Urysohn equation

A non-linear integral equation of the form

$$ \tag{* } \phi ( x) = \lambda \int\limits _ \Omega K ( x , s , \phi ( s) ) d s + f ( x) ,\ \ x \in \Omega , $$

where $ \Omega $ is a bounded closed set in a finite-dimensional Euclidean space and $ K ( x , s , t ) $ and $ f ( x) $ are given functions for $ x , s \in \Omega $, $ - \infty < t < \infty $. Suppose that $ K ( x , s , t ) $ is continuous for the set of variables $ x , s \in \Omega $, $ | t | \leq \rho $( where $ \rho $ is some positive number), and let

$$ \left | \frac{\partial K ( x , s , t ) }{\partial t } \right | \leq M = \textrm{ const } ,\ \ x , s \in \Omega ,\ \ | t | \leq \rho . $$

If

$$ | \lambda | M \mathop{\rm meas} ( \Omega ) < 1 , $$

$$ | \lambda | \max _ {x \in \Omega } \int\limits _ \Omega \max _ {| t | \leq \rho } | K ( x , s , t ) | d s \leq \rho , $$

then the equation

$$ \phi ( x) = \lambda \int\limits _ \Omega K ( x , s , \phi ( s) ) d s $$

has a unique continuous solution $ \phi ( x) $, $ x \in \Omega $, satisfying the inequality $ | \phi ( x) | \leq \rho $. If $ \phi _ {0} $ is any continuous function satisfying $ | \phi _ {0} ( x) | \leq \rho $( $ x \in \Omega $), then the sequence of approximations

$$ \phi _ {n} ( x) = \lambda \int\limits _ \Omega K ( x , s , \phi _ {n-} 1 ( s) ) d s ,\ \ n = 1 , 2 \dots $$

converges uniformly on $ \Omega $ to $ \phi ( x) $.

Let the Urysohn operator

$$ A \phi ( x) = \ \int\limits _ \Omega K ( x , s , \phi ( s) ) d s $$

act in the space $ L _ {p} ( \Omega ) $, $ p > 1 $, and let for all $ t _ {1} , t _ {2} $, $ x , s \in \Omega $ the inequality

$$ | K ( x , s , t _ {1} ) - K ( x , s , t _ {2} ) | \leq K _ {1} ( x , s ) | t _ {1} - t _ {2} | $$

be fulfilled, where $ K _ {1} $ is a measurable function satisfying

$$ \Delta ^ {p} = \ \int\limits _ \Omega \left ( \int\limits _ \Omega K _ {1} ^ {p / ( p - 1 ) } ( x , s ) d s \right ) ^ {p-} 1 d x < \infty . $$

Then for $ | \lambda | < \Delta ^ {-} 1 $ and $ f \in L _ {p} ( \Omega ) $, equation (*) has a unique solution in $ L _ {p} ( \Omega ) $.

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)