Ishikawa Iteration Research Articles

Faster Multistep Iterations for the Approximation of Fixed Points Applied to Zamfirescu Operators

By taking a counterexample, we prove that the multistep iteration process is faster than the Mann and Ishikawa iteration processes for Zamfirescu operators.

Bregman Asymptotic Pointwise Nonexpansive Mappings in Banach Spaces

We first introduce a new class of mappings called Bregman asymptotic pointwise nonexpansive mappings and investigate the existence and the approximation of fixed points of such mappings defined on a nonempty, bounded, closed, and convex subset <i>C</i> of a real Banach space <i >E</i>. Without using the original Opial property of a Banach space <i >E</i>, we prove weak convergence theorems for the sequences produced by generalized Mann and Ishikawa iteration processes for Bregman asymptotic pointwise nonexpansive mappings in a reflexive Banach space <i >E</i>. Our results are applicable in the function spaces <svg style="vertical-align:-0.0pt;width:17.512501px;" id="M1" height="14.45" version="1.1" viewBox="0 0 17.512501 14.45" width="17.512501" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,14.387)"><path id="x1D43F" d="M559 163q-23 -66 -68 -163h-474l6 26q62 4 79.5 19.5t28.5 75.5l78 409q7 35 8.5 49t-8 25t-24 13t-51.5 5l5 28h266l-6 -28q-65 -5 -79.5 -18t-25.5 -74l-76 -406q-10 -57 14 -75q12 -13 96 -13q93 0 126 29q41 40 76 109z" /></g> <g transform="matrix(.012,-0,0,-.012,9.763,6.225)"><path id="x1D45D" d="M570 304q0 -108 -87 -199q-40 -42 -94.5 -74t-105.5 -43q-41 0 -65 11l-29 -141q-9 -45 -1.5 -58t45.5 -16l26 -2l-5 -29l-241 -10l4 26q51 10 67.5 24t26.5 60l113 520q-54 -20 -89 -41l-7 26q38 28 105 53l11 49q20 25 77 58l8 -7l-17 -77q39 14 102 14q82 0 119 -36&#xA;t37 -108zM482 289q0 114 -113 114q-26 0 -66 -7l-70 -327q12 -14 32 -25t39 -11q59 0 118.5 81.5t59.5 174.5z" /></g> </svg>, where <svg style="vertical-align:-3.50804pt;width:74.550003px;" id="M2" height="15.2875" version="1.1" viewBox="0 0 74.550003 15.2875" width="74.550003" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,10.862)"><path id="x31" d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" /></g><g transform="matrix(.017,-0,0,-.017,12.931,10.862)"><path id="x3C" d="M512 -3l-437 233v51l437 233v-58l-378 -200v-2l378 -199v-58z" /></g><g transform="matrix(.017,-0,0,-.017,27.634,10.862)"><use xlink:href="#x1D45D"/></g><g transform="matrix(.017,-0,0,-.017,42.44,10.862)"><use xlink:href="#x3C"/></g><g transform="matrix(.017,-0,0,-.017,57.144,10.862)"><path id="x221E" d="M983 225q0 -112 -67 -174.5t-150 -62.5q-91 0 -154.5 43.5t-113.5 129.5q-49 -85 -104 -129t-138 -44q-98 0 -158.5 66t-60.5 154q0 59 21 106.5t54.5 75.5t70.5 43t73 15q90 0 152.5 -43.5t112.5 -128.5q48 84 104.5 128t140.5 44q93 0 155 -65t62 -158zM478 196&#xA;q-27 49 -47 80t-50 67t-64 54t-73 18q-48 0 -81.5 -47t-33.5 -128q0 -96 37.5 -157.5t99.5 -61.5q68 0 117.5 47t94.5 128zM889 204q0 91 -35.5 151t-99.5 60q-68 0 -119 -47t-95 -127q27 -49 47 -80.5t50 -67.5t65 -54t74 -18q113 0 113 183z" /></g> </svg> is a real number.

Implicit Ishikawa Approximation Methods for Nonexpansive Semigroups in CAT(0) Spaces

This paper is devoted to the convergence of the implicit Ishikawa iteration processes for approximating a common fixed point of nonexpansive semigroup in CAT(0) spaces. We obtain theΔ-convergence results of the implicit Ishikawa iteration sequences for a family of nonexpansive mappings in CAT(0) spaces. Under certain and different conditions, we also get the strong convergence theorems of implicit Ishikawa iteration sequences for nonexpansive semigroups in the CAT(0) spaces. The results presented in this paper extend and generalize some previous results.

