The Duhamel product for two suitable functions f and g is defined as follows:

$$\begin{aligned} (f\circledast g)(x)={\frac{\mathrm{{d}}}{\mathrm{d}x}} {\textstyle \int \limits _{0}^{x}} f(x-t)g(t)\mathrm{{d}}t. \end{aligned}$$

We consider the integration operator J, \(Jf(x)={\textstyle \int \limits _{0}^{x}} f(t)\mathrm{{d}}t\), on the Frechet space \(C^{\infty }\) of all infinitely differentiable functions in \(\left[ 0,1\right] \) and describe in terms of Duhamel operators its commutant. Also, we consider the Duhamel equation \(\varphi \circledast f=g\) and prove that it has a unique solution if and only if \(\varphi (0)\ne 0\).

中文翻译：

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Duhamel方程解的可交换性和唯一性

具有两个合适的函数f和g的Duhamel乘积定义如下：

$$ \ begin {aligned}（f \ circledast g）（x）= {\ frac {\ mathrm {{d}}} {\ mathrm {d} x}} {\ textstyle \ int \ limits _ {0} ^ {x}} f（xt）g（t）\ mathrm {{d}} t。\ end {aligned} $$

我们考虑在Frechet空间\上的积分算子J，\（Jf（x）= {\ textstyle \ int \ limits _ {0} ^ {x}} f（t）\ mathrm {{d}} t \）\（\ left [0,1 \ right] \）中所有无限微分函数的（C ^ {\ infty} \），并用Duhamel算子描述其可交换性。此外，我们考虑了Duhamel方程\（\ varphi \ circledast f = g \）并证明，当且仅当\（\ varphi（0）\ ne 0 \）时，它具有唯一解。