Abstract:Let $f : [a,b] \times \Bbb R^{n+1}\rightarrow \Bbb R$ be a Carath\'{e}odory's function. Let $ \{t_{h}\} $, with $t_{h} \in [a,b]$, and $\{x_{h}\}$ be two real sequences. In this paper, the family of boundary value problems $$ \cases x^{(k)} = f \left ( t,x,x',\ldots ,x^{(n)} \right ) x^{(i)}(t_{i}) = x_{i} , \hskip 1em\relax i=0,1, \ldots , k-1 \endcases \hskip 2em\relax (k=n+1,n+2,n+3,\ldots ) $$ is considered. It is proved that these boundary value problems admit at least a solution for each $k \geq \nu $, where $\nu \geq n+1$ is a suitable integer. Some particular cases, obtained by specializing the sequence $\{t_{h}\}$, are pointed out. Similar results are also proved for the Picard problem.
Keywords: higher order ordinary differential equations, Nicoletti problem, Picard \newline problem
AMS Subject Classification: 34B15, 34B10, 34A12