Difference between revisions of "De la Vallée-Poussin multiple-point problem"
Ulf Rehmann (talk | contribs) m (tex encoded by computer) |
m (AUTOMATIC EDIT (latexlist): Replaced 1 formulas out of 1 by TEX code with an average confidence of 2.0 and a minimal confidence of 2.0.) |
||
| Line 1: | Line 1: | ||
| − | <!-- | + | <!--This article has been texified automatically. Since there was no Nroff source code for this article, |
| − | + | the semi-automatic procedure described at https://encyclopediaofmath.org/wiki/User:Maximilian_Janisch/latexlist | |
| − | + | was used. | |
| − | + | If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category. | |
| − | |||
| − | |||
| − | |||
| − | |||
| + | Out of 1 formulas, 1 were replaced by TEX code.--> | ||
| + | |||
| + | {{TEX|semi-auto}}{{TEX|done}} | ||
{{TEX|auto}} | {{TEX|auto}} | ||
{{TEX|done}} | {{TEX|done}} | ||
| Line 27: | Line 26: | ||
where $ x \in [ a, b] $, | where $ x \in [ a, b] $, | ||
| − | $ | y ^ {(} s) | | + | $ | y ^ {(} s) | < + \infty $, |
$ s = 0 \dots n - 1 $, | $ s = 0 \dots n - 1 $, | ||
subject to the conditions | subject to the conditions | ||
| Line 45: | Line 44: | ||
l _ {k} | l _ {k} | ||
\frac{h ^ {k} }{k!} | \frac{h ^ {k} }{k!} | ||
| − | + | < 1, | |
$$ | $$ | ||
| Line 66: | Line 65: | ||
====References==== | ====References==== | ||
| − | <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 $n$" ''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==== | ||
| Line 72: | Line 71: | ||
====References==== | ====References==== | ||
| − | <table>< | + | <table><tr><td valign="top">[a1]</td> <td valign="top"> P. Hartman, "Ordinary differential equations" , Birkhäuser (1982)</td></tr></table> |
Revision as of 16:57, 1 July 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 $n$" 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) |
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=46591