Dynamics of Infinite Dimensional Systems pp 151-159 | Cite as

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

Conference paper

## 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.
uniformly in x ∈ ℝ, then there exists s has exactly zero, one or two T-periodic solutions according to s < s

$$
f(x,u) \to + \infty as|u| \to \infty
$$

_{1}∈ ℝ such that the equation$$
\{ u\} '(x) + f(x,u(x)) = s
$$

(1)

_{1}, s = s_{1}or s > s_{1}. 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.P. HARTMAN, “Ordinary Differential Equations”, Wiley, 1964.Google Scholar
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- 4.J. MAWHIN, First order ordinary differential equations with several periodic solutions, to appear.Google Scholar
- 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