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The problem of finding a solution to an ordinary non-linear differential equation of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030250/d0302501.png" />,
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<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/d/d030/d030250/d0302502.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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The problem of finding a solution to an ordinary non-linear differential equation of order  $  n $,
 +
 
 +
$$ \tag{1 }
 +
y  ^ {(} n)  = \
 +
f ( x, y, y  ^  \prime  \dots y ^ {( n - 1) } )
 +
$$
  
 
or to a linear equation
 
or to a linear equation
  
<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/d/d030/d030250/d0302503.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
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$$ \tag{2 }
 +
y  ^ {(} n) + p _ {1} ( x)
 +
y ^ {( n - 1) } + \dots +
 +
p _ {n} ( x) y  = 0,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030250/d0302504.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030250/d0302505.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030250/d0302506.png" />, subject to the conditions
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where $  x \in [ a, b] $,  
 +
$  | y  ^ {(} s) | < + \infty $,  
 +
$  s = 0 \dots n - 1 $,  
 +
subject to the conditions
  
<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/d/d030/d030250/d0302507.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
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$$ \tag{3 }
 +
y ( x _ {i} )  = c _ {i} ,\ \
 +
i = 1 \dots n; \ \
 +
x _ {i} \in [ a, b].
 +
$$
  
It was shown by Ch.J. de la Vallée-Poussin [[#References|[1]]] that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030250/d0302508.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030250/d0302509.png" />, and if the inequality
+
It was shown by Ch.J. de la Vallée-Poussin [[#References|[1]]] that if $  p _ {k} ( x) \in C [ a, b] $,  
 +
$  k = 1 \dots n $,  
 +
and if the inequality
  
<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/d/d030/d030250/d03025010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
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$$ \tag{4 }
 +
\sum _ {k = 1 } ^ { n }
 +
l _ {k}
 +
\frac{h  ^ {k} }{k!}
 +
  < 1,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030250/d03025011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030250/d03025012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030250/d03025013.png" />, is met, there exists a unique solution of the problem (2), (3). He also showed that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030250/d03025014.png" /> is continuous in all its arguments and satisfies a Lipschitz condition with constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030250/d03025015.png" /> in the variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030250/d03025016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030250/d03025017.png" />, then, if equation (4) is satisfied, there can be only one solution of the problem (1), (3).
+
where $  l _ {k} \geq  | p _ {k} ( x) | $,
 +
$  x \in [ a, b] $,  
 +
$  h= b - a $,  
 +
is met, there exists a unique solution of the problem (2), (3). He also showed that if $  f( x, u _ {1} \dots u _ {n} ) $
 +
is continuous in all its arguments and satisfies a Lipschitz condition with constant $  l _ {k} $
 +
in the variable $  u _ {n+ 1- k }  $,  
 +
$  k = 1 \dots n $,  
 +
then, if equation (4) is satisfied, there can be only one solution of the problem (1), (3).
  
The following aspects of the de la Vallée-Poussin multiple point problem are studied: improvement of an estimate of the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030250/d03025018.png" /> by changing the coefficients of (4); extension of the class of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030250/d03025019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030250/d03025020.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030250/d03025021.png" />; and generalization of the conditions (3). A main problem is to prove that the solution exists and that it is unique. As far as the problem (2), (3) is concerned, this is equivalent with the following statement: Any non-trivial solution of equation (2) has at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030250/d03025022.png" /> zeros on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030250/d03025023.png" /> (non-oscillation of solutions or separation of zeros).
+
The following aspects of the de la Vallée-Poussin multiple point problem are studied: improvement of an estimate of the number $  h $
 +
by changing the coefficients of (4); extension of the class of functions $  p _ {k} ( x) $,  
 +
$  k= 1 \dots n $,  
 +
or $  f( x, u _ {1} \dots u _ {n} ) $;  
 +
and generalization of the conditions (3). A main problem is to prove that the solution exists and that it is unique. As far as the problem (2), (3) is concerned, this is equivalent with the following statement: Any non-trivial solution of equation (2) has at most $  n - 1 $
 +
zeros on $  [ a, b] $(
 +
non-oscillation of solutions or separation of zeros).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Ch.J. de la Vallée-Poussin,  "Sur l'equation différentielle linéaire du second ordre. Détermination d'une intégrale par deux valeurs assignées. Extension aux équations d'ordre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030250/d03025024.png" />"  ''J. Math. Pures Appl.'' , '''8'''  (1929)  pp. 125–144</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G. Sansone,  "Equazioni differenziali nel campo reale" , '''1''' , Zanichelli  (1948)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Ch.J. de la Vallée-Poussin,  "Sur l'equation différentielle linéaire du second ordre. Détermination d'une intégrale par deux valeurs assignées. Extension aux équations d'ordre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030250/d03025024.png" />"  ''J. Math. Pures Appl.'' , '''8'''  (1929)  pp. 125–144</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G. Sansone,  "Equazioni differenziali nel campo reale" , '''1''' , Zanichelli  (1948)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Revision as of 17:32, 5 June 2020


