## Abstract

In this paper the disconjugate linear differential operator of n-th order D^{1/(n)} given by

is considered together with other n−1 operators, which are obtained from 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

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 (*).

## References

- [1]
M. Cecchi -M. Marini -Gab. Villari,

*On the monotonicity property for a certain class of second order differential equations*, J. Diff. Equat.,**82**, 1 (1989), pp. 15–27. - [2]
M. Cecchi -M. Marini -Gab. Villari,

*Integral criteria for a classification of solutions of linear differential equations*, J. Diff. Equat.,**99**, 2 (1992), pp. 381–397. - [3]
M.Cecchi - M.Marini - Gab.Villari,

*Integral criteria for the asymptotic behavior of solutions of linear differential equations: the duality principle*, in*Proc. Equadiff. '91*, World Scientific (1993), pp. 385–389. - [4]
W. A. Coppel,

*Disconjugacy*, Lect. Notes Math.,**220**, Springer, Berlin (1971). - [5]
J. M. Dolan -G. A. Klaasen,

*Strongly oscillatory and nonoscillatory subspaces of linear equations*, Can. J. Math.,**27**, 1 (1975), pp. 106–110. - [6]
J. Dzurina,

*Comparison theorems for functional differential equations with advanced argument*, Boll. Un. Mat. Ital.,**7-A**(1993), pp. 461–470. - [7]
U. Elias,

*Nonoscillation and eventual disconjugacy*, Proc. Amer. Math. Soc.,**66**, 2 (1977), pp. 269–275. - [8]
U. Elias,

*A classification of the solutions of a differential equation according to their asymptotic behaviour*, Proc. R. Soc. Edinb.,**83**A (1979), pp. 25–38. - [9]
U.Elias - H.Gingold,

*Oscillation of two-term differential equations through asymptotics*, in*Proc. Equadiff. '91*, World Scientific 1993), pp. 463–467. - [10]
M. Gaudenzi,

*On the Sturm-Picone theorem for the n-th-order differential equations*, SIAM J. Math. Anal.,**21**, 4 (1990), pp. 980–994. - [11]
M. Gaudenzi,

*On the comparison of the m-th eigenvalue for the equation Ly+λq(x)y=0*, Results Math.,**20**(1991), pp. 481–498. - [12]
M. Gaudenzi,

*Comparison theorems for disconjugate linear equations*, Res. Notes Math.,**272**(1992), pp. 34–38. - [13]
P. Hartman,

*Ordinary Differential Equations*, 2nd edition, Birkhauser, Boston (1982). - [14]
I. T. Kiguradze -T. A. Chanturia,

*Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations*, Kluwer Acad. Publ., Dordrecht (1993). - [15]
T. Kusano -M. Naito,

*Boundedness of solutions of a class of higher order ordinary differential equations*, J. Diff. Equat.,**46**(1982), pp. 32–45. - [16]
T. Kusano -M. Naito -K. Tanaka,

*Oscillatory and asymptotic behaviour of solutions of a class of linear ordinary differential equations*, Proc. R. Soc. Edinb.,**90**A (1981), pp. 25–40. - [17]
T. Kusano -B. Singh,

*Positive solutions of functional differential equations with singular nonlinear terms*, Nonlinear Anal., T.M.A., 8, 9 (1984), pp. 1081–1090. - [18]
V. Liberto Jannelli,

*Proprietà di parziale e completa oscillatorietà per le soluzioni di equazioni differenziali lineari ordinarie del secondo e del terzo ordine*, Boll. Un. Mat. Ital.,**3**C (1984), pp. 171–187. - [19]
M. Naito,

*Nonoscillatory solutions of linear differential equations with deviating argument*, Ann. Mat. Pura Appl. IV,**136**(1984), pp. 1–13. - [20]
R. L. Potter,

*On self-adjoint differential equations of second order*, Pacif. J. Math.,**3**(1953), pp. 467–491. - [21]
M. Rab,

*Criteria fur die Oszillation der Losungen der Differentialgleichung [p(t)x′]′++q(t)x=0*, Cas. Pest. Math.,**84**(1959), pp. 335–370 (*Erratum*, ibid.,**85**, (1960), p. 91). - [22]
M. Svec,

*Behavior of nonoscillatory solutions of some nonlinear differential equations*, Acta Math. Univ. Comen.,**39**(1980), pp. 115–129. - [23]
C. A. Swanson,

*Comparison and Oscillation Theory of Linear Differential Equations*, Academic Press, New York (1968).

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Cecchi, M., Marini, M. & Villari, G. On a cyclic disconjugate operator associated to linear differential equations.
*Annali di Matematica pura ed applicata* **170, **297–309 (1996). https://doi.org/10.1007/BF01758992

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### Keywords

- Differential Equation
- Differential Operator
- Linear Differential Equation
- Interesting Consequence
- Cyclic Permutation