Solvability of the finite-difference equations of the implicit scheme for a nonlinear second-order parabolic equation

We consider the following initial–boundary-value difference problem \begin{eqation}
\begin{gather} \rho(t_i,x_j,u_{ij},\dfrac{u_{ij+1}-u_{ij-1}}{2h})\cdot\dfrac{u_{ij}-u_{i-1j}}{\tau}= a(t_i,x_j,u_{ij},\dfrac{u_{ij+1}-u_{ij-1}}{2h})\cdot \dfrac{u_{ij+1}-2u_{ij}+u_{ij-1}}{h^2}+b(t_i,x_j,u_{ij},\dfrac{u_{ij+1}-u_{ij-1}}{2h}), \quad i=1,\dots,m,\quad j=1,\dots,n \\ u_{0j}=\omega(x_j)\quad j=1,\dots,n \\ u_{i0}=u_{in+i}=0\quad i=1,\dots,m \\ x_j=jh;\quad t_i=i\tau\quad h=\dfrac{1}{n+1},\quad \tau=\dfrac{T}{m} \end{gather}
\end{eqation} which is the approximation of the corresponding problem for a differential equation. We assume that for $o<t\leqslant T$, $0<x<1$, $-\infty<u,p<=\infty$ the functions $\rho(t,x,u,p)$, $a(t,x,u,p)$ и $(t,x,u,p)$ are continuous and \begin{eqation}
\begin{gather*} \rho(t,x,u,p)>0,\quad a(t,x,u,p)\geqslant0 \\ |b(t,x,u,p)-b(t,x,u,0)|\leqslant\biggl[\dfrac{\sigma}{x}+\dfrac{\sigma_1}{1-x}+\dfrac{M}{2} \biggl(\dfrac{1}{x^y}+\dfrac{1}{(1-x)^y}\biggr)\biggr]p/a(t,x,u,p) \\ \sigma_1,\sigma\geqslant0,\quad M\geqslant0,\quad\sigma_1+\sigma\leqslant2,\quad0\leqslant y<1 \\ \dfrac{1}{u}\biggl[b(t,x,u,0)-b(t,x,0,0)\biggr]\leqslant\biggl[\dfrac{\alpha}{t}+l+\alpha_1\tau^\mu\biggr] \rho(t,x,u,p),\quad 0\leqslant\alpha<1,\quad \alpha_1\geqslant0,\quad 0\leqslant\mu<1 \\ |b(t,x,0,0)|\leqslant A(t)\rho(t,x,u,p). \end{gather*}
\end{eqation} Then, for $h^{1-y}M\leqslant M\leqslant3-\sigma_1-\sigma, 1-\alpha-\tau l-\alpha_1\tau^{1-\mu}>0$ the initial–boundary-value difference problem (1)–(3) is solvable. We also give various modifications and generalizations of the mentioned statement, related to various difference approximations of an initial-boundary-value problem for the equations
$$ \rho(t,x,u,\dfrac{\partial u}{\partial x})\dfrac{\partial u}{\partial t}=F\biggl(t,x,u,\dfrac{\partial u}{\partial x},\dfrac{d}{dx}K(t,x,\dfrac{\partial u}{\partial x})\biggr),\quad 0<t\leqslant T,\quad0<x<1 $$
and for a system of weakly connected equations of this type.