# Article

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Keywords:
higher order ordinary differential equations; Nicoletti problem; Picard \newline problem
Summary:
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.
References:
[1] Abramowitz M., Stegun I.A.: Handbook of Mathematical functions with Formulas, Graphs, and Mathematical Tables. Dover Publ., New York, 1972. MR 0208797
[2] Agarwal R.P.: Boundary Value Problems for Higher Order Differential Equations. World Sci. Publ., Singapore, 1986. MR 1021979 | Zbl 0921.34021
[3] Bernfeld S.R., Lakshmikantham V.: An Introduction to Nonlinear Boundary Value Problems. Academic Press, New York, 1974. MR 0445048
[4] Bernstein S.N.: Sur les fonctions régulierèment monotones. Atti Congresso Int. Mat. Bologna 1928, vol. 2 (1930), 267-275.
[5] Bernstein S.N.: On some properties of cyclically monotonic functions. Izvestiya Akad. Nauk SSSR, Ser. Mat. 14 (1950), 381-404. MR 0037885
[6] Bonanno G., Marano S.A.: Higher order ordinary differential equations. Differential Integral Equations 6 (1993), 1119-1123. MR 1230485
[7] Miranda C.: Istituzioni di Analisi Funzionale Lineare - I. Unione Matematica Italiana, 1978.
[8] Piccinini L.C., Stampacchia G., Vidossich G.: Ordinary Differential Equations in $\Bbb R^n$ (Problems and Methods). Springer-Verlag, New York, 1984. MR 0740539 | Zbl 1220.68090
[9] Schoenberg I.J.: On the zeros of successive derivatives of integral functions. Trans. Amer. Math. Soc. 40 (1936), 12-23. MR 1501863
[10] Whittaker J.M.: Interpolatory Function Theory. Stechert-Hafner Service Agency, New York, 1964. MR 0185330
[11] Zwirner G.: Su un problema di valori al contorno per equazioni differenziali ordinarie di ordine $n$. Rend. Sem. Mat. Univ. Padova 12 (1941), 114-122. MR 0017834

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