Urysohn equation

A non-linear integral equation of the form

(*)

where is a bounded closed set in a finite-dimensional Euclidean space and and are given functions for , . Suppose that is continuous for the set of variables , (where is some positive number), and let

If

then the equation

has a unique continuous solution , , satisfying the inequality . If is any continuous function satisfying (), then the sequence of approximations

converges uniformly on to .

Let the Urysohn operator

act in the space , , and let for all , the inequality

be fulfilled, where is a measurable function satisfying

Then for and , equation (*) has a unique solution in .

Under certain assumptions, equation (*) was first studied by P.S. Urysohn (cf. Non-linear integral equation).

References

[1]	M.A. Krasnosel'skii, "Topological methods in the theory of nonlinear integral equations" , Pergamon (1964) (Translated from Russian)
[2]	P.P. Zabreiko (ed.) A.I. Koshelev (ed.) M.A. Krasnoselskii (ed.) S.G. Mikhlin (ed.) L.S. Rakovshchik (ed.) V.Ya. Stet'senko (ed.) T.O. Shaposhnikova (ed.) R.S. Anderssen (ed.) , Integral equations - a reference text , Noordhoff (1975) (Translated from Russian)