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Asymptotic Behavior of Solutions of Certain Differential Equations

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

The study of behavioral properties of solutions of differential equations near infinity is of immense importance and hence it continues to attract many researchers. Therefore, in this chapter we shall present some recent contributions on the asymptotic behavior of solutions of second order differential equations as well as the behavioral properties of positive solutions of singular Emden-Fowler-type equations. In Section 8.1 it is shown that for a large class of differential equations, not only can the existence of nonoscillatory solutions be proved, but also an explicit asymptotic form of the nonoscillatory solutions may be provided. Then, we shall impose more restrictions on the sign of the integrable coefficient of the equation, and get necessary and sufficient conditions so that the solutions have the specified asymptotic behavior as t → ∞, i.e., solutions which behave asymptotically like a nonzero constant and also those which behave asymptotically like ct, c ≠ 0. For this, various averaging techniques of the types employed in the previous chapters to study the oscillatory behavior of such equations have been used. Section 8.2 is devoted to the study of existence, uniqueness and asymptotic behavior of positive solutions of singular Emden-Fowler-type equations. The cases when the coefficient of the equation under consideration is of constant sign, or of an alternating sign are systematically discussed. Then the existence as well as nonexistence results for the positive solutions of Emden-Fowler-type systems are proved.

Suggested Citation

  • Ravi P. Agarwal & Said R. Grace & Donal O’Regan, 2002. "Asymptotic Behavior of Solutions of Certain Differential Equations," Springer Books, in: Oscillation Theory for Second Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations, chapter 0, pages 558-624, Springer.
    • Handle: RePEc:spr:sprchp:978-94-017-2515-6_8
    DOI: 10.1007/978-94-017-2515-6_8
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