Ishikawa Iteration Research Articles

On stability of iteration processes convergent to stationary points of semigroups of nonlinear operators in metric spaces

Let C be a closed subset of a complete metric space. In this paper we study the stability problem for some iteration processes that approximate stationary points of pointwise Lipschitzian semigroups of nonlinear mappings acting within C. We show that a large class of well-controlled processes is stable under summable errors. These results are then applied to the stability of some known processes including the generalized Krasnosel'skii-Mann and the two-step Ishikawa iteration processes.

Distributed Ishikawa algorithms for seeking the fixed points of multi-agent global operators over time-varying communication graphs

In this article, the problem of seeking fixed points for global operators over the time-varying graphs in a real Hilbert space is studied. The global operator is a linear combination of local operators, each local operator being accessed privately by one agent for less resource consumption. All agents form a network and they need to cooperate to solve problems. To this end, on the basis of the centralized Ishikawa iteration, the distributed Ishikawa algorithm (D-I) is first proposed. In the sequel, to predigest the calculational complexity, further considering the situation that only the random part of each operator coordinate is calculated in each iteration, the distributed block coordinate Ishikawa algorithm (D-BI) is also designed. The results indicate that the proposed D-I and D-BI algorithms can weakly converge to a fixed point of the multi-agent global operator. Eventually, we give a few numerical examples to illustrate practical benefits of the proposed algorithms.

Convergence of Fibonacci–Ishikawa iteration procedure for monotone asymptotically nonexpansive mappings

In uniformly convex Banach spaces, we study within this research Fibonacci–Ishikawa iteration for monotone asymptotically nonexpansive mappings. In addition to demonstrating strong convergence, we establish weak convergence result of the Fibonacci–Ishikawa sequence that generalizes many results in the literature. If the norm of the space is monotone, our consequent result demonstrates the convergence type to the weak limit of the sequence of minimizing sequence of a function. One of our results characterizes a family of Banach spaces that meet the weak Opial condition. Finally, using our iterative procedure, we approximate the solution of the Caputo-type nonlinear fractional differential equation.

Ishikawa iteration convergence to fixed points of a multi-valued mapping in modular function spaces

Ishikawa iteration convergence to fixed points of a multi-valued mapping in modular function spaces

Existence and approximating of common best proximity points of relatively nonexpansive mappings via Ishikawa iteration method

Existence and approximating of common best proximity points of relatively nonexpansive mappings via Ishikawa iteration method

On Chaos Controlling Mechanism for Ishikawa Iteration and Its Traffic Flow Model in Discrete Dynamical Systems

On Chaos Controlling Mechanism for Ishikawa Iteration and Its Traffic Flow Model in Discrete Dynamical Systems

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.

On the convergence of Ishikawa iterates defined by nonlinear quasi-contractions

In this study, we establish the convergence of Ishikawa iterates defined by nonlinear quasicontractive mappings on TVS-cone metric space. Further, our results generalize many existing results in the literature.

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.

Extended multi-valued pseudocontractive mappings and extended Mann and Ishikawa iteration schemes for finite family of mappings

Extended multi-valued pseudocontractive mappings and extended Mann and Ishikawa iteration schemes for finite family of mappings

Convergence and Stability of the Ishikawa Iterative Process for a class of ϕ-quasinonexpansive Mappings

The paper analyzes the convergence of Ishikawa iteration to the fixed point of a class of '-quasinonexpansive mappings in uniformly convex Banach spaces, as well as the stability of the Ishikawa iteration used in approximating the fixed point. The work not only confirmed Ishikawa iteration’s convergence and stability to the fixed point of '-quasinonexpansive mappings, but it also pointed the way for future research in the estimate of fixed points of ϕ-quasinonexpansive mappings.

ON COMPARISON OF SOLUTION OF ORDINARY DIFFERENTIAL EQUATION WITH HAAR WAVELET METHOD AND THE MODIFIED ISHIKAWA ITERATION METHOD

In this study, we have used a newly modified Ishikawa iteration method and the Haar wavelet method to solve an ordinary linear differential equation with initial conditions. Using the modified Ishikawa iteration approach, we derive approximate solutions to the issue as well as the related iterative schemes. For this problem, the Ishikawa Iteration Method is applied for different lambda and gamma values and approximation solutions for these values are compared with the approximate solution of Haar wavelet collocation and its exact solution. Finally, the error tables are written and the graphs are shown.

Approximating Fixed Points of Relatively Nonexpansive Mappings via Thakur Iteration

The study of symmetry is a major tool in the nonlinear analysis. The symmetricity of distance function in a metric space plays important role in proving the existence of a fixed point for a self mapping. In this work, we approximate a fixed point of noncyclic relatively nonexpansive mappings by using a three-step Thakur iterative scheme in uniformly convex Banach spaces. We also provide a numerical example where the Thakur iterative scheme is faster than some well known iterative schemes such as Picard, Mann, and Ishikawa iteration. Finally, we provide a stronger version of our proposed theorem via von Neumann sequences.

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.

Convergence theorems using Ishikawa iteration for finding common fixed points of demiclosed and 2-demiclosed mappings in Hilbert spaces

Convergence theorems using Ishikawa iteration for finding common fixed points of demiclosed and 2-demiclosed mappings in Hilbert spaces

An Accelerated Fixed-Point Algorithm with an Inertial Technique for a Countable Family of G-Nonexpansive Mappings Applied to Image Recovery

Many authors have proposed fixed-point algorithms for obtaining a fixed point of G-nonexpansive mappings without using inertial techniques. To improve convergence behavior, some accelerated fixed-point methods have been introduced. The main aim of this paper is to use a coordinate affine structure to create an accelerated fixed-point algorithm with an inertial technique for a countable family of G-nonexpansive mappings in a Hilbert space with a symmetric directed graph G and prove the weak convergence theorem of the proposed algorithm. As an application, we apply our proposed algorithm to solve image restoration and convex minimization problems. The numerical experiments show that our algorithm is more efficient than FBA, FISTA, Ishikawa iteration, S-iteration, Noor iteration and SP-iteration.

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.

Convergence of Mann and Ishikawa Iterative Processes for Some Contractions in Convex Generalized Metric Space

In this paper, we consider a JS metric space endowed with convexity structure, which will allow us to examine and study convergence of Mann iteration and Ishikawa iteration for Banach type and Chatterjea type contractions defined on JS metric space.

Weak stability of Ishikawa iterations for strongly demicontractive mappings in Hilbert spaces

In this article, we establish a weak stability theorem for Ishikawa iterations in Hilbert spaces. Moreover, a strongly demicontractive mapping is presented to illustrate that the Ishikawa iteration of T is weakly T-stable but not T-stable. Our results are new and extend several known results.

Novel Noor iterations technique for solving nonlinear equations

<abstract><p>The aim of this paper is to propose a novel Noor iteration technique, called the CT-iteration for approximating a fixed point of continuous functions on closed interval. Then, a necessary and sufficient condition for the convergence of the CT-iteration of continuous functions on closed interval is established. We also compare the rate of convergence between the proposed iteration and some other iteration processes in the literature. Specifically, our main result shows that CT-iteration converges faster than CP-iteration to the fixed point. We finally give numerical examples to compare the result with Mann, Ishikawa, Noor, SP and CP iterations. Our findings improve corresponding results in the contemporary literature.</p></abstract>