Common fixed points Noor iteration for a finite family of asymptotically quasi-nonexpansive mappings in Banach spaces

In this paper, we introduce a general iteration scheme for a finite family of asymptotically quasi-nonexpansive mappings. The new iterative scheme includes the modified Mann and Ishikawa iterations, three-step iterative scheme of Xu and Noor and Khan and Takahashi scheme as special cases. Our results are generalizations as well as refinement of several known results in the current literature.     

An alternative technique for solving a coupled PDE system in interphase heat transfer

Recently an accelerated iterative procedure was studied for solving a coupled partial differential equation system in interphase heat transfer to improve some existing iterative procedures in the literature. In that procedure, at each step of the iteration one has to evaluate the derivative of a well-known function at a new point. In this paper, an alternative approach is proposed in which one has to evaluate the derivative only once throughout the procedure. The proposed new iterative scheme also has the same order of convergence and takes lesser number of iterations for certain benchmark problems. An interesting theoretical study on the monotone convergence as well as error estimate of the proposed iterative procedure are provided for continuous as well as discretized problems. The proposed iterative procedure also supplements the existence and uniqueness of the solution in both the cases. A comparative numerical study is also done to demonstrate the efficacy of the proposed scheme.     

On the efficiency index of one-point iterative processes

In this paper, we analyze the index of efficiency of one-point iterative processes, which are in practice the most used methods to solve a nonlinear equation. We obtain the best situation for one-point iterative processes with cubic convergence: Chebyshev’s method, Halley’s method, the super-Halley method and many others classical iterative methods with order of convergence three. By means of a construction of particular multipoint iterations, we get to improve the best situation obtained for one-point methods. Moreover, these type of multipoint iterations, can be considered as quasi-one-point iterations, since they only depend on one initial approximation. Numerical examples are given and the computed results support this theory. Partly supported by the Ministry of Education and Science (MTM 2005-03091) and the University of La Rioja (ATUR-05/43).     

On the convergence of right transforming iterations for the numerical solution of PDE‐constrained optimization problems

We present an iterative solver, called right transforming iterations (or right transformations), for linear systems with a certain structure in the system matrix, such as they typically arise in the framework of Karush–Kuhn–Tucker (KKT) conditions for optimization problems under PDE constraints. The construction of the right transforming scheme depends on an inner approximate solver for the underlying PDE subproblems. We give a rigorous convergence proof for the right transforming iterative scheme in dependence on the convergence properties of the inner solver. Provided that a fast subsolver is available, this iterative scheme represents an efficient way of solving first‐order optimality conditions. Numerical examples endorse the theoretically predicted contraction rates. Copyright © 2011 John Wiley & Sons, Ltd.     

Banach空间中渐近非扩张非自身映射带误差Noor迭代的收敛性定理

本文主要对三个渐近非扩张非自身映射引入了一种新的投影型Noor迭代程序,并在一致凸Banach空间中给出了该Noor迭代序列的弱与强收敛性定理.我们的主要结果推广和改进了该领域许多近期的结果.     

带扰动映像的广义平衡问题和不动点问题的混合粘性逼近问题

引入一个用于寻求带扰动映像的广义平衡问题解集以及可数无穷多非扩张映像之族公共不动点集的公共解的新的迭代算法. 证明了由此算法生成的序列的强收敛性. 所得的结果推广改进了先前许多作者的结果.     

Weak and Strong Convergence Theorems of Quasi-Nonexpansive Mappings in a Hilbert Spaces

In this paper, we first prove the weak convergence for the Moudafi’s iterative scheme of two quasi-nonexpansive mappings. Then we prove the weak convergence for the Moudafi’s iterative scheme of quasi-nonexpansive mapping and nonexpansive mapping. Finally, we prove the strong convergence for the Moudafi’s iterative scheme of two quasi-nonexpansive mappings. Our results generalize the recent results due to Iemoto and Takahashi.     

Numerical methods for fourth‐order nonlinear elliptic boundary value problems

The aim of this article is to present several computational algorithms for numerical solutions of a nonlinear finite difference system that represents a finite difference approximation of a class of fourth‐order elliptic boundary value problems. The numerical algorithms are based on the method of upper and lower solutions and its associated monotone iterations. Three linear monotone iterative schemes are given, and each iterative scheme yields two sequences, which converge monotonically from above and below, respectively, to a maximal solution and a minimal solution of the finite difference system. This monotone convergence property leads to upper and lower bounds of the solution in each iteration as well as an existence‐comparison theorem for the finite difference system. Sufficient conditions for the uniqueness of the solution and some techniques for the construction of upper and lower solutions are obtained, and numerical results for a two‐point boundary‐value problem with known analytical solution are given. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17:347–368, 2001     

A PARALLEL COMPUTATION SCHEME FOR IMPLICIT RUNGE-KUTTA METHODS AND THE ITERATIVELY B-CONVERGENCE OF ITS NEWTON ITERATIVE PROCESS

In this paper, based on the implicit Runge-Kutta(IRK) methods, we derive a class of parallel scheme that can be implemented on the parallel computers with Ns(N is a positive even number) processors efficiently, and discuss the iteratively B-convergence of the Newton iterative process for solving the algebraic equations of the scheme, secondly we present a strategy providing initial values parallelly for the iterative process. Finally, some numerical results show that our parallel scheme is higher efficient as N is not so large.     

