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Oscillation Results for Differential Systems

In: Oscillation Theory for Second Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations

Author

Listed:
  • Ravi P. Agarwal

    (Florida Institute of Technology)

  • Said R. Grace

    (Cairo University)

  • Donal O’Regan

    (National University of Ireland)

Abstract

In this chapter we are concerned with the oscillation of nonlinear two-dimensional differential systems and second order vector-matrix differential equations. In Section 7.1 we shall present criteria for the oscillation of nonlinear two-dimensional differential systems. This includes the superlinear, linear, and sublinear cases. Section 7.2 deals with the oscillation of linear second order differential systems. Here, first the system considered will be reduced to a certain scalar Riccati inequality, so that the known results from the literature can be applied to obtain oscillation criteria. Then, we shall employ the notation and definitions of Section 6.1 to present some general results. Finally, we shall use Riccati and variational techniques which involve assumptions on the behavior of the eigenvalues of the coefficient matrix (or of its integral) to present a number of sufficient conditions which guarantee the oscillation of linear second order systems. In Section 7.3 we shall discuss the oscillation of nonlinear second order differential systems with functionally commutative matrix coefficients. Here, we shall show that the oscillation theory of such systems can be effectively reduced to the study of diagonal systems of scalar second order differential equations. In Section 7.4 we shall prove some comparison theorems of Hille-Wintner type for second order operator-valued linear differential equations. In Section 7.5 some oscillation results for second order differential systems with a forcing term are given.

Suggested Citation

  • Ravi P. Agarwal & Said R. Grace & Donal O’Regan, 2002. "Oscillation Results for Differential Systems," Springer Books, in: Oscillation Theory for Second Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations, chapter 0, pages 491-557, Springer.
    • Handle: RePEc:spr:sprchp:978-94-017-2515-6_7
    DOI: 10.1007/978-94-017-2515-6_7
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