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 ...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).展开更多
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....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.展开更多
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 analyt...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.展开更多
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 equat...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.展开更多
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 b...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.展开更多
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.
Iterative root problem can be regarded as a weak version of the problem of embedding a homeomorphism into a flow. There are many results on iterative roots of monotone functions. However, this problem gets more diffcu...Iterative root problem can be regarded as a weak version of the problem of embedding a homeomorphism into a flow. There are many results on iterative roots of monotone functions. However, this problem gets more diffcult in non-monotone cases. Therefore, it is interesting to find iterative roots of linear fractional functions (abbreviated as LFFs), a class of non-monotone functions on ?. In this paper, iterative roots of LFFs are studied on ?. An equivalence between the iterative functional equation for non-constant LFFs and the matrix equation is given. By means of a method of finding matrix roots, general formulae of all meromorphic iterative roots of LFFs are obtained and the precise number of roots is also determined in various cases. As applications, we present all meromorphic iterative roots for functions z and 1/z.展开更多
In [l], a property of roots of polynomials is considered, which involves the existence of local analytic solutions of polynomial-like functional iterative equations. In this paper we discuss this property and obtain a...In [l], a property of roots of polynomials is considered, which involves the existence of local analytic solutions of polynomial-like functional iterative equations. In this paper we discuss this property and obtain a succinct condition to decide whether this property holds. Our main result is: A polynomialλnzn+''' + λ2z2 + λlz + λ0 of degree n has a root or such that inf{|λnanm +... + λ2a2m + λ1am+ λ0|: m = 2, 3,.. .} > 0 if and only if at least one of the following two conditions holds: (i) the polynomial has a root β satisfying ｜β｜ > 1; (ii) the polynomial has a root β satisfying ｜β｜ < 1, and λ0≠0展开更多
文摘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).
文摘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.
基金the Natural Science Foundation of Shandong Province (No.2006ZRB01066)
文摘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.
文摘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.
文摘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.
基金The NSF(11326120 and 11501069)of Chinathe Foundation(KJ1400528 and KJ1600320)of Chongqing Municipal Education Commissionthe Foundation(02030307-00039)of Youth Talent of Chongqing Normal University
文摘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.
基金supported by the Youth Fund of Sichuan Provincial Education Department of China (Grant No.07ZB042)
文摘Iterative root problem can be regarded as a weak version of the problem of embedding a homeomorphism into a flow. There are many results on iterative roots of monotone functions. However, this problem gets more diffcult in non-monotone cases. Therefore, it is interesting to find iterative roots of linear fractional functions (abbreviated as LFFs), a class of non-monotone functions on ?. In this paper, iterative roots of LFFs are studied on ?. An equivalence between the iterative functional equation for non-constant LFFs and the matrix equation is given. By means of a method of finding matrix roots, general formulae of all meromorphic iterative roots of LFFs are obtained and the precise number of roots is also determined in various cases. As applications, we present all meromorphic iterative roots for functions z and 1/z.
基金Supported by the National Natural Science Foundation of China.
文摘In [l], a property of roots of polynomials is considered, which involves the existence of local analytic solutions of polynomial-like functional iterative equations. In this paper we discuss this property and obtain a succinct condition to decide whether this property holds. Our main result is: A polynomialλnzn+''' + λ2z2 + λlz + λ0 of degree n has a root or such that inf{|λnanm +... + λ2a2m + λ1am+ λ0|: m = 2, 3,.. .} > 0 if and only if at least one of the following two conditions holds: (i) the polynomial has a root β satisfying ｜β｜ > 1; (ii) the polynomial has a root β satisfying ｜β｜ < 1, and λ0≠0