On Numerov's method for a class of strongly nonlinear two-point boundary value problems

The purpose of this paper is to give a numerical treatment for a class of strongly nonlinear two-point boundary value problems. The problems are discretized by fourth-order Numerov's method, and a linear monotone iterative algorithm is presented to compute the solutions of the resulting discrete problems. All processes avoid constructing explicitly an inverse function as is often needed in the known treatments. Consequently, the full potential of Numerov's method for strongly nonlinear two-point boundary value problems is realized. Some applications and numerical results are given to demonstrate the high efficiency of the approach.     

Convergence of Nonstationary Iterative Methods for Solving Singular Linear Equations with Index One

In this paper, we investigate the nonstationary iterative scheme for solving consistent singular linear system with index one. We utilize the group inverse to present a sufficient condition for the convergence of the nonstationary iterative method. Our result extends the known results of the stationary iterative scheme. Finally, we present a sufficient condition for the multisplitting algorithm and provide numerical examples to illustrate the advantages of nonstationary method.     

关于渐近非扩张映象的强收敛定理

在具有一致正规结构且其范数是一致Gateaux可微的实Banach空间中,为寻求渐近非扩张半群的公共不动点,引入了一种新的迭代序列.在适当的条件下,用迭代逼近算法,证明了逼近于这一公共不动点的某些强收敛定理.其结果也推广和改进了引文中相应的结果.     

A nonlinear iteration method for solving a two-dimensional nonlinear coupled system of parabolic and hyperbolic equations

A nonlinear iteration method for solving a class of two-dimensional nonlinear coupled systems of parabolic and hyperbolic equations is studied. A simple iterative finite difference scheme is designed; the calculation complexity is reduced by decoupling the nonlinear system, and the precision is assured by timely evaluation updating. A strict theoretical analysis is carried out as regards the convergence and approximation properties of the iterative scheme, and the related stability and approximation properties of the nonlinear fully implicit finite difference (FIFD) scheme. The iterative algorithm has a linear constringent ratio; its solution gives a second-order spatial approximation and first-order temporal approximation to the real solution. The corresponding nonlinear FIFD scheme is stable and gives the same order of approximation. Numerical tests verify the results of the theoretical analysis. The discrete functional analysis and inductive hypothesis reasoning techniques used in this paper are helpful for overcoming difficulties arising from the nonlinearity and coupling and lead to a related theoretical analysis for nonlinear FI schemes.     

A strong convergence theorem for common fixed points of two relatively nonexpansive mappings and its applications

In this paper, a projection iterative scheme is introduced for the approximation method for finding common fixed points of two relatively nonexpansive mappings in a real uniformly smooth and uniformly convex Banach space and, by using the iterative scheme, we obtain a strong convergence theorem and some applications of the main result. Our results extend the corresponding works given by some authors.     

Three-step Iterations with Errors for Nonlinear Strongly Accretive Operator Equations

In this paper, we suggest and analyse a three-step iterative scheme with errors for solving nonlinear strongly accretive operator equation Tx = f without the Lipshitz condition. The results presented in this paper improve and extend current results in the more general setting.     

On Sequences of Nested Triangles

Summary For a given triangle, we consider several sequences of nested triangles obtained via iterative procedures. We are interested in the limiting behavior of these sequences. We briefly mention the relevant known results and prove that the triangle determined by the feet of the angle bisectors converges in shape towards an equilateral one. This solves a problem raised by Trimble~[5].     

Three-step iterations for nonlinear accretive operator equations

In this paper, we suggest and analyze a three-step iterative scheme for solving nonlinear strongly accretive operator equation Tx=f without continuous condition in a uniformly smooth Banach space. Our results include the Ishikawa, Mann and Noor iterations as special cases. The results presented in this paper improve and extend almost all the current results in the more general setting.     

A general iterative scheme for k-strictly pseudo-contractive mappings and optimization problems

In this paper, we introduce a new general iterative scheme for finding fixed points of a strictly pseudo-contractive mapping and then prove that the sequence generated by the proposed iterative scheme converges strongly to a fixed point of the mapping, which is a solution of a certain optimization problem related to a strongly positive bounded linear operator. Additional results of the main result are also obtained.     

Iterative approximation of a common solution of a split equilibrium problem,a variational inequality problem and a fixed point problem

In this paper, we introduce an iterative method to approximate a common solution of a split equilibrium problem, a variational inequality problem and a fixed point problem for a nonexpansive mapping in real Hilbert spaces. We prove that the sequences generated by the iterative scheme converge strongly to a common solution of the split equilibrium problem, the variational inequality problem and the fixed point problem for a nonexpansive mapping. The results presented in this paper extend and generalize many previously known results in this research area.     

两个渐近demi-压缩映射的Ishikawa迭代过程的收敛性

研究了Banach空间中两个不同渐近demi-压缩映射的Ishikawa迭代过程,给出了此Ishikawa迭代序列收敛的充要条件.推广了以前的结论.