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Strong convergence theorems that approximate common fixed points of two nonlinear mappings are presented. Our method is based on the Martinez-Yanes–Xu iteration, which extends Nakajo and Takahashi’s CQ method. In this paper, by exploiting the mean-valued iteration procedure, we further develop Nakajo and Takahashi’s CQ method and Takahashi, Takeuchi, and Kubota’s shrinking projection method. The approach of this paper does not require that the two mappings be continuous or commutative. The types of mappings considered in this paper include nonexpansive mappings and other well-known classes of mappings as special cases.

It is well known that the classical trapezoidal rule (TR2) is only second order of convergence. However, Leonhard Euler and Colin Maclaurin had independently discovered that the error terms of TR2 can be represented as an infinite sum of terms in even powers of the mesh size. In this article, we exploit this error representation to construct a nonconventional fourth-order trapezoid rule (TR4) and combine it with different iterative schemes to solve nonlinear Fredholm integral equations. First, the problem is projected into a discrete space and approximated with TR2. Then, the leading error term of the TR2 is approximated with finite difference formula, giving rise to TR4. Next, we derive three discrete fixed point methods (Picard, Ishikawa, and Argawal) and a Newton–Raphson method to approximate the solution of the resulting nonlinear system. This led to four different algorithms for solving nonlinear Fredholm equation. The algorithms were tested and compared for accuracy and efficiency by using six example problems with known exact solutions. The results show that (i) all the methods converge with fourth order accuracy, but only the Newton scheme maintained the fourth order accuracy at all levels of grid refinement and for all the six examples considered in this work, (ii) the Ishikawa and Argawal iterative processes have approximately equal performance, (iii) whenever the Picard scheme is applicable, it is fast, but may degrade the quality of the solution at very fine mesh more than the others, and (iv) the Newton approach, though may be slower in some problems, is guaranteed to produce very reliable results and maintain fourth order of accuracy. [ABSTRACT FROM AUTHOR]

In this article, we establish a concept of fixed point result in Uniformly convex Banach space. Our main finding uses the Ishikawa iteration technique in uniformly convex Banach space to demonstrate strong convergence. Additionally, we use our primary result to demonstrate some corollaries.

This paper addresses the approximation problem for fixed points of Zamfirescu mappings (Z-mapping) [Arch. Math. 23(1) (1972), 292-298]. We use a three-step mean iteration that combines Noor's iteration as well as mean-valued iteration, and we prove a general theorem that extends Berinde's strong convergence theorem [Acta Math. Univ. Comenianae 73(1) (2004), 119-126]. Our results are obtained in arbitrary real Banach space setting. Furthermore, an application to a variational inequality problem is presented in a framework of real Hilbert spaces. [ABSTRACT FROM AUTHOR]

In this paper, we establish a strong convergence theorem that approximates a common fixed point of two nonlinear mappings by comprehensively using an Ishikawa iterative method, a hybrid method, and a mean-valued iterative method. The shrinking projection method is also developed. The nonlinear mappings are a general type that includes nonexpansive mappings and other classes of well-known mappings. The two mappings are not assumed to be continuous or commutative. The main theorems in this paper generate a variety of strong convergence theorems including a type of "three-step iterative method". An application to the variational inequality problem is also given. [ABSTRACT FROM AUTHOR]

