Boundary-value problem for singularly perturbed integro-differential equation with singularly perturbed Neumann boundary condition (Q6052498)

The paper is focused on the following boundary value problem for singularly perturbed ordinary integro-differential equation subject to singularly perturbed Neumann boundary conditions: \[ \left\{\begin{array}{l} \varepsilon^2 u''=L(u,u,x,\varepsilon),\,\,\,0<x<1,\\ \varepsilon u'(0,\varepsilon)=a,\,\,\,u'(1,\varepsilon)=b, \end{array}\right. \] where the integral operator \(L(u,u,x,\varepsilon)\) is defined by \[ L(u,u,x,\varepsilon)=\displaystyle\int_0^1 g(u(x),u(s),x,s,\varepsilon)\,ds. \] Under some conditions on the function \(g\) and the operator \(L\), the authors prove the existence of a boundary layer solution. Then by using the modified Vasil'eva's boundary layer function method, they construct the asymptotic approximation of this solution, and establish its asymptotic Lyapunov stability.