x, y)$.

The inequality $ 0\leqslant F_{1}(z; t, x, y) \leqslant 1 $ results from nonnegativeness of the random variable $\eta_{t}(y)$ and the definition of the generating function. Taking into account all possible evolutions of the system on the interval $[t, t+h]$ and using the Markov property we obtain: $$\begin{aligned}
F_{1}(z; t+h, x, y) &= \sum_{x'\neq x}p(h, x, x')E e^{-z \tilde{\eta}_{t}(y)}+ \delta_{0}(x) \sum_{n\neq 1}p_{*}(h, n)Ee^{-z(\tilde{\eta}_{t}^{1}(y) + \cdots + \tilde{\eta}_{t}^{n}(y))}\\
&\quad+ [(1-\delta_{0}(x))p(h, x, x) + \delta_{0}(x)p_{*}(h, 1)]F_{1}(z; t, x, y)\\
&=  \sum_{x'\neq x}a(x, x')hE e^{-z \tilde{\eta}_{t}(y)} + \delta_{0}(x) \sum_{n\neq 1}b_{n}hE e^{-z(\tilde{\eta}_{t}^{1}(y) + \cdots + \tilde{\eta}_{t}^{n}(y))}\\
&\quad+ [1 + a(x, x)h + \delta_{0}(x)b_1 h]F_{1}(z; t, x, y)\\
&\quad + S_1(h) + S_2(h) + S_3(h) + S_4(h)\\
&= F_{1}(z; t, x, y) + [(\mathscr{A}F_{1}(z; t, \cdot, y))(x) + \delta_{0}(x)f(F_{1}(z; t, x, y))]h\\
&\quad+ S_1(h) + S_2(h) + S_3(h) + S_4(h),\end{aligned}$$ where $$\begin{aligned}
S_1(h) &= \sum_{x'\neq x}[p(h, x, x') - a(x, x')h ]E e^{-z \tilde{\eta}_{t}(y)}x, \\
S_2(h) &= \delta_{0}(x)\sum_{n\neq 1}[p_{*}(h, n) - b_n h ]E_{\infty}e^{-z(\tilde{\eta}_{t}^{1}(y) + \cdots + \tilde{\eta}_{t}^{n}(y))},\\
S_3(h) &= [p(h, x, x) - 1- a(x, x)h]F_{1}(z; t, x, y),\\
S_4(h) &= \delta_{0}(x) [p_{*}(h, 1) - 1 - b_1 h]F_{1}(z; t, x, y).\end{aligned}$$ Here $ \tilde{\eta}_{t}(y) $ is a new process, starting at 