Singularly perturbed integral equations (Q2577156)

The following singularly perturbed integral equation \[ \varepsilon u_\varepsilon(x) + \int_a^b K(x,y)u_\varepsilon(y)\,dy =f(x), \quad x\in[a,b]\tag{1} \] is considered, which becomes a Fredholm equation of first kind for \(\varepsilon=0\). The kernels \(K_\pm(x,y):=K(x,y)\) for \(\pm(x-y)>0\) are both smooth on \([a,b]\times[a,b]\) and might have jump on the diagonal \(K_+(x,x)-K_-(x,x)=a(x)\), or their derivatives have the jump \([\partial^n_yK_+(x,y)-\partial^n_yK_-(x,y)]| _{y=x} =a(x)\) and \(a\in C^\infty([a,b])\), \(a(x)\not=0\) for \(x\in[a,b]\) (an ellipticity condition). The author proves the unique solvability of equation (1) under condition that the ``unperturbed'' equation with \(\varepsilon=0\) is uniquely solvable. Moreover, the principal term of the asymptotic expansion with respect to \(\varepsilon\) is given by adapting the technique developed by \textit{G. I. Eskin} [Dokl. Akad. Nauk SSSR 211, 547--550 (1973; Zbl 0292.35068)] for similar problems for pseudodifferential equations. The results are applied to several examples including a Volterra equation.