Finite-difference method for solving the first boundary-value problem for a second-order nonlinear ordinary differential equation with a divergent principal part

The approximation is studied of the first boundary-value problem for the equation
\begin{equation} -\dfrac{d}{dx}K\biggl(x,\dfrac{du}{dx}\biggr)+f(x,u)=0,\quad 0<x<1, \tag{1} \end{equation}
with boundary conditions
\begin{equation} u(0)=u(1)=0 \tag{2} \end{equation}
by difference boundary-value problems of form
\begin{gather} -[a(x,W_{\overline x})]_x+\varphi(x,W)=0,\quad x\in\omega_n, \tag{3} \\ W(0)=W(1)=0. \tag{4} \end{gather}
Theorems are established on the solvability of problem (3), (4). Theorems are proved on uniform convergence and on the order of uniform convergence. Here, as usual, boundedness is not assumed, but just the summability of the corresponding derivatives of the solutions of problem (1), (2). Also considered are singular boundary-value problems of form (1), (2), where uniform convergence with order h is proved under assumption of piecewise absolute continuity of the function $f(x,u(x))$.