Combining the ideas of Ishikawa iteration and mean iteration, we establish weak convergence theorems for finding common fixed points of nonlinear mappings. The mappings are not necessarily continuous or commutative. We consider a class of mappings which includes nonexpansive mappings, generalized hybrid mappings and normally 2-generalized hybrid mappings as special cases. Our result generates many alternative iteration schemes to approximate common fixed points of nonlinear mappings and improves many existing theorems in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we prove some weak and strong convergence results for mappings satisfying condition (CC) using Ishikawa iteration process in uniformly convex Banach spaces. The results obtained in the paper extend and improve many well-known results in the iterations theory. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Here, we develop escape criteria for pc(z) = sin(z n)−az +c, a, c ∈ C, n ≥ 2, exploiting four different iterations of fixed point theory to explore various Mandelbrot sets which are different than the classical Mandelbrot set . Our concern is to utilize the lesser number of iterations that are necessary to attain the fixed point of the transcendental complex-valued sine function. Further, we investigate the effect of variables on the shape, size, color, and dynamics of fractals. Noticeably, some of the obtained fractals symbolize the Swastika (a symbol of spirituality and divinity in Indian religions), Shivling (an abstract representation of the Hindu God Shiva), flowers, spiders, butterflies, Rangoli (made mainly in the festive season in India), art on glass, and so on. Interestingly, the higher-order Mandelbrot set in Picard-orbit has a resemblance to Corona-virus. [ABSTRACT FROM AUTHOR]

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.

Abstract In this paper, some new formula of error estimations of Ishikawa iteration and some strong convergence theorems of strongly demicontractive mappings are first obtained. And then, convergence speeds and error estimations of the Ishikawa iteration and the Mann iteration are discussed in some examples. Finally, T-stability of the Ishikawa iteration is proved.

In this paper, we introduce the notion of Osilike-Berinde-G-nonexpansive mappings in metric spaces and show that every Osilike-Berinde-G-nonexpansive mapping with nonempty fixed point set is a G-quasinonexpansive mapping. We also prove the demiclosed principle and apply it to obtain a fixed point theorem for Osilike-Berinde-G-nonexpansive mappings. Strong and Δ-convergence theorems of the Ishikawa iteration process for G-quasinonexpansive mappings are also discussed. [ABSTRACT FROM AUTHOR]

Let X be a complete CAT(0) space, K ≠ ∅ be a closed and convex subset of X and T : K → C (K) be a multivalued nonexpansive mapping. We prove that the sequence of Ishikawa iteration converges to an endpoint of T. This improves, extends and unifies some recently announced results of the current literature. [ABSTRACT FROM AUTHOR]

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The aim of this paper is two fold: the first is to define two new classes of mappings and show the existence and iterative approximation of their fixed points; the second is to show that the Ishikawa, Mann, and Krasnoselskij iteration methods defined for such classes of mappings are equivalent. An application of the main results to solve split feasibility and variational inequality problems are also given.

We extend the (CN) inequality of Bruhat and Tits in CAT(0) spaces to the general setting of uniformly convex hyperbolic spaces. We also show that, under some appropriate conditions, the sequence of Ishikawa iteration defined by Panyanak converges to a strict fixed point of a multi-valued Suzuki mapping. [ABSTRACT FROM AUTHOR]

The aim of the present paper is to state and prove some convergence theorems for the Mann and Ishikawa iteration schemes involving C * -algebra-multi-valued contractive mappings in the setting of convex C * -algebra-valued metric spaces. The convergence theorems of the proposed iterations to a common fixed point of finite and infinite family of such mappings are also established.

A new class of α-hemicontractive maps T for which the strong convergence of the Ishikawa iteration algorithm to a fixed point of T is assured is introduced and studied. The study is a continuation of a recent study of a new class of α-demicontractive mappings T by L. Mărușter and Ș. Mărușter, Mathematical and Computer Modeling 54 (2011) 2486-2492 in which they proved strong convergence of the Mann iteration scheme to a fixed point of T. Our class of α-hemicontractive maps is more general than the class of α-demicontractive maps. No compactness assumption is imposed on the operator or it’s domain, and no additional requirement is imposed on the set of fixed points.

