ABSTRACT

Let A be a set of positive reals, I be a real interval and $\mathcal{A}\{f_{\alpha }:\alpha \in A\}$ be a set of functions of I into itself. We determine necessary and sufficient conditions on A, $\mathcal{A}$ and I for the system of Poincaré functional equation $\psi \left(\alpha x\right)=f_{\alpha }\left(\psi \right(x\left)\right)(\alpha \in A,x\in {\mathbb{R}}^{+})$to have a continuous increasing solution $\psi :{\mathbb{R}}^{+}\to I$ which is non-constant. It turns out that such a solution exists whenever A is closed under multiplication and $\mathcal{A}$ is a multiplicative iteration semigroup of increasing continuous functions satisfying a specific density condition and I is a half-open interval. Finally, we show that such a solution is unique up to an internal multiplicative constant.

摘要

设A为一组正实数，我为一个实数区间，$\mathcal{一个}\{F_{\alpha }：\alpha \in \mathrm{一个}\}$成为我自身的一套功能。我们确定的充分必要条件一，$\mathcal{一个}$和I为庞加莱泛函方程组$\psi （\alpha X）=F_{\alpha }（\psi （X））（\alpha \in \mathrm{一个}，X\in {\mathbb{[R}}^{+}）$不断增加解决方案 $\psi ：{\mathbb{[R}}^{+}\to \mathrm{一世}$这是非恒定的。事实证明，只要A在乘法和运算中闭合，这种解决方案就存在。$\mathcal{一个}$是满足特定密度条件的连续函数递增的乘法迭代半群，I是半开区间。最后，我们证明了这种解决方案在内部乘法常数之前都是唯一的。