# Article

 Title: Boundary value problems for higher order ordinary differential equations (English) Author: Majorana, Armando Author: Marano, Salvatore A. Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 35 Issue: 3 Year: 1994 Pages: 451-466 . Category: math . 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. (English) Keyword: higher order ordinary differential equations Keyword: Nicoletti problem Keyword: Picard \newline problem MSC: 34A12 MSC: 34B10 MSC: 34B15 idZBL: Zbl 0809.34034 idMR: MR1307273 . Date available: 2009-01-08T18:12:31Z Last updated: 2012-04-30 Stable URL: http://hdl.handle.net/10338.dmlcz/118686 . Reference: [1] Abramowitz M., Stegun I.A.: Handbook of Mathematical functions with Formulas, Graphs, and Mathematical Tables.Dover Publ., New York, 1972. MR 0208797 Reference: [2] Agarwal R.P.: Boundary Value Problems for Higher Order Differential Equations.World Sci. Publ., Singapore, 1986. Zbl 0921.34021, MR 1021979 Reference: [3] Bernfeld S.R., Lakshmikantham V.: An Introduction to Nonlinear Boundary Value Problems.Academic Press, New York, 1974. MR 0445048 Reference: [4] Bernstein S.N.: Sur les fonctions régulierèment monotones.Atti Congresso Int. Mat. Bologna 1928, vol. 2 (1930), 267-275. Reference: [5] Bernstein S.N.: On some properties of cyclically monotonic functions.Izvestiya Akad. Nauk SSSR, Ser. Mat. 14 (1950), 381-404. MR 0037885 Reference: [6] Bonanno G., Marano S.A.: Higher order ordinary differential equations.Differential Integral Equations 6 (1993), 1119-1123. MR 1230485 Reference: [7] Miranda C.: Istituzioni di Analisi Funzionale Lineare - I.Unione Matematica Italiana, 1978. Reference: [8] Piccinini L.C., Stampacchia G., Vidossich G.: Ordinary Differential Equations in $\Bbb R^n$ (Problems and Methods).Springer-Verlag, New York, 1984. Zbl 1220.68090, MR 0740539 Reference: [9] Schoenberg I.J.: On the zeros of successive derivatives of integral functions.Trans. Amer. Math. Soc. 40 (1936), 12-23. MR 1501863 Reference: [10] Whittaker J.M.: Interpolatory Function Theory.Stechert-Hafner Service Agency, New York, 1964. MR 0185330 Reference: [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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