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On a cyclic disconjugate operator associated to linear differential equations. (English) Zbl 0870.34004

Summary: The disconjugate linear differential operator of \(n\)-th order \(D_1^{(n)}\) given by \[ {\mathbf D}_1^{(n)} (x)(t)= {1\over a_n(t)} {d\over dt} {1\over a_{n-1} (t)} \cdots {1\over a_1(t)} {d\over dt} x(t) \] is considered together with other \(n-1\) operators, which are obtained from \({\mathbf D}_1^{(n)}\) by an ordered cyclic permutation of the functions \(a_i\). Such operators play an important role in the study of oscillation of the associated linear differential equation (*) \({\mathbf D}_1^{(n)} (x)(t) \pm x(t)=0\).
Some properties of these operators suggest the new idea of “isomorphism of oscillation”. The existence of an isomorphism of oscillation allows to describe the oscillatory or nonoscillatory behavior of solutions of (*) by the oscillatory or nonoscillatory behavior of solutions of other \(n-1\) suitable linear differential equations. From this fact one can easily obtain new results about oscillation or nonoscillation of (*) that might be hard to prove directly. Several interesting consequences concerning the classification of solutions of (*) are also presented together with some new applications to the structure of the set of nonoscillatory solutions of (*).

MSC:

34A30 Linear ordinary differential equations and systems
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
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[1] Cecchi, M.; Marini, M.; Villari, Gab, On the monotonicity property for a certain class of second order differential equations, J. Diff. Equat., 82, 1, 15-27 (1989) · Zbl 0694.34035
[2] Cecchi, M.; Marini, M.; Villari, Gab, Integral criteria for a classification of solutions of linear differential equations, J. Diff. Equat., 99, 2, 381-397 (1992) · Zbl 0761.34009
[3] M.Cecchi - M.Marini - Gab.Villari,Integral criteria for the asymptotic behavior of solutions of linear differential equations: the duality principle, inProc. Equadiff. ’91, World Scientific (1993), pp. 385-389. · Zbl 0938.34519
[4] Coppel, W. A., Disconjugacy, Lect. Notes Math.,220 (1971), Berlin: Springer, Berlin · Zbl 0224.34003
[5] Dolan, J. M.; Klaasen, G. A., Strongly oscillatory and nonoscillatory subspaces of linear equations, Can. J. Math., 27, 1, 106-110 (1975) · Zbl 0313.34028
[6] Dzurina, J., Comparison theorems for functional differential equations with advanced argument, Boll. Un. Mat. Ital., 7-A, 461-470 (1993) · Zbl 0803.34066
[7] Elias, U., Nonoscillation and eventual disconjugacy, Proc. Amer. Math. Soc., 66, 2, 269-275 (1977) · Zbl 0367.34024
[8] Elias, U., A classification of the solutions of a differential equation according to their asymptotic behaviour, Proc. R. Soc. Edinb., 83 A, 25-38 (1979) · Zbl 0407.34042
[9] U.Elias - H.Gingold,Oscillation of two-term differential equations through asymptotics, inProc. Equadiff. ’91, World Scientific 1993), pp. 463-467. · Zbl 0938.34520
[10] Gaudenzi, M., On the Sturm-Picone theorem for the n-th-order differential equations, SIAM J. Math. Anal., 21, 4, 980-994 (1990) · Zbl 0703.34043
[11] Gaudenzi, M., On the comparison of the m-th eigenvalue for the equation Ly+λq(x)y=0, Results Math., 20, 481-498 (1991) · Zbl 0749.34048
[12] Gaudenzi, M., Comparison theorems for disconjugate linear equations, Res. Notes Math., 272, 34-38 (1992) · Zbl 0792.34031
[13] Hartman, P., Ordinary Differential Equations (1982), Boston: Birkhauser, Boston · Zbl 0125.32102
[14] Kiguradze, I. T.; Chanturia, T. A., Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations (1993), Dordrecht: Kluwer Acad. Publ., Dordrecht · Zbl 0782.34002
[15] Kusano, T.; Naito, M., Boundedness of solutions of a class of higher order ordinary differential equations, J. Diff. Equat., 46, 32-45 (1982) · Zbl 0532.34027
[16] Kusano, T.; Naito, M.; Tanaka, K., Oscillatory and asymptotic behaviour of solutions of a class of linear ordinary differential equations, Proc. R. Soc. Edinb., 90 A, 25-40 (1981) · Zbl 0486.34021
[17] Kusano, T.; Singh, B., Positive solutions of functional differential equations with singular nonlinear terms, Nonlinear Anal., T.M.A., 8, 9, 1081-1090 (1984) · Zbl 0558.34057
[18] Jannelli, V. Liberto, Proprietà di parziale e completa oscillatorietà per le soluzioni di equazioni differenziali lineari ordinarie del secondo e del terzo ordine, Boll. Un. Mat. Ital., 3C, 171-187 (1984)
[19] Naito, M., Nonoscillatory solutions of linear differential equations with deviating argument, Ann. Mat. Pura Appl. IV, 136, 1-13 (1984) · Zbl 0548.34035
[20] Potter, R. L., On self-adjoint differential equations of second order, Pacif. J. Math., 3, 467-491 (1953) · Zbl 0051.06502
[21] Rab, M., Criteria fur die Oszillation der Losungen der Differentialgleichung [p(t)x′]′++q(t)x=0, Cas. Pest. Math., 84, 335-370 (1959) · Zbl 0087.29505
[22] Svec, M., Behavior of nonoscillatory solutions of some nonlinear differential equations, Acta Math. Univ. Comen., 39, 115-129 (1980) · Zbl 0525.34029
[23] Swanson, C. A., Comparison and Oscillation Theory of Linear Differential Equations (1968), New York: Academic Press, New York · Zbl 0191.09904
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