# De la Vallée-Poussin multiple-point problem

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)