AN ITERATIVE EQUATION ON THE UNIT CIRCLE

A functional equation of nonlinear iterates is discussed on the circle S^1 for its continuous solutions and differentiable solutions. By lifting to R, the existence, uniqueness and stability of those solutions are obtained. Techniques of continuation are used to guarantee the preservation of continuity and differentiability in lifting.

CONTINUOUSLY DECREASING SOLUTIONS FOR A GENERAL ITERATIVE EQUATION

Most known results on polynomial-like iterative equations are concentrated to increasing solutions. Without the uniformity of orientation and monotonicity, it becomes much more difficult for decreasing cases. In this paper, we prove the existence of decreasing solutions for a general iterative equation, which was proposed as an open problem in [J. Zhang, L. Yang, W. Zhang, Some advances on functional equations, Adv. Math. （China） 24 （1995） 385-405] （or [W. Zhang, J.A. Baker, Continuous solutions of a polynomial-like iterative equation with variable coefficients, Ann. Polon. Math. 73 （2000） 29-36]）.

The C1 Solutions .of a Class of a Non-extended Iterative Equation

By using the Schauder＇s fixed point theorem, the existence, uniqueness and sta- bility of the C1 solutions of a class non-extended iterative equation are discussed.

Continuously decreasing solutions for polynomial-like iterative equations

Most known results on existence, uniqueness and stability for solutions of the polynomial-like iterative equation ∑ni=1λifi（x） = F（x） were obtained in the case of λ1 ≠0. In this paper, we construct Co decreasing solutions of the iterative equation in the case that A1 can vanish to answer the Leading Coeffi- cient Problem. Moreover, we also give the necessary and sufficiently condition for uniqueness of solutions.

Multivalued Solutions of an Iterative Equation with Variable Coefficients

In this paper, we consider a second order multivalued iterative equation with variable coefficients and the results on increasing solution and decreasing solution are obtained.

ANALYTIC SOLUTIONS OF AN ITERATIVE EQUATION WITH FIRST ORDER DERIVATIVE

This paper is concerned with a nonlinear iterative equation with first order derivative. By construction a convergent power series solution, analytic solutions for the original equation are obtained.

Existence and Uniqueness of Analytic Solutions of Systems of Iterative Functional Equations

Let r be a given positive number. Denote by D=D r the closed disc in the complex plane C whose center is the origin and radius is r. Write A(D,D)={f: f is a continuous map from D into itself, and f|D ° is analytic}. Suppose G,H: D 2n+1 →C are continuous maps (n≥2), and G|(D 2n+1 ) °, H|(D 2n+1 ) ° are analytic. In this paper, we study the system of iterative functional equationsG(z,f(z),…,f n(z), g(z),…,g n(z))=0, H(z,f(z),…,f n(z), g(z),…,g n(z))=0, for any z∈D,and give some conditions for the system of equations to have a solution or a unique solution in A(D,D) ×A(D,D).

DISCUSSION ON THE C^r－SOLUTIONS OF THE ITERATED EQUATION λ_1f（x）＋λ_2f^2（x）＝F（x）

In this paper, we consider the iterated equationλ1f(x) + λ2f2(x)=F(x)where f2(x)= f(f(x)), F (x) denotes known function and f(x) denotes the unknown function. There are given conditions for the existence, uniqueness and stability of C'-solutions ofthe iterated equation (*) and also there is a proved theorem for the continuous dependence of Cr-solutions of iterated equation (*) on the given function.

CONVERGENCE ANALYSIS OF THE LOPING OS-EM ITERATIVE VERSION OF THE CIRCULAR RADON TRANSFORM

The loping OS-EM iteration is a numerically efficient regularization method for solving ill-posed problems. In this article we investigate the loping OS-EM iterative method in connection with the circular Radon transform. We show that the proposed method converges weakly for the noisy data. Numerical tests are presented for a linear problem related to photoacoustic tomography.

AN APPLICATION OF HARDY-BOEDEWADT'S THEOREM TO ITERATED FUNCTIONAL EQUATIONS

In this paper an iterated functional equation of polynomial type which does not possess the firt order iterative term g(x) is to be discussed. The difficulties resulted from loss of the first order term are overcome by utilization of Hardy-Boedewadt's theorem.

A Note on a Functional Differential Equation with State Dependent Argument

This paper is concerned with solutions of a functional differential equation. Using Krasnoselskii’s fixed point theorem, the solutions can be obtained from periodic solutions of a companion equation.

TRANSONIC FLOW CALCULATION OF EULER EQUATIONS BY IMPLICIT ITERATING SCHEME WITH FLUX SPLITTING

Three dimensional Euler equations are solved in the finite volume form with van Leer's flux vector splitting technique. Block matrix is inverted by Gauss-Seidel iteration in two dimensional plane while strongly implicit alternating sweeping is implemented in the direction of the third dimension. Very rapid convergence rate is obtained with CFL number reaching the order of 100. The memory resources can be greatly saved too. It is verified that the reflection boundary condition can not be used with flux vector splitting since it will produce too large numerical dissipation. The computed flow fields agree well with experimental results. Only one or two grid points are there within the shock transition zone.

