## The solution representation formula

Dear friends,

I have a problem with the solution representation formula for first-order (delay) differential equations.
Let $t_{0}\in\mathbb{R}$, $A,f\in C([t_{0},\infty),\mathbb{R})$ and $\alpha\in C([t_{0},\infty),\mathbb{R})$ be a function such that $\alpha(t)\leq t$ for all $t\geq t_{0}$ and $\lim\nolimits_{t\to\infty}\alpha(t)=\infty$.
Let $t_{-1}:=\min\{\alpha(t):t\in[t_{0},\infty)\}$, and $\varphi\in C([t_{-1},t_{0}],\mathbb{R})$.
Consider the following differential equation:
$\begin{cases}
\end{cases}\rule{3.85cm}{0cm}(1)$

Then, $x$ can be represented in the following form uniquely:
$x(t)=\varphi(t_{0})\mathcal{X}(t,t_{0})+\int_{t_{0 }}^{t}\mathcal{X}(t,\eta)\big[f(\eta)-A(\eta)\varphi(\alpha(\eta))\big]\mathrm{d}\eta\rule{2cm}{0cm}(2)$
for $t\geq t_{0}$, where $f,A,\varphi$ are assumed to be identically zero out of their domains, and the fundamental solution $\mathcal{X}=\mathcal{X}(t,s)$ is the solution of the following differential equation:
$\begin{cases}
In (3), $A,\alpha$ are same as in (1).