Analytic solution to a class of integro-differential equations

In this paper, we consider the integro-differential equation $$ epsilon^2 y''(x)+L(x)mathcal{H}(y)=N(epsilon,x,y,mathcal{H}(y)), $$ where $mathcal{H}(y)[x]=frac{1}{pi}(P)int_{-infty}^{infty} frac{y(t)}{t-x}dt$ is the Hilbert transform. The existence and uniqueness of analytic solution in appropriately chosen space is proved. Our method consists of extending the equation to an appropriately chosen region in the complex plane, then use the Contraction Mapping Theorem.