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)


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
$$ \{ u\} '(x) + f(x,u(x)) = s $$
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.


Periodic Solution Cauchy Problem Deterministic Model Riccati Equation Qualitative Behavior 
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    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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