Grid approximation of singularly perturbed quasi-linear elliptic equations in a case of multiple solutions of the reduced equation

The first boundary value problem for quasi-linear elliptic equations $\varepsilon^2Lu(x)-g(x,u(x))=0$ is considered on a strip. Here $L$ is linear second order operator, the parameter $\varepsilon$ takes arbitrary values in the interval $(0,1]$. The reduced equation $g(x,u(x))=0$ has an even number of solutions. For solving boundary value problems the special noniterative and iterative finite difference schemes are constructed. These schemes converge uniformly with respect to the parameter. For the construction of schemes classical difference approximations on grids, condensed in boundary layers, are used.