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 solutions of the following two-scale difference equations: f(x)=NΣ anf(2x-n). The conditions for the solvability and the iteractive solvability of this kind ofequstions in certain spaces ...This paper is concerned with solutions of the following two-scale difference equations: f(x)=NΣ anf(2x-n). The conditions for the solvability and the iteractive solvability of this kind ofequstions in certain spaces are obtained respectively.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
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 solutions of the following two-scale difference equations: f(x)=NΣ anf(2x-n). The conditions for the solvability and the iteractive solvability of this kind ofequstions in certain spaces are obtained respectively.
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
