A boundary value problem for a second-order nonlinear equation with delta-like potential (Q2631307)

The authors consider the boundary value problem \[ \begin{aligned} {\mathcal L}^{\kappa}_{\epsilon}u^{\epsilon}&=f,\quad x \in (a,b),\;u^{\epsilon}(a)=u^{\epsilon}(b)=0, \\ \text{where }{\mathcal L}^{\kappa}(u)&=-\frac{d}{dx}(k(x) \frac{du}{dx})+ \kappa \frac{d}{dx}p(u) + q(x)u \text{ and } {\mathcal L}^{\kappa}_{\epsilon}={\mathcal L}^{\kappa} + \epsilon^{-1}Q(\frac{x}{\epsilon}).\end{aligned} \] Here, \(k \in C^2[a,b], q \in C[a,b], p\in C^2[-\infty,+\infty], Q\in C[-\infty,+\infty]\) and \(\{0\}\in (a,b)\). Moreover, \(\kappa \geq 0, 0<\epsilon \ll 1\), \(k(x) \geq k_0, q(x) \geq q_0 >0\) and \(Q(\tau)\geq 0.\) In the main result, it is proved that for any fixed \(f\in L^2[a,b]\) and sufficiently small \(\epsilon >0\), the solution \(u^{\epsilon}\) to the given boundary value problem (which, by previous results, is known to exist) satisfies the equality \(\|u^{\epsilon}-u_0\|_{C[a,b]}=O(\epsilon)\), where \(u_0\) is the solution to the boundary value problem \[ {\mathcal L}^{\kappa}u_0=f, x \in (a,0) \cup (0,b), u_0(a)=u_0(b)=0, k(0)u'_0(0)=\langle Q \rangle u_0(0). \] The proof is performed using the method of matched asymptotic approximation (based on the use of the contraction theorem).