Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On the extremal solutions of $ n$th-order linear differential equations


Author: W. J. Kim
Journal: Proc. Amer. Math. Soc. 33 (1972), 62-68
MSC: Primary 34C10
DOI: https://doi.org/10.1090/S0002-9939-1972-0294780-9
MathSciNet review: 0294780
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Distribution of zeros of extremal solutions of linear nth-order differential equations is discussed. Existence and nonexistence of extremal solutions with certain zero distributions are established. For instance, it is proved that every extremal solution for $ [\alpha, {\eta _1}(\alpha )]$ of the equation $ {y^{(n)}} + {p_{n - 1}}{y^{(n - 1)}} + \cdots + {p_0}y = 0$ has a zero of order 2 at $ {\eta _1}(\alpha )$ and has no more than $ n - 2$ zeros on $ [\alpha, {\eta _1}(\alpha ))\;{\text{if}}\;{p_i} \leqq 0,i = 0,1, \cdots, n - 2$.


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

  • [1] R. G. Aliev, On certain properties of solutions of ordinary differential equations of fourth order, Sb. Aspirantsh. Kazan Institute 1964, 15-30. (Russian)
  • [2] N. V. Azbelev and Z. B. Caljuk, On the question of distribution of zeros of solutions of linear differential equations of third order, Mat. Sb. 51 (93) (1960), 475-486; English transl., Amer. Math. Soc. Transl. (2) 42 (1964), 233-245. MR 22 #12266. MR 0121529 (22:12266)
  • [3] J. H. Barrett, Third-order differential equations with nonnegative coefficients, J. Math. Anal. Appl. 24 (1968), 212-224. MR 38 #365. MR 0232039 (38:365)
  • [4] -, Oscillation theory of ordinary linear differential equations, Advances in Math. 3 (1969), 415-509. MR 41 #2113. MR 0257462 (41:2113)
  • [5] P. Hartman, Unrestricted n-parameter families, Rend. Circ. Mat. Palermo (2) 7 (1958), 123-142. MR 21 #4211. MR 0105470 (21:4211)
  • [6] W. J. Kim, Oscillatory properties of linear third-order differential equations, Proc. Amer. Math. Soc. 26 (1970), 286-293. MR 0264162 (41:8758)
  • [7] W. J. Kim, Simple zeros of solutions of nth-order linear differential equations, Proc. Amer. Math. Soc. 28 (1971), 557-561. MR 0274861 (43:619)
  • [8] W. Leighton and Z. Nehari, On the oscillation of solutions of self-adjoint linear differential equations of the fourth order, Trans. Amer. Math. Soc. 89 (1958), 325-377. MR 21 #1429. MR 0102639 (21:1429)
  • [9] A. Ju. Levin, The non-oscillation of solutions of the equation $ {x^{(n)}} + {p_1}(t){x^{(n - 1)}} + \cdots + {p_n}(t)x = 0$, Uspehi Mat. Nauk 24 (1969), no. 2 (146), 43-96 = Russian Math. Surveys 24 (1969), no. 2, 43-99. MR 40 #7537. MR 0254328 (40:7537)
  • [10] Z. Opial, On a theorem of O. Aramă, J. Differential Equations 3 (1967), 88-91. MR 34 #6194. MR 0206375 (34:6194)
  • [11] A. C. Peterson, Distribution of zeros of solutions of a fourth order differential equation, Pacific J. Math. 30 (1969), 751-764. MR 40 #5975. MR 0252758 (40:5975)
  • [12] -, A theorem of Aliev, Proc. Amer. Math. Soc. 23 (1969), 364-366. MR 40 #2961. MR 0249720 (40:2961)
  • [13] V. Pudei, Über die Eigenschaften der Lösungen linearer Differentialgleichungen gerader Ordnung, Časopis Pěst. Mat. 94 (1969), 401-425. MR 0273112 (42:7993)
  • [14] T. L. Sherman, Properties of solutions of nth order linear differential equations, Pacific J. Math. 15 (1965), 1045-1060. MR 32 #2654. MR 0185185 (32:2654)
  • [15] -, Conjugate points and simple zeros for ordinary linear differential equations, Trans. Amer. Math. Soc. 146 (1969), 397-411. MR 41 #572. MR 0255912 (41:572)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 34C10

Retrieve articles in all journals with MSC: 34C10


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0294780-9
Keywords: nth-order linear equations with real-valued continuous coefficients, zero distribution of extremal solutions, existence and nonexistence
Article copyright: © Copyright 1972 American Mathematical Society

American Mathematical Society