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

In a recent paper [Pata, V., A fixed point theorem in metric spaces, J. Fixed Point Theory Appl., 10 (2011), No. 2, 299–305], the author stated and proved a fixed point theorem that is intended to generalize the well known Banach's contraction mapping principle. In this note we show that the main result in the above paper does not hold at least in two extremal cases for the parameter #949; involved in the contraction condition used there. We also present some illustrative examples and related results. [ABSTRACT FROM AUTHOR]

In this paper, we prove the weak and strong convergence of a sequence $\{x_{n}\}$ generated by the Ishikawa iteration to some common fixed points of two G-nonexpansive mappings defined on a Banach space endowed with a graph. [ABSTRACT FROM AUTHOR]

The use of Mann iteration in the stable inversion of NARX models that have been converted to state space form is investigated to either recover the convergence or improve the accuracy of the best approximate solution under conditions when Picard iteration fails to converge. Attention is given to the use of filtering and time-varying iteration gains. The results are potentially of use in response reconstruction for fatigue testing purposes where the inverse of a NARX model, obtained by system identification, may be used to achieve the reconstruction. [ABSTRACT FROM AUTHOR]

The aim of this paper is to present some modifications of the biomorphs generation algorithm introduced by Pickover in 1986. A biomorph stands for biological morphologies. It is obtained by a modified Julia set generation algorithm. The biomorph algorithm can be used in the creation of diverse and complicated forms resembling invertebrate organisms. In this paper the modifications of the biomorph algorithm in two directions are pro- posed. The first one uses different types of iterations (Picard, Mann, Ishikawa). The second one uses a sequence of parameters instead of one fixed parameter used in the original biomorph algorithm. Biomorphs generated by the modi- fied algorithm are essentially different in comparison to those obtained by the standard biomorph algorithm, i.e., the algorithm with Picard iteration and one fixed constant. [ABSTRACT FROM AUTHOR]

We prove two strong convergence results of the Ishikawa iteration for a multivalued quasi-nonexpansive mapping in a complete geodesic space with curvature bounded above. Our results improve significantly the recent results of Panyanak (Fixed Point Theory Appl. 2014:1, ) in the sense that many restrictions in his results are weakened. In particular, we can conclude that the convergence result of the Mann iteration which cannot be deduced from Panyanak's results. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and uniqueness of fixed points for a class of self-mappings satisfying certain rational expressions on closed, bounded and convex subsets with normal structures in reflexive Banach spaces. We show that, in particular, this class extends that introduced by Ray and Singh (Indian J. Pure Appl. Math. 9:216-221, 1978). As an application, we give an investigation of the convergence and stability of some iterative processes associated to these mappings. [ABSTRACT FROM AUTHOR]

Let the set $C \subset L_{1}([0,1])$ be nonempty, convex and compact for the convergence almost everywhere and $T: C \rightarrow C$ be a monotone nonexpansive mapping. In this paper, we study the behavior of the Krasnoselskii-Ishikawa iteration sequence $\{f_{n}\}$ defined by $f_{n+1} = \lambda f_{n} + (1-\lambda)T(f_{n})$, $n = 1, 2, \ldots$ , $\lambda\in(0,1)$. Then we prove a fixed point theorem for these mappings. Our result is new and was never investigated. [ABSTRACT FROM AUTHOR]

In this paper, by means of the concept of attractive points of a nonlinear mapping, we prove strong convergence theorem of the Ishikawa iteration for an (α, β)-generalized hybrid mapping in a uniformly convex Banach space, and obtain weak convergence theorem of the Ishikawa iteration for such a mapping in a Hilbert space. [ABSTRACT FROM AUTHOR]

Let X be a uniformly convex Banach space, and S, T be pair of mean non-expansive mappings. Some necessary and sufficient conditions are given for Ishikawa iterative sequence converge to common fixed points, and we prove that the sequence of Ishikawa iterations associated with S and T converges to the common fixed point of S and T. This generalizes former results proved by Z. Gu and Y. Li [4]. [ABSTRACT FROM AUTHOR]

We prove the equivalence between the \(T\)-stabilities of Krasnoselskij and Ishikawa iterations; a consequence is the equivalence with the \(T\)-stability of Picard-Banach iteration.

