View source for Lyapunov-Schmidt equation

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A non-linear integral equation of the form

$$ \tag{1 }
u ( x) - \int\limits _  \Omega  K ( x , s ) u ( s)  d s =
$$

$$ 
= \ 
U _ {01} \left ( \begin{array}{c}
x \\
 v 
\end{array}
 \right ) + \sum
_ { m+ } n\geq  2 U _ {mn} \left ( \begin{array}{c}
x \\
 u,v 
\end{array}
 \right ) ,\  x \in \Omega ,
$$

where

$$ 
U _ {01} \left ( \begin{array}{c}
x \\
 v 
\end{array}
 \right )  =  K _ {0} ( x) v ( x) + \int\limits _  \Omega  K _ {1} ( x , s ) v ( s)  d s ,
$$

$$ 
U _ {mn} \left ( \begin{array}{c}
x \\
 u,v 
\end{array}
 \right )  =  \sum _
{\nu = 1 } ^ { {n }  } \int\limits _  \Omega  \dots \int\limits _  \Omega 
K  ^ {(} v) ( x , s _ {1} \dots s _ {i} ) \times
$$

$$ 
\times u ^ {\alpha _ {0} } ( x) u ^ {\alpha _ {1} } ( s _ {1} ) \dots u ^ {\alpha _ {i} } ( s _ {i} ) v ^ {\beta _ {0}  } ( x) v ^ {\beta _ {1} } ( s _ {1} ) \dots v ^ {\beta _ {i} } ( s _ {i} )  d s _ {1} \dots d s _ {i} ,
$$

$  \alpha _ {0} \dots \alpha _ {i} $, 
$  \beta _ {0} \dots \beta _ {i} $
are non-negative integers,

$$ 
\alpha _ {0} + \dots + \alpha _ {i}  =  m ,\ \ 
\beta _ {0} + \dots + \beta _ {i}  =  n ,
$$

$  \Omega $
is a closed bounded set in a finite-dimensional Euclidean space,  $  v $
and the function  $  K $
are given continuous functions of their arguments,  $  s _ {1} \dots s _ {i} \in \Omega $, 
and  $  u $
is the unknown function. The sum on the right-hand side of (1) may be finite or it may represent an infinite series. In the latter case the series is called an integro-power series of two functional arguments. It is assumed that the series converges absolutely and uniformly.

If 1 is not a characteristic number of the kernel  $  K ( x , s ) $, 
then equation (1) has a unique small solution in the class of continuous functions for sufficiently small  $  | v ( x) | $. 
This solution can be represented as an integro-power series. The case when 1 is a characteristic number of the kernel  $  K $
is more complicated. In this case one constructs a system of equations — the branching equations (bifurcation equations):

$$ \tag{2 }
\omega _ {k} ( \xi _ {1} \dots \xi _ {n} , v )  =  0 ,\ \ 
k = 1 \dots n ,
$$

where  $  \omega _ {k} $
are known power series and  $  n $
is the multiplicity of the characteristic number 1. In the general case the system (2) has a non-unique solution. Whatever the fixed sufficiently small function  $  v $, 
to every small continuous solution of (2) (a continuous solution of (2) is said to be small if  $  \xi _ {i} ( 0) = 0 $) 
there corresponds a small solution of (1) that can be represented as an integro-power series.

An equation of the type (1) was first considered by A.M. Lyapunov in 1906, and later — in a more general form — by E. Schmidt in 1908.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.M. Vainberg,   V.A. Trenogin,   "Theory of branching of solutions of non-linear equations" , Noordhoff  (1974)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.S. Smirnov,   "Introduction to the theory of integral equations" , Moscow-Leningrad  (1936)  (In Russian)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S.-N. Chow,   J.K. Hale,   "Methods of bifurcation theory" , Springer  (1982)</TD></TR></table>