Improvement of CORDIC Algorithm

Aim To discuss the basic CORDIC algorithm that can be applied to digital signal processing and its applying condition called convergence range.Methods In addition to the original basic equation, another group iterative equation was used to evaluate the correspondent values of input data that did not lie within the convergence range. Results and Conclusion The improved CORDIC algorithm removes the limits of the range of convergence and can adapt itself to the variations of input values. The correctness of improved CORDIC algorithms has been proved by calculating examples.

Existence,uniqueness and stability of C^m solutions of iterative functional equations

In this paper we discuss a relatively general kind of iterative functional equation G(x,f(x), ...,f n (x)) = 0 (for allx ∈J), whereJ is a connected closed subset of the real number axis ?,G∈C m (J n+1, ?) andn ≥ 2. Using the method of approximating fixed points by small shift of maps, choosing suitable metrics on functional spaces and finding a relation between uniqueness and stability of fixed points of maps of general spaces, we prove the existence, uniqueness and stability ofCm solutions of the above equation for any integer m ≥ 0 under relatively weak conditions, and generalize related results in reference in different aspects.

Analytic Solutions of a Polynomial-Like Iterative Functional Equation near Resonance

In this paper existence of local analytic solutions of a polynomial-like iterative functional equation is studied. As well as in previous work, we reduce this problem with the SchrSder transformation to finding analytic solutions of a functional equation without iteration of the unknown function f. For technical reasons, in previous work the constant α given in the Schroder transformation, i.e., the eigenvalue of the linearized f at its fixed point O, is required to fulfill that α is off the unit circle S^1 or lies on the circle with the Diophantine condition. In this paper, we obtain results of analytic solutions in the case of α at resonance, i.e., at a root of the unity and the case of α near resonance under the Brjuno condition.

Analytic Solutions of an Iterative Differential Equation under Brjuno Condition

In this paper, the differential equation involving iterates of the unknown function,x＇（z）=[a^2-x^2（z）]x^[m]（z）with a complex parameter a, is investigated in the complex field C for the existence of analytic solutions. First of all, we discuss the existence and the continuous dependence on the parameter a of analytic solution for the above equation, by making use of Banach fixed point theorem. Then, as well as in many previous works, we reduce the equation with the SchrSder transformation x（z） = y（αy^-1（z）） to the following another functional differential equation without iteration of the unknown functionαy＇（αz）=[a^2-y^2（αz）]y＇（z）y（α^mz）,which is called an auxiliary equation. By constructing local invertible analytic solutions of the auxiliary equation, analytic solutions of the form y（αy^-1 （z）） for the original iterative differential equation are obtained. We discuss not only these α given in SchrSder transformation in the hyperbolic case 0 〈 ｜α｜ 〈 1 and resonance, i.e., at a root of the unity, but also those α near resonance （i.e., near a root of the unity） under Brjuno condition. Finally, we introduce explicit analytic solutions for the original iterative differential equation by means of a recurrent formula, and give some particular solutions in the form of power functions when a = 0.

ANALYTIC SOLUTIONS OF AN ITERATIVE FUNCTIONAL EQUATION

This paper is concerned with the iterative functional equation. By constructing the solution of a companion equation in the form of a convergent power series, the analytic solutions for the original differential equation are obtained.

Fast Solvers of Fredholm Optimal Control Problems

The formulation of optimal control problems governed by Fredholm integral equations of second kind and an efficient computational framework for solving these control problems is presented. Existence and uniqueness of optimal solutions is proved.A collective Gauss-Seidel scheme and a multigrid scheme are discussed. Optimal computational performance of these iterative schemes is proved by local Fourier analysis and demonstrated by results of numerical experiments.

Efcient Iterative Solutions to General Coupled Matrix Equations

Linear matrix equations are encountered in many systems and control applications.In this paper,we consider the general coupled matrix equations（including the generalized coupled Sylvester matrix equations as a special case）l t=1EstYtFst = Gs,s = 1,2,···,l over the generalized reflexive matrix group（Y1,Y2,···,Yl）.We derive an efcient gradient-iterative（GI） algorithm for fnding the generalized reflexive solution group of the general coupled matrix equations.Convergence analysis indicates that the algorithm always converges to the generalized reflexive solution group for any initial generalized reflexive matrix group（Y1（1）,Y2（1）,···,Yl（1））.Finally,numerical results are presented to test and illustrate the performance of the algorithm in terms of convergence,accuracy as well as the efciency.

The Real Solutions of Functional Equationf^[m]=1/f

The authors study the functional equation f^[m]=1/f and analyze the featuresof its piecewise continuous solutions. All the piecewise continuous real solutions are obtained constructively. The results generalize the ones in [2]. Moreover, the conclusion is drawn that there is no circuit iterative roots for those functions not satisfying Babbage equation.