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A non-linear integral equation of the form
 
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>
+
$$ \tag{1 }
 +
u ( x) - \int\limits _  \Omega  K ( x , s ) u ( s) d s =
 +
$$
  
<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>
+
$$
 +
= \
 +
U _ {01} \left ( \begin{array}{c}
 +
x \\
 +
v
 +
\end{array}
 +
\right ) + \sum
 +
_ { m+ } n\geq  2 U _ {mn} \left ( \begin{array}{c}
 +
x \\
 +
u,v
 +
\end{array}
 +
\right ) ,\  x \in \Omega ,
 +
$$
  
 
where
 
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>
+
$$
 +
U _ {01} \left ( \begin{array}{c}
 +
x \\
 +
v
 +
\end{array}
 +
\right )  = K _ {0} ( x) v ( x) + \int\limits _  \Omega  K _ {1} ( x , s ) v ( s)  d s ,
 +
$$
  
<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>
+
$$
 +
U _ {mn} \left ( \begin{array}{c}
 +
x \\
 +
u,v
 +
\end{array}
 +
\right )  = \sum _
 +
{\nu = 1 } ^ { {n }  } \int\limits _  \Omega  \dots \int\limits _  \Omega
 +
K  ^ {(} v) ( x , s _ {1} \dots s _ {i} ) \times
 +
$$
  
<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>
+
$$
 +
\times u ^ {\alpha _ {0} } ( x) u ^ {\alpha _ {1} } ( s _ {1} ) \dots u ^ {\alpha _ {i} } ( s _ {i} ) v ^ {\beta _ {0}  } ( x) v ^ {\beta _ {1} } ( s _ {1} ) \dots v ^ {\beta _ {i} } ( s _ {i} )  d s _ {1} \dots d s _ {i} ,
 +
$$
  
<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,
+
$  \alpha _ {0} \dots \alpha _ {i} $,  
 +
$  \beta _ {0} \dots \beta _ {i} $
 +
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>
+
$$
 +
\alpha _ {0} + \dots + \alpha _ {i}  = m ,\ \
 +
\beta _ {0} + \dots + \beta _ {i}  = n ,
 +
$$
  
<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.
+
$  \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 <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):
+
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):
  
<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>
+
$$ \tag{2 }
 +
\omega _ {k} ( \xi _ {1} \dots \xi _ {n} , v )  = 0 ,\ \
 +
k = 1 \dots n ,
 +
$$
  
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.
+
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.
 
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.
Line 29: Line 91:
 
====References====
 
====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>
 
<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====
 
====Comments====
 
  
 
====References====
 
====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>
 
<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>

Latest revision as of 04:11, 6 June 2020


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)

Comments

References

[a1] S.-N. Chow, J.K. Hale, "Methods of bifurcation theory" , Springer (1982)
How to Cite This Entry:
Lyapunov-Schmidt equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lyapunov-Schmidt_equation&oldid=22781
This article was adapted from an original article by B.V. Khvedelidze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article