Fixed point theorems and spproximating fixed points of generalized (C)-condition mappings

In this paper, we present fixed point theorems of generalized (C)condition mappings defined on weakly compact convex subsets of Banach spaces satisfying property (D). Moreover, we prove weak and strong convergence theorems for Ishikawa iteration of generalized (C)condition mappings in uniformly convex Banach spaces.

On Ishikawa-type iteration with errors for a continuous real function on an arbitrary interval

We consider the Ishikawa-type iteration process with errors for a continuous real function on an arbitrary interval and prove the convergence theorem. Furthermore, we give numerical examples to compare with Mann and Ishikawa iteration processes with error sequences. Mathematics Subject Classification: 47H09, 47H10

Stability of the Ishikawa iteration scheme with errors for two strictly hemicontractive operators in Banach spaces

Abstract The main purpose of this paper is to establish the convergence, almost common-stability and common-stability of the Ishikawa iteration scheme with error terms in the sense of Xu (J. Math. Anal. Appl. 224:91-101, 1998) for two Lipschitz strictly hemicontractive operators in arbitrary Banach spaces.

A generalized f-projection method for countable families of weak relatively nonexpansive mappings and the system of generalized Ky Fan inequalities

The purpose of this paper is to present new hybrid Ishikawa iteration process by the generalized f-projection operator for finding a common element of the fixed point set for two countable families of weak relatively nonexpansive mappings and the set of solutions of the system of generalized Ky Fan inequalities in a uniformly convex and uniformly smooth Banach space. Furthermore, we show that our new iterative scheme converges strongly to a common element of the afore mentioned sets. As applications, we apply our results to obtain some new results for finding a solution of a common fixed point of two countable in finite families, a system of generalized Ky Fan inequalities and a common zero-point problem for general B-monotone and maximal monotone operators in Banach spaces. The results presented in this paper improve and extend important recent results.

An iteration process for common fixed points of two nonself asymptotically nonexpansive mappings

Abstract In this paper, we introduce an iteration process for approximating common fixed points of two nonself asymptotically nonexpansive map- pings in Banach spaces. Our process contains Mann iteration process and some other processes for nonself mappings but is independent of Ishikawa iteration process. We prove some weak and strong convergence theorems for this iteration process. Our results generalize and improve some results in contemporary literature.

A Study of New Fractals Complex Dynamics for Inverse and Logarithmic Functions

The object Mandelbrot set given by Mandelbrot in 1979 and its relative object Julia set have become a wide and elite area of research nowadays due to their beauty and complexity of their nature. Many researchers and authors have worked to study and reveal the new concepts unexplored in the complexities of these two most popular sets of fractal geometry. In this paper we review the recently done work on complex functions for producing beautiful fractal graphics, by few eminent researchers contributing a lot to the field of fractal geometry. The reviewed work mainly emphasizes on the complex functional dynamics of Ishikawa iterates for inverse and logarithmic function and existence of relative superior Mandel-bar set.

A Relative Superior Julia Set and Relative Superior Tricorn and Multicorns of Fractals

In this paper we investigate the new Julia set and a new Tricorn and Multicorns of fractals. The beautiful and useful fractal images are generated using Ishikawa iteration to study many of their properties. The paper mainly emphasizes on reviewing the detailed study and generation of Relative Superior Tricorn and Multicorns along with Relative Superior Julia Set.

Complex and Inverse Complex Dynamics of Fractals using Ishikawa Iteration

Complex graphics of dynamical system have been a subject of intense research nowadays. The fractal geometry is the base of these beautiful graphical images. Many researchers and authors have worked to study the complex nature of the two most popular sets in fractal geometry, the Julia set and the Mandelbrot set, and proposed their work in various forms using existing tools and techniques. Still researches are being conducted to study and reveal the new concepts unexplored in the complexities of these two most popular sets of fractal geometry. Recently, Ashish Negi, Rajeshri Rana and Yashwant S. Chauhan are among those researchers who have contributed a lot in the area of Fractal Geometry applications. In this paper we review the recently done work on complex and inverse complex functions for producing beautiful fractal graphics. The reviewed work mainly emphasizes on the study of the nature of complex and inverse complex functional dynamics using Ishikawa iterates and existence of relative superior Mandel-bar set.

