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)

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)