The problem of finding a solution to an ordinary non-linear differential equation of order $ n $,

$$ \tag{1 } y ^ {(} n) = \ f ( x, y, y ^ \prime \dots y ^ {( n - 1) } ) $$

or to a linear equation

$$ \tag{2 } y ^ {(} n) + p _ {1} ( x) y ^ {( n - 1) } + \dots + p _ {n} ( x) y = 0, $$

where $ x \in [ a, b] $, $ | y ^ {(} s) | < + \infty $, $ s = 0 \dots n - 1 $, subject to the conditions

$$ \tag{3 } y ( x _ {i} ) = c _ {i} ,\ \ i = 1 \dots n; \ \ x _ {i} \in [ a, b]. $$

It was shown by Ch.J. de la Vallée-Poussin [1] that if $ p _ {k} ( x) \in C [ a, b] $, $ k = 1 \dots n $, and if the inequality

$$ \tag{4 } \sum _ {k = 1 } ^ { n } l _ {k} \frac{h ^ {k} }{k!} < 1, $$

where $ l _ {k} \geq | p _ {k} ( x) | $, $ x \in [ a, b] $, $ h= b - a $, is met, there exists a unique solution of the problem (2), (3). He also showed that if $ f( x, u _ {1} \dots u _ {n} ) $ is continuous in all its arguments and satisfies a Lipschitz condition with constant $ l _ {k} $ in the variable $ u _ {n+ 1- k } $, $ k = 1 \dots n $, then, if equation (4) is satisfied, there can be only one solution of the problem (1), (3).

The following aspects of the de la Vallée-Poussin multiple point problem are studied: improvement of an estimate of the number $ h $ by changing the coefficients of (4); extension of the class of functions $ p _ {k} ( x) $, $ k= 1 \dots n $, or $ f( x, u _ {1} \dots u _ {n} ) $; and generalization of the conditions (3). A main problem is to prove that the solution exists and that it is unique. As far as the problem (2), (3) is concerned, this is equivalent with the following statement: Any non-trivial solution of equation (2) has at most $ n - 1 $ zeros on $ [ a, b] $( non-oscillation of solutions or separation of zeros).

References

[1] Ch.J. de la Vallée-Poussin, "Sur l'equation différentielle linéaire du second ordre. Détermination d'une intégrale par deux valeurs assignées. Extension aux équations d'ordre " J. Math. Pures Appl. , 8 (1929) pp. 125–144
[2] G. Sansone, "Equazioni differenziali nel campo reale" , 1 , Zanichelli (1948)

Comments

This problem is also known as the multipoint boundary value problem; it is mostly of historical interest. In [a1] an extension of de la Vallée-Poussin's result is given.

References

[a1] P. Hartman, "Ordinary differential equations" , Birkhäuser (1982)
How to Cite This Entry:
De la Vallée-Poussin multiple-point problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=De_la_Vall%C3%A9e-Poussin_multiple-point_problem&oldid=23246
This article was adapted from an original article by L.N. Eshukov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article