Ishikawa Iteration Methods Research Articles

Iterative scheme generating method beyond Ishikawa iterative method

Iterative scheme generating method beyond Ishikawa iterative method

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

STRONG CONVERGENCE TO COMMON FIXED POINTS USING ISHIKAWA AND HYBRID METHODS FOR MEAN-DEMICLOSED MAPPINGS IN HILBERT SPACES

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.

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.

Some fixed point results for enriched nonexpansive type mappings in Banach spaces

In this paper, we introduce two new classes of nonlinear mappings and present some new existence and convergence theorems for these mappings in Banach spaces. More precisely, we employ the Krasnosel'skii iterative method to obtain fixed points of Suzuki-enriched nonexpansive mappings under different conditions. Moreover, we approximate the fixed point of enriched-quasinonexpansive mappings via Ishikawa iterative method.

On the convergence of Ishikawa iterative process for a pair of single-valued and multi-valued nonexpansive mappings

Let K be a nonempty compact convex subset of a complete CAT(0) space X and let t: K → K and T: K → CB(K) be a single-valued nonexpansive mapping and a multi-valued nonexpansive mapping, respectively. Assume, in addition, that F(t) ∩ F(T) ≠ ∅ and Tp = {p} for all p ∈ F(t) ∩ F(T). We prove that the sequence of modified Ishikawa iteration method generated from an arbitrary x0 ∈ K by yn = (1 - βn) xn ⊕ βn zn, xn+1 = (1-αn) xn ⊕ αntyn, where zn ∈ Txn and {αn}, {βn} are sequences of positive numbers satisfying 0< a ≤ αn, βn ≤ b < 1 converges strongly to a common fixed point of t and T; that is, there exists x ∈ K such that x = tx ∈ Tx. Our results extends and generalizes some theorems of T. Puttasontiphot [24].

New-Fangled Mandelbrot and Julia Sets for Logarithmic Function

In this paper we explore the dynamics of complex logarithmic function for integer and non-integer values. We have used Ishikawa iteration method for generating fractals and analyzed them.

The new modified Ishikawa iteration method for the approximate solution of different types of differential equations

Abstract In this article, the new Ishikawa iteration method is presented to find the approximate solution of an ordinary differential equation with an initial condition. Additionally, some numerical examples with initial conditions are given to show the properties of the iteration method. Furthermore, the results of absolute errors are compared with Euler, Runge-Kutta and Picard iteration methods. Finally, the present method, namely the new modified Ishikawa iteration method, is seen to be very effective and efficient in solving different type of the problem. MSC:65K15, 65L07, 65L06, 65L70.

Hybrid Ishikawa iterative methods for a nonexpansive semigroup in Hilbert space

In this paper, on the base of the Ishikawa iteration method and the hybrid method in mathematical programming, we give two new strong convergence methods for finding a point in the common fixed point set of a nonexpansive semigroup in Hilbert space.

Strong Convergence Theorems of Modified Ishikawa Iterative Method for an Infinite Family of Strict Pseudocontractions in Banach Spaces

We introduce a new modified Ishikawa iterative process and a new W‐mapping for comput‐ ing fixed points of an infinite family of strict pseudocontractions mapping in the framework of q‐uniformly smooth Banach spaces. Then, we establish the strong convergence theorem of the proposed iterative scheme under some mild conditions. The results obtained in this paper extend and improve the recent results of Cai and Hu 2010, Dong et al. 2010, Katchang and Kumam 2011 and many others in the literature.

Corrigendum to “Strong convergence of the modified Ishikawa iterative method for infinitely many nonexpansive mappings in Banach spaces” [Comput. Math. Appl. 59 (2010) 1473–1483

In this communication we give the following corrigendum for the paper ‘‘Strong convergence of the modified Ishikawa iterativemethod for infinitelymany nonexpansivemappings in Banach spaces’’ [Comput. Math. Appl. 59 (2010) 1473–1483, doi: 10.1016/j.camwa.2010.01.025]: The conditions (ii) in Theorem3.1 should be revised to 0 a for some a ∈ (0, 1). Also assume that subset C is such that C + C ⊂ C . The proof on page 1480, lines 23–27, should have the following inserted: ‖xn − Wnxn‖ ≤ ‖xn − yn‖ + ‖yn − Wnzn‖ + ‖Wnzn − Wnxn‖

Strong convergence of the modified Ishikawa iterative method for infinitely many nonexpansive mappings in Banach spaces

In this paper, we introduce a new modified Ishikawa iterative process for computing fixed points of an infinite family nonexpansive mapping in the framework of Banach spaces. Then, we establish the strong convergence theorem of the proposed iterative scheme under some mild conditions which solves a variational inequality. The results obtained in this paper extend and improve on the recent results of Qin et al. [Strong convergence theorems for an infinite family of nonexpansive mappings in Banach spaces, Journal of Computational and Applied Mathematics 230 (1) (2009) 121–127], Cho et al. [Approximation of common fixed points of an infinite family of nonexpansive mappings in Banach spaces, Computers and Mathematics with Applications 56 (2008) 2058–2064] and many others.

Ishikawa Iterative Process for a Pair of Single-valued and Multivalued Nonexpansive Mappings in Banach Spaces

AbstractLet "Equation missing" be a nonempty compact convex subset of a uniformly convex Banach space "Equation missing", and let "Equation missing" and "Equation missing" be a single-valued nonexpansive mapping and a multivalued nonexpansive mapping, respectively. Assume in addition that "Equation missing" and "Equation missing" for all "Equation missing". We prove that the sequence of the modified Ishikawa iteration method generated from an arbitrary "Equation missing" by "Equation missing", "Equation missing", where "Equation missing" and "Equation missing", "Equation missing" are sequences of positive numbers satisfying "Equation missing", "Equation missing", converges strongly to a common fixed point of "Equation missing" and "Equation missing"; that is, there exists "Equation missing" such that "Equation missing".

