Geodesic
Analytic solution to a class of integro-differential equations
Electronic journal of differential equations, Tome 2003 (2003)
In this paper, we consider the integro-differential equation

$ \epsilon^2 y''(x)+L(x){\cal H}(y)=N(\epsilon,x,y,{\cal H}(y)), $

where ${cal H}(y)[x]=\frac{1}{\pi}(P)\int_{-\infty}^{\infty} \frac{y(t)}{t-x}dt$ 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.
Classification : 34A20, 45E05
Keywords: analytic solution, singular integro-differential equation 
@article{EJDE_2003__2003__a195,
     author = {Xie,  Xuming},
     title = {Analytic solution to a class of integro-differential equations},
     journal = {Electronic journal of differential equations},
     year = {2003},
     volume = {2003},
     zbl = {1019.45004},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2003__2003__a195/}
}
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Xie,  Xuming. Analytic solution to a class of integro-differential equations. Electronic journal of differential equations, Tome 2003 (2003). http://geodesic.mathdoc.fr/item/EJDE_2003__2003__a195/