%0 Journal Article
%T Analytic solution to a class of integro-differential equations
%A Xuming Xie
%J Electronic Journal of Differential Equations
%D 2003
%I Texas State University
%X 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.
%K Analytic solution
%K singular integro-differential equation.
%U http://ejde.math.txstate.edu/Volumes/2003/33/abstr.html