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Generalized de la Vallée Poussin Disconjugacy Tests for Linear Differential Equations(1)

Published online by Cambridge University Press:  20 November 2018

D. Willett
D. Willett*
Affiliation:
University of Utah, Salt Lake City, Utah; University of Alberta, Edmonton, Alberta
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In this paper, we study the oscillatory behavior of the solutions of the linear differential equation

(1.1)

where

(1.2)

and all functions are assumed to be continuous on a bounded interval [a, b). An «th-order linear equation is said to be disconjugate on an interval I provided it has no nontrivial solution with more than n — 1 zeros, counting multiplicities, in I.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 14 , Issue 3 , September 1971 , pp. 419 - 428
Copyright
Copyright © Canadian Mathematical Society 1971

Footnotes

(1)

Research supported, in part, by NSF Grant GP-19425.

References

0. Ararna, O. and Ripianu, D., Quelques recherches actuelles concernant l'équation de Ch. de la Vallée Poussin relative au problème polylocal dans la théorie des équations différentielles, Mathematica (Cluj) (31) 8 (1966), 19–28.Google Scholar
1. Bessmertnykh, G. A. and Levin, A. Yu., Some estimates of differentiable functions of one variable, Dokl. Akad. Nauk SSSR 144 (1962), 471-474.Google Scholar
2. Erdélyi, A., et al., Higher transcendental functions, Vol. I, McGraw-Hill, New York, 1953.Google Scholar
3. Hartman, P., On disconjugacy criteria, Proc. Amer. Math. Soc. 24 (1970), 374-381.Google Scholar
4. Hukuhara, M., Senkei Zyôbibunhôteisiki no Zero-ten ni tuite, Sûgaku 15 (1963), 108-109.Google Scholar
5. Hukuhara, M., Une propriété de l'application f{x, y, y', …, yn), Funkcial Ekvac. 5 (1963), 135-144.Google Scholar
6. Kondrat'ev, V. A., Oscillating properties of solutions of the equation y(n)+p(x)y=0, Trudy Moskov. Mat. Obšč. 10 (1961), 419-436.Google Scholar
7. Lasota, A., Sur la distance entre les zéros de l'équation diff. linéaire du troisième ordre, Ann. Polon. Math. 13 (1963), 129-132.Google Scholar
8. Levin, A. Yu., An estimate for a function with monotonically distributed zeros of successive derivatives, Mat. Sb. 64 (1964), 396-409.Google Scholar
9. Levin, A. Yu., A Fredholm equation with a smooth kernel and boundary-value problems for a linear differential equation, Dokl. Akad. Nauk SSSR 159, 1964. (Soviet Math. Dokl. 5 (1964), 1415–1419.)Google Scholar
10. Levin, A. Yu., Non-oscillation of solutions of the equation xn+p1(t)xn-1+…+pn(t)x = 0, Uspehi Mat. Nauk 24 (1969), 43-96. (Russian Math. Surveys 24 (1969), 43-100.)Google Scholar
11. Martelli, M., Sul criterio di unicità di de la Vallée Poussin, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 45 (1968), 7-12.Google Scholar
12. Richard, U., Metodi diver si per ottenere disequaglianze alla de la Vallée Poussin nelle equazioni differ enziali ordinarie del secondo e terzo or dine, Univ. e Politec. Torino Rend. Sem. Mat. 27 (1967–68), 35-68.Google Scholar
13. Satô, T., Seizi Senkei Bibunhôteisiki no Zero-ten no Bunpu ni tuite, Kansû Hoteisiki 22 (1940), 39-43.Google Scholar
14. Sherman, T. L., Conjugate points and simple zeros for ordinary linear differential equations, Trans. Amer. Math. Soc. 146 (1969), 397-411.Google Scholar
15. Sherman, T. L., Properties of solutions of nth order linear differential equations, Pacific J. Math. 15 (1965), 1045-1060.Google Scholar
16. Tumura, M., Kôkai Zyôbibunhôteisiki ni tuite, Kansû Hôteisiki 30 (1941), 20-35.Google Scholar
17. de la Vallée Poussin, C., Sur l'équation différentielle linéaire du second ordre, détermination d'une intégrale par deux valeurs assignées. Extension aux équations d'ordre n, J. Math. Pures Appl. 8 (1929), 125-144.Google Scholar
18. Willett, D., Oscillation on finite or infinite intervals of second order linear differential equations, Canad. Math. Bull, (to appear).Google Scholar
19 Willett, D. Asymptotic behavior of disconfugate nth order differential equations, Canad. J. Math. 23 (1971), 293-314.Google Scholar
20. Zaǐceva, G. S., A multipoint boundary value problem, Dokl. Akad. Nauk SSSR 176 (1967), 763-765. (Soviet Math. Dokl. 8 (1967), 1183–1185.)Google Scholar