Uniform convergence of the implicit difference scheme of a nonlinear boundary-value problem for a second-order nonlinear parabolic equation

The nonlinear boundary-value problem for the parabolic equation
\begin{gather} \dfrac{\partial u}{\partial t}=F(t,x,u,\dfrac{\partial u}{\partial x},\dfrac{\partial^2u}{\partial x^2})\quad 0<t\leqslant T,\quad 0\leqslant x<1 \tag{1} \\ u(0,x)=\omega(x),\quad 0<x\leqslant1 \tag{2} \\ \dfrac{\partial u(t,0)}{\partial x}=\varphi(t,u(t,0)),\quad u(t,1)=0,\quad 0<t\leqslant T \tag{3} \end{gather}
is approximated by the boundary-value difference problem
\begin{gather} P_{i0}(u_{ij})=\dfrac{u_{i0}-u_{i-1,0}}{\tau}-F(t_1,0,u_{i0},\varphi(t_i,u_{i0}),\quad \dfrac{2}{h}\biggl[\dfrac{u_{i1}-u_{i0}}{h}-\varphi(t_i,u_{i0})\biggr]\quad i=1,\dots,m \tag{4} \\ P_{ij}(u_{ij})=\dfrac{u_{ij}-u_{i-1,j}}{\tau}-F(t_i,x_j,\delta u_{ij},\Delta u_{ij}),\quad i=1,2,\dots,m,\quad j=1,\dots,n \tag{5} \\ u_{0j}=\omega_j\quad j=0,1,\dots,n;\quad u_{i,n+1}=0\quad i=1,\dots,m \tag{6} \\ \delta u_{ij}=\dfrac{1}{2h}[u_{i,j+1}-u_{i,j-1}],\quad \Delta u_{ij}=\dfrac{1}{h^2}[u_{i,j+1}-2u_{ij}+u_{i,j-1}]. \tag{7} \end{gather}
Under certain assumptions on the solutions of the original problem and functions $F$ and $\varphi$, for small $\tau$ and $h$ we prove the existence of a solution of problem (4)–(6) and derive a bound on the approximation error. Under certain restrictions on the steps $h$ and $\tau$ and the functions $F$ and $\varphi$, we prove that the problem (4)–(6) has a nonnegative solution.