Electronic Journal of Differential Equations,
Vol. 2003(2003), No. 33, pp. 1-25.

Title: Analytic solution to a class of integro-differential equations

Author: Xuming Xie (Univ. of Delaware, Newark, USA)

Abstract: 
 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.
 
Submitted August 13, 2002. Published March 28, 2003.
Math Subject Classifications: 34A20, 45E05.
Key Words: Analytic solution; singular integro-differential equation.