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)

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)