Cauchy problem in the Banach space $l^{\infty}(\mathbb{Z}^d)$ dependent on the parameter $z$: $$\label{E1.4}
 \frac{dF_{1}(z; t, \cdot)}{dt} = \mathscr{A}F_{1}(z; t, \cdot) + \Delta_{0}f(F_{1}(z; t, \cdot))$$ with the initial condition $ F_{1}(z; 0, \cdot) = e^{-z} $.

Similarly, since by Lemma \[L1\] the function $ F_{x}(z; t, x, y) $ is uniformly (with respect to $ x, y\in \mathbb{Z}^d $) continuously differentiabile in $t$ for each $0\leqslant z \leqslant \infty $, then  can also be treated as the Cauchy problem in the Banach space $l^{2}(\mathbb{Z}^d)$ dependent on the parameters $z$ and $y$: $$\label{E1.5}
\frac{dF_{1}(z; t, \cdot, y)}{dt} = \mathscr{A}F_{1}(z; t, \cdot, y) +  \Delta_{0}f(F_{1}(z; t, \cdot, y))$$ with the initial condition $ F_{1}(z; 0, x, y) = e^{-z\delta_{x}(y)} $.

Because the right sides of equations and are continuous and (under the assumption $\beta = f'(1) < \infty$) satisfy the Lipschitz condition as operators in $l^{\infty}(\mathbb{Z}^d)$ and $l^{2}(\mathbb{Z}^d)$, owing to and by the theorem on continuous dependence of the solutions of differential equation on initial conditions and parameters [@Hartman82] the functions $ F_{1}(z; t, \cdot) $ and $ F_{1}(z; t, \cdot, y) $ depend continuously with respect to the norm on the spaces $l^{\infty}(\mathbb{Z}^d)$ and $l^{2}(\mathbb{Z}^d)$ on the parameter $z$ for each $y$ 