View source for De la Vallée-Poussin multiple-point problem

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" />,

<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>

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>

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

<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>

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

<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>

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).

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).

====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>



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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Hartman,   "Ordinary differential equations" , Birkhäuser  (1982)</TD></TR></table>