WEAK AND STRONG CONVERGENCE FOR COMMON FIXED POINTS OF A PAIR OF NONEXPANSIVE AND ASYMPTOTICALLY NONEXPANSIVE MAPPINGS

In this paper, we establish some weak and strong convergence theorems of modified Ishikawa iteration methods with errors with respect to a pair of nonexpansive and asymptotically nonexpansive mappings in uniformly convex Banach spaces. The results presented in this paper generalize, improve and unify a few results due to Chang [2], Osilike and Aniagbosor [10], Rhoades [11] and Schu [13, 14] and others.

Strong convergence of Ishikawa iterative method for nonexpansive mappings in Hilbert spaces

In this paper, we introduce a modified Ishikawa iterative process for approximating a fixed point of nonexpansive mappings in Hilbert spaces. we establish the strong convergence theorem of the general iteration scheme under some mild conditions. Our results extend and improve the results announced by many others.

The equivalence between the convergences of Mann and Ishikawa iteration methods with errors for demicontinuous φ‐strongly accretive operators in uniformly smooth Banach spaces

We investigate the equivalence between the convergences of the Mann iteration method and the Ishikawa iteration method with errors for demicontinuous φ‐strongly accretive operators in uniformly smooth Banach spaces. A related result deals with the equivalence of the Mann iteration method and the Ishikawa iteration method for φ‐pseudocontractive operators in nonempty closed convex subsets of uniformly smooth Banach spaces. The results presented in this paper extend and improve the corresponding results in the literature.

ISHIKAWA ITERATION WITH ERRORS FOR APPROXIMATING FIXED POINTS OF STRICTLY PSEUDOCONTRACTIVE MAPPINGS OF BROWDER-PETRYSHYN TYPE

Let q > 1 and E be a real q−uniformly smooth Banach space. Let K be a nonempty closed convex subset of E and T : K → K be a strictly pseudocontractive mapping in the sense of F. E. Browder and W. V. Petryshyn [1]. Let {un} be a bounded sequence in K and {αn}, {βn}, {γn} be real sequences in [0,1] satisfying some restrictions. Let {xn} be the bounded sequence in K generated from any given x1 ∈ K by the Ishikawa iteration method with errors: yn = (1 − βn)xn + βnTxn, xn+1 = (1 −αn − γn)xn + αnTyn + γnun, n ≥ 1. It is shown in this paper that if T is compact or demicompact, then {xn} converges strongly to a fixed point of T.

Stability of iterative procedures with errors for approximating common fixed points of a couple of q-contractive-like mappings in Banach spaces

Recently, Agarwal, Cho, Li and Huang [R.P. Agarwal, Y.J. Cho, J. Li, N.J. Huang, Stability of iterative procedures with errors approximating common fixed points for a couple of quasi-contractive mappings in q-uniformly smooth Banach spaces, J. Math. Anal. Appl. 272 (2002) 435–447] introduced the new iterative procedures with errors for approximating the common fixed point of a couple of quasi-contractive mappings and showed the stability of these iterative procedures with errors in Banach spaces. In this paper, we introduce a new concept of a couple of q-contractive-like mappings ( q > 1 ) in a Banach space and apply these iterative procedures with errors for approximating the common fixed point of the couple of q-contractive-like mappings. The results established in this paper improve, extend and unify the corresponding ones of Agarwal, Cho, Li and Huang [R.P. Agarwal, Y.J. Cho, J. Li, N.J. Huang, Stability of iterative procedures with errors approximating common fixed points for a couple of quasi-contractive mappings in q-uniformly smooth Banach spaces, J. Math. Anal. Appl. 272 (2002) 435–447], Chidume [C.E. Chidume, Approximation of fixed points of quasi-contractive mappings in L p spaces, Indian J. Pure Appl. Math. 22 (1991) 273–386], Chidume and Osilike [C.E. Chidume, M.O. Osilike, Fixed points iterations for quasi-contractive maps in uniformly smooth Banach spaces, Bull. Korean Math. Soc. 30 (1993) 201–212], Liu [Q.H. Liu, On Naimpally and Singh's open questions, J. Math. Anal. Appl. 124 (1987) 157–164; Q.H. Liu, A convergence theorem of the sequence of Ishikawa iterates for quasi-contractive mappings, J. Math. Anal. Appl. 146 (1990) 301–305], Osilike [M.O. Osilike, A stable iteration procedure for quasi-contractive maps, Indian J. Pure Appl. Math. 27 (1996) 25–34; M.O. Osilike, Stability of the Ishikawa iteration method for quasi-contractive maps, Indian J. Pure Appl. Math. 28 (1997) 1251–1265] and many others in the literature.

Iterative solutions of nonlinear equations with φ-strongly accretive operators in uniformly smooth Banach spaces

Suppose that X is a uniformly smooth Banach space and T : X → X is a demicontinuous (not necessarily Lipschitz) φ-strongly accretive operator. It is proved that the Ishikawa iterative method with errors converges strongly to the solutions of the equations f = Tx and f = x+ Tx, respectively. A related result deals with the approximation of fixed points of φ-strongly pseudocontractive operators. Our results extend, improve, and unify the recent results obtained by Chidume [1,2] and Zhou [3].

On the equivalence of Mann and Ishikawa iteration methods

We show that certain Mann and Ishikawa iteration schemes are equivalent for various classes of functions.