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Qualitative Behavior of the Solutions of Periodic First Order Scalar Differential Equations with Strictly Convex Coercive Nonlinearity

  • J. Mawhin
Part of the NATO ASI Series book series (volume 37)

Abstract

It has been proved in [4] that if f: ℝ × ℝ → ℝ is continuous, f(., u) is T-periodic for each u ∈ ℝ, f(x,.) is strictly convex on ℝ for each x ∈ ℝ, and if f(x,.) is uniformly coercive, i.e.
$$ f(x,u) \to + \infty as|u| \to \infty $$
uniformly in x ∈ ℝ, then there exists s1 ∈ ℝ such that the equation
$$ \{ u\} '(x) + f(x,u(x)) = s $$
(1)
has exactly zero, one or two T-periodic solutions according to s < s1, s = s1 or s > s1. The aim of this note is to complete the result by getting a fairly complete picture of the trajectoires of (1) under the same assumptions upon f.

Keywords

Periodic Solution Cauchy Problem Deterministic Model Riccati Equation Qualitative Behavior 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    P. HARTMAN, “Ordinary Differential Equations”, Wiley, 1964.Google Scholar
  2. 2.
    A.C. LAZER and D.A. SANCHEZ, Periodic equilibria under periodic harvesting, Math. Magazine 57 (1984) 156–158.MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    J.L. MASSERA, The existence of periodic solutions of systems of differential equations, Duke Math. J. 17 (1950) 457–475.MathSciNetMATHGoogle Scholar
  4. 4.
    J. MAWHIN, First order ordinary differential equations with several periodic solutions, to appear.Google Scholar
  5. 5.
    D.A. SANCHEZ, Periodic environments, harvesting and a Riccati equation, in “Nonlinear Phenomena in Mathematical Sciences”, Lakshmikantam ed., Academic Press, 1982, 883–886.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1987

Authors and Affiliations

  • J. Mawhin
    • 1
  1. 1.Institut MathématiqueUniversité de LouvainLouvain-la-NeuveBelgium

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