Difference between revisions of "De la Vallée-Poussin multiple-point problem"
From Encyclopedia of Mathematics
Ulf Rehmann (talk | contribs) m (moved De la Vallee-Poussin multiple-point problem to De la Vallée-Poussin multiple-point problem over redirect: accented title) |
Ulf Rehmann (talk | contribs) m (tex encoded by computer) |
||
| Line 1: | Line 1: | ||
| − | + | <!-- | |
| + | d0302501.png | ||
| + | $#A+1 = 23 n = 1 | ||
| + | $#C+1 = 23 : ~/encyclopedia/old_files/data/D030/D.0300250 de la Vall\Aeee\AAnPoussin multiple\AAhpoint problem | ||
| + | Automatically converted into TeX, above some diagnostics. | ||
| + | Please remove this comment and the {{TEX|auto}} line below, | ||
| + | if TeX found to be correct. | ||
| + | --> | ||
| − | + | {{TEX|auto}} | |
| + | {{TEX|done}} | ||
| + | |||
| + | 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 | ||
| − | + | $$ \tag{2 } | |
| + | y ^ {(} n) + p _ {1} ( x) | ||
| + | y ^ {( n - 1) } + \dots + | ||
| + | p _ {n} ( x) y = 0, | ||
| + | $$ | ||
| − | where | + | 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 [[#References|[1]]] that if | + | 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 | ||
| − | + | $$ \tag{4 } | |
| + | \sum _ {k = 1 } ^ { n } | ||
| + | l _ {k} | ||
| + | \frac{h ^ {k} }{k!} | ||
| + | < 1, | ||
| + | $$ | ||
| − | where | + | 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 | + | 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
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