Geodesic
Boundary value problems for higher order ordinary differential equations
Commentationes Mathematicae Universitatis Carolinae, Tome 35 (1994) no. 3, pp. 451-466.

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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 $$ \cases x^{(k)} = f \left( t,x,x',\ldots ,x^{(n)} \right) \ x^{(i)}(t_{i}) = x_{i} \,, \quad i=0,1, \ldots , k-1 \endcases \qquad (k=n+1,n+2,n+3,\ldots ) $$ 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.
Classification : 34A12, 34B10, 34B15
Keywords: higher order ordinary differential equations; Nicoletti problem; Picard \newline problem 
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     title = {Boundary value problems for higher order ordinary differential equations},
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Majorana, Armando; Marano, Salvatore A. Boundary value problems for higher order ordinary differential equations. Commentationes Mathematicae Universitatis Carolinae, Tome 35 (1994) no. 3, pp. 451-466. http://geodesic.mathdoc.fr/item/CMUC_1994__35_3_a3/