Stability results for Jungck-type iterative processes in convex metric spaces

In this paper, we obtain some stability results in complete convex metric spaces for nonselfmappings satisfying certain general contractivity condition. We prove our results for Jungck–Mann and Jungck–Ishikawa iterations. Our results extend several stability results in the literature.

The Equivalence of Convergence Results between Ishikawa and Mann Iterations with Errors for Uniformly Continuous Generalized Φ‐Pseudocontractive Mappings in Normed Linear Spaces

We prove the equivalence of the convergence of the Mann and Ishikawa iterations with errors for uniformly continuous generalized Φ‐pseudocontractive mappings in normed linear spaces. Our results extend and improve the corresponding results of Xu, 1998, Kim et al., 2009, Ofoedu, 2006, Chidume and Zegeye, 2004, Chidume, 2001, Chang et al. 2002, Liu, 1995, Hirano and huang, 2003, C. E. Chidume and C. O. Chidume, 2005, and huang, 2007.

Dynamics of Mandelbrot set with Transcendental Function

These days Mandelbrot set with transcendental function is an interesting area for mathematicians. New equations have been created for Mandelbrot set using trigonometric, logarithmic and exponential functions. Earlier, Ishikawa iteration has been applied to these equations and generate new fractals named as Relative Superior Mandelbrot Set with transcendental function. In this paper, the Mann iteration is being applied on Mandelbrot set with sine function i.e. sin(zn)+c and new fractals with the concept of Superior Transcendental Mandelbrot Set will be shown. Our goal is to focus on the less number of iterations which are required to obtain fixed point of function sin(zn)+c.

The Equivalence of Convergence Results of Modified Mann and Ishikawa Iterations with Errors without Bounded Range Assumption

Let E be an arbitrary uniformly smooth real Banach space, let D be a nonempty closed convex subset of E, and let T : D → D be a uniformly generalized Lipschitz generalized asymptotically Φ‐strongly pseudocontractive mapping with q ∈ F(T) ≠ ∅. Let {an}, {bn}, {cn}, {dn} be four real sequences in [0,1] and satisfy the conditions: (i) an + cn ≤ 1, bn + dn ≤ 1; (ii) an, bn, dn → 0 as n → ∞ and cn = o(an); (iii) . For some x0, z0 ∈ D, let {un}, {vn}, {wn} be any bounded sequences in D, and let {xn}, {zn} be the modified Ishikawa and Mann iterative sequences with errors, respectively. Then the convergence of {xn} is equivalent to that of {zn}.

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.

On Multivalued Nonexpansive Mappings in ℝ‐Trees

The relationships between nonexpansive, weakly nonexpansive, *‐nonexpansive, proximally nonexpansive, proximally continuous, almost lower semicontinuous, and ɛ‐semicontinuous mappings in ℝ‐trees are studied. Convergence theorems for the Ishikawa iteration processes are also discussed.

An iterative Algorithm for Hemicontractive Mappings in Banach Spaces

We First introduce a three‐step iterative algorithm for approximating the fixed points of the hemicontractive mappings in Banach spaces. Consequently, we prove the strong convergence of the proposed algorithm under some assumptions. Since three‐step iterations include Ishikawa iterations as special cases, our result continue to hold for these problems. Our main results can be viewed as an important refinement of the previously known results.

APPROXIMATING COMMON FIXED POINTS FOR TOTAL ASYMPTOTICALLY NONEXPANSIVE MAPPINGS

In this paper, we first show the weak convergence of the modified Ishikawa iteration process with errors of two total asymptotically nonexpansive mappings, which generalizes the result due to Khan and Fukhar-ud-din [1]. Next, we show the strong convergence of the modified Ishikawa iteration process with errors of two total asymptotically nonexpansive mappings satisfying Condition (), which generalizes the result due to Fukhar-ud-din and Khan [2].

On Convergence Results for Lipschitz Pseudocontractive Mappings

We establish the strong convergence for the Ishikawa iteration scheme associated with Lipschitz pseudocontractive mappings in real Banach spaces. Moreover, our technique of proofs is of independent interest.