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
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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) |
Urysohn equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Urysohn_equation&oldid=49100