View source for Lyapunov-Schmidt equation

A non-linear integral equation of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061150/l0611501.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061150/l0611502.png" /></td> </tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061150/l0611503.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061150/l0611504.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061150/l0611505.png" /></td> </tr></table>

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061150/l0611506.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061150/l0611507.png" /> are non-negative integers,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061150/l0611508.png" /></td> </tr></table>

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061150/l0611509.png" /> is a closed bounded set in a finite-dimensional Euclidean space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061150/l06115010.png" /> and the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061150/l06115011.png" /> are given continuous functions of their arguments, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061150/l06115012.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061150/l06115013.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061150/l06115014.png" />, then equation (1) has a unique small solution in the class of continuous functions for sufficiently small <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061150/l06115015.png" />. This solution can be represented as an integro-power series. The case when 1 is a characteristic number of the kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061150/l06115016.png" /> is more complicated. In this case one constructs a system of equations — the branching equations (bifurcation equations):

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061150/l06115017.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061150/l06115018.png" /> are known power series and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061150/l06115019.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061150/l06115020.png" />, to every small continuous solution of (2) (a continuous solution of (2) is said to be small if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061150/l06115021.png" />) 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>