we study Ishikawa iterative scheme with error terms for a finite family of Iasymptotically quasi-nonexpansive mappings in a convex metric space. We established strong convergence theorems and their applications for the proposed algorithms in a convex metric space. Our theorems improve and extend the corresponding known results in Banach spaces.

SOME STRONG CONVERGENCE RESULTS OF RANDOM ITERATIVE ALGORITHMS WITH ERRORS IN BANACH SPACES

In this paper, we study the strong convergence and stability of a new two step random iterative scheme with errors for accretive Lipschitzian mapping in real Banach spaces. The new iterative scheme is more acceptable because of much better convergence rate and less restrictions on parameters as compared to random Ishikawa iterative scheme with errors. We support our analytic proofs by providing numerical examples. Applications of random iterative schemes with errors to variational inequality are also given. Our results improve and establish random generalization of results obtained by Chang [4], Zhang [31] and many others.

Fractals through modified iteration scheme

In this paper we study the geometry of relative superior Mandelbrot sets through S-iteration scheme. Our results are quit significant from other Mandelbrot sets existing in the literature. Besides this, we also observe that S-iteration scheme converges faster than Ishikawa iteration scheme. We believe that the results of this paper can be inspired thosewho are interested in creating automatically aesthetic patterns.

On an open problem of Kyung Soo Kim

AbstractWe prove a convergence theorem of the Mann iteration scheme for a uniformlyL-Lipschitzian asymptotically demicontractive mapping in a$\operatorname{CAT}(\kappa)$CAT(κ)space with$\kappa>0$κ>0. We also obtain a convergence theorem of the Ishikawa iteration scheme for a uniformlyL-Lipschitzian asymptotically hemicontractive mapping. Our results provide a complete solution to an open problem raised by Kim (Abstr. Appl. Anal. 2013:381715, 2013).

Convergence Theorems for Nonspreading Type Mappings in Hilbert Spaces

In this paper first we give a weak convergence ergodic theorem for K-strictly pseudo-nonspreading and nonspreading mapping in Ishikawa iteration scheme, motivated by Kurokawa and Takahashi in9. Then we deal with a strong convergence theorem for nonspreading mappings in a Hilbert space. Our results improve and extend Kurokawa and Takahashi result.

$$\Delta $$ Δ -Convergence for Generalized Hybrid Mappings in CAT(0) Spaces

In this paper, we first give some properties for generalized hybrid mappings in complete CAT(0) spaces. Then we prove $$\Delta $$ -convergence and strong convergence theorems of the proposed Picard, Mann, and Ishikawa iterative schemes for such mappings in complete CAT(0) spaces.

Stability and Fractal Patterns of Complex Logistic Map

Abstract The intent of this paper is to study the fractal patterns of one dimensional complex logistic map by finding the optimum values of the control parameter using Ishikawa iterative scheme. The logistic map is shown to have bounded and stable behaviour for larger values of the control parameter. This is well depicted via time series analysis and interesting fractal patterns as well are presented