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On a cyclic disconjugate operator associated to linear differential equations

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Abstract

In this paper the disconjugate linear differential operator of n-th order D1/(n) given by

$$D_1^{(n)} (x)(t) = \frac{1}{{a_n (t)}}\frac{d}{{dt}}\frac{1}{{a_{n - 1} (t)}} \ldots \frac{1}{{a_1 (t)}}\frac{d}{{dt}}x(t)$$

is considered together with other n−1 operators, which are obtained from D 1 (n) by an ordered cyclic permutation of the functions ai. Such operators play an important role in the study of oscillation of the associated linear differential equation

$$D_1^{(n)} (x)(t) \pm x(t) = 0.$$
((*))

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 (*).

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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