Lyapunov-Schmidt equation

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

[1]	M.M. Vainberg, V.A. Trenogin, "Theory of branching of solutions of non-linear equations" , Noordhoff (1974) (Translated from Russian)
[2]	N.S. Smirnov, "Introduction to the theory of integral equations" , Moscow-Leningrad (1936) (In Russian)

References

[a1]	S.-N. Chow, J.K. Hale, "Methods of bifurcation theory" , Springer (1982)