Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On $ N$-parameter families and interpolation problems for nonlinear ordinary differential equations


Author: Philip Hartman
Journal: Trans. Amer. Math. Soc. 154 (1971), 201-226
MSC: Primary 34A99
DOI: https://doi.org/10.1090/S0002-9947-1971-0301277-X
MathSciNet review: 0301277
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ y = ({y_0}, \ldots ,{y_{N - 1}})$. This paper is concerned with the existence of solutions of a system of ordinary differential equations $ y' = g(t,y)$ satisfying interpolation conditions $ (^\ast )\,{y_0}({t_j}) = {c_j}$ for $ j = 1, \ldots $, N and $ {t_1} < \cdots < {t_N}$. It is shown that, under suitable conditions, the assumption of uniqueness for all such problems and of ``local'' solvability (i.e., for $ {t_1}, \ldots ,{t_N}$ on small intervals) implies the existence for arbitrary $ {t_1}, \ldots ,{t_N}$ and $ {c_1}, \ldots ,{c_N}$. A result of Lasota and Opial shows that, in the case of a second order equation for $ {y_0}$, the assumption of uniqueness suffices, but it will remain undecided if the assumption of ``local'' solvability can be omitted in general. More general interpolation conditions involving N points, allowing coincidences, are also considered.

Part I contains the statement of the principal results for interpolation problems and those proofs depending on the theory of differential equations. Actually, the main theorems are consequences of results in Part II dealing with ``N-parameter families'' and ``N-parameter families with pseudoderivatives.'' A useful lemma states that if F is a family of continuous functions $ \{ {y^0}(t)\} $ on an open interval (a, b), then F is an N-parameter family (i.e., contains a unique solution of the interpolation conditions $ (^\ast)$ for arbitrary $ {t_1} < \cdots < {t_N}$ on (a, b) and $ {c_1}, \ldots ,{c_N}$) if and only if (i) $ {y^0},{z^0} \in F$ implies $ {y^0} - {z^0} \equiv 0$ or $ {y^0} - {z^0}$ has at most N zeros; (ii) the set $ \Omega \equiv \{ ({t_1}, \ldots ,{t_N},{y^0}({t_1}), \ldots ,{y^0}({t_N})):a < {t_1} < \cdots < {t_N} < b$ and $ {y^0} \in F\} $ is open in $ {R^{2N}}$; (iii) $ {y^1},{y^2}, \ldots , \in F$ and the inequalities $ {y^n}(t) \leqq {y^{n + 1}}(t)$ for $ n = 1,2, \ldots $ or $ {y^n}(t) \geqq {y^{n + 1}}(t)$ for $ n = 1,2, \ldots $ on an interval $ [\alpha ,\beta ] \subset (a,b)$ imply that either $ {y^0}(t) = \lim {y^n}(t)$ exists on (a, b) and $ {y^0} \in F$ or $ \lim \vert{y^n}(t)\vert = \infty $ on a dense set of (a, b); and finally, (iv) the set $ S(t) = \{ {y^0}(t):{y^0} \in F\} $ is not bounded from above or below for $ a < t < b$. The notion of pseudoderivatives permits generalizations to interpolation problems involving some coincident points.


References [Enhancements On Off] (What's this?)

  • [1] O. Arama, Rezultate comparative asupra unor probleme la limita pentru ecuatii differentiale lineare, Acad. R. P. Romîne. Fil. Cluj. Stud. Circ. Mat. 10 (1959), 207-257.
  • [2] E. F. Beckenbach, Generalized convex functions, Bull. Amer. Math. Soc. 43 (1937), 363-371. MR 1563543
  • [3] R. Conti, Recent trends in the theory of boundary value problems for ordinary differential equations, Boll. Un. Mat. Ital. (3) 22 (1967), 135-178. MR 36 #1734. MR 0218650 (36:1734)
  • [4] M. Fukuhara, Sur l'ensemble des courbes intégrales d'un système d'équations différentielles ordinaires, Proc. Imp. Acad. Japan 6 (1930), 360-362. MR 1568301
  • [5] P. Hartman, Unrestricted n-parameter families, Rend. Circ. Mat. Palermo (2) 7 (1958), 123-142. MR 21 #4211. MR 0105470 (21:4211)
  • [6] -, Ordinary differential equations, Wiley, New York, 1964. MR 30 #1270. MR 0171038 (30:1270)
  • [7] E. Kamke, Zur Theorie der Systeme gewöhnlicher Differentialgleichungen. II, Acta Math. 58 (1932), 57-85. MR 1555344
  • [8] A. Lasota and M. Łuczyński, A note on the uniqueness of two point boundary value problems. I, Zeszyty Nauk. Uniw. Jagiello, Prace Mat. No. 12 (1968), 27-29. MR 37 #499. MR 0224900 (37:499)
  • [9] A. Lasota and Z. Opial, L'existence et l'unicité des solutions du problème d'interpolation pour l'équation différentielle ordinaire d'ordre n, Ann. Polon. Math. 15 (1964), 253-271. MR 30 #4012. MR 0173804 (30:4012)
  • [10] -, On the existence and uniqueness of solutions of a boundary value problem for an ordinary second-order differential equation, Colloq. Math. 18 (1967), 1-5. MR 36 #2871. MR 0219792 (36:2871)
  • [11] L. S. Nicolson, Boundary value problems for systems of ordinary differential equations, J. Differential Equations 6 (1969), 397-407. MR 0249710 (40:2951)
  • [12] Z. Opial, On a theorem of O. Arama, J. Differential Equations 3 (1967), 88-91. MR 34 #6194. MR 0206375 (34:6194)
  • [13] M. M. Peixoto, Generalized convex functions and second order differential inequalities, Bull. Amer. Math. Soc. 55 (1949), 563-572. MR 10, 686. MR 0029949 (10:686a)
  • [14] M. C. Peixoto, On the inequalities $ y''' \geqq G(x,y,y',y'')$, An. Acad. Brasil Ci. 21 (1949), 205-218. MR 0031969 (11:235a)
  • [15] L. Tornheim, On n-parameter families of functions and associated convex functions, Trans. Amer. Math. Soc. 69 (1950), 457-467. MR 12, 395. MR 0038383 (12:395d)
  • [16] W. M. Whyburn, Differential systems with boundary conditions at more than two points, Proc. Conference Differential Equations, Univ. of Maryland Book Store, College Park, Md., 1956. MR 18, 481. MR 0082002 (18:481f)
  • [17] K. W. Schrader and P. Waltman, An existence theorem for non-linear boundary value problems, Proc. Amer. Math. Soc. 21 (1969), 653-656. MR 39 #533. MR 0239176 (39:533)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 34A99

Retrieve articles in all journals with MSC: 34A99


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1971-0301277-X
Keywords: Ordinary nonlinear differential equations, multiple point conditions, interpolation, N-parameter family, pseudoderivatives
Article copyright: © Copyright 1971 American Mathematical Society

American Mathematical Society