# De la Vallée-Poussin multiple-point problem

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

 (1)

or to a linear equation

 (2)

where , , , subject to the conditions

 (3)

It was shown by Ch.J. de la Vallée-Poussin [1] that if , , and if the inequality

 (4)

where , , , is met, there exists a unique solution of the problem (2), (3). He also showed that if is continuous in all its arguments and satisfies a Lipschitz condition with constant in the variable , , 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 by changing the coefficients of (4); extension of the class of functions , , or ; 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 zeros on (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)