New proofs of existence and uniqueness results for the solution of the Cauchy problem with delay are obtained by use of Mann-Ishikawa iteration.

In this paper we consider the dynamics of complex transcendental function i.e. tan function using iterative procedure known as Ishikawa Method. Fractals are created and examined for integer and non-integer values. New Julia sets are generated for different values of parameters defined by Ishikawa iteration and transcendental controlled function. Here different dendrites, spiral lobes are considered and our study relies on the analysis of Julia sets. [ABSTRACT FROM AUTHOR]

In this paper, we prove Δ–convergence theorems for Ishikawa iteration of single-valued and multivalued nonexpansive mappings in CAT(0) spaces. This results we obtain are analogs of CAT(0) spaces results of Laokul and Panyanak [11]. [ABSTRACT FROM AUTHOR]

We show the equivalence bewteen the convergences of Mann and Ishikawa iterations dealing with various classes of non-Lipschitzian operators.

The convergence rates of Picard, Mann and Ishikawa iterations have been compared by several authors for a class of quasi-contractive maps defined on an arbitrary closed convex subset of a Banach space (e.g. [1], [3] and [10]). In this paper, a comparison of the convergence rates of those iterations are studied for a more general class of operators called the generalized contractive operators. [ABSTRACT FROM AUTHOR]

We give an affirmative answer to the following question: are Mann and Ishikawa iterations equivalent under the assumptions that \(\lim_{n\rightarrow\infty }\alpha_{n}\neq0\;\) and \(\lim_{n\rightarrow\infty}\beta_{n}\neq0\)?

Abstract: Let E be a real q-uniformly smooth Banach space which is also uniformly convex (for example, or spaces, ), and K a nonempty closed convex (not necessarily bounded) subset of E. Let be a k-strictly asymptotically pseudocontractive map with a nonempty fixed-point set. It is proved that is demiclosed at 0. Furthermore, weak and strong convergence of an averaging iteration method to a fixed point of T are proved. [Copyright &y& Elsevier]

Abstract: As an important iteration, the Mann and Ishikawa iteration has extensive application in fixed point theory. In 1991, David Borwein and Jonathan Borwein proved the convergence of the Mann iteration on a closed bounded interval in their paper. In this paper, we will extend their result to an arbitrary interval and to the Ishikawa iteration, indicating the necessary and sufficient condition for the convergence of Ishikawa iteration of continuous functions on an arbitrary interval. [Copyright &y& Elsevier]

Abstract: We show that T-stability of Mann and Ishikawa iterations are equivalent. [Copyright &y& Elsevier]

In this paper, we have shown that the convergence of one-step, two-step and three-step iterations is equivalent, which are known as Mann, Ishikawa and Noor iteration procedures, for a special class of Lipschitzian operators defined in a closed, convex subset of an arbitrary Banach space. [ABSTRACT FROM AUTHOR]

Abstract: In the present paper, motivated by Stević''s iteration method, we show the stability result of the following unified iterative scheme for a ϕ-strongly pseudocontractive selfmapping T. This stability result implies the stability of Stević''s iteration scheme naturally. [Copyright &y& Elsevier]

Abstract: Necessary and sufficient conditions for the convergence of Picard iteration to a fixed point for a continuous mapping in metric spaces are established. As application, we prove the convergence theorem of Ishikawa iteration to a fixed point for a nonexpansive mapping in Banach spaces. [Copyright &y& Elsevier]

We show that the convergence of Mann, Ishikawa iterations are equivalent to the convergence of a multistep iteration, for various classes of operators. [Copyright &y& Elsevier]

The stability of the Ishikawa iteration procedures was studied for one class of continuity strong pseudocontraction and continuity strongly accretive operators with bounded range in real uniformly smooth Banach space. Under parameters satisfying certain conditions, the convergence of iterative sequences was proved. The results improve and extend the recent corresponding results, and supply the basis of theory for further discussing convergence of iteration procedures with errors. [ABSTRACT FROM AUTHOR]