Boundary-value problem for a second-order nonlinear equation with delta-like potential

A Dirichlet nonlinear problem for a second-order equation is considered on an interval. The problem is perturbed by the delta-like potential $\varepsilon^{-1}Q\left(\varepsilon^{-1}x\right)$, where the function $Q(\xi)$ is compactly supported and $0<\varepsilon\ll1$. A solution of this boundary-value problem is constructed with accuracy up to $O(\varepsilon)$ with the use of the method of matched asymptotic expansions. The obtained asymptotic approximation is validated by means of the fixed-point theorem. All types of boundary conditions are considered for a linear boundary-value problem.