## Differential and Integral Equations

### Necessary and sufficient conditions for existence and uniqueness of bounded or almost-periodic solutions for differential systems with convex potential

Philippe Cieutat

#### Abstract

We give necessary and sufficient conditions for the existence and uniqueness of bounded or almost-periodic solutions of the first-order differential system: $u' + \nabla \Phi (u) = e(t)$, when $\nabla \Phi$ denotes the gradient of a convex function on $\mathbb R^N$. We also study the relations of continuity between the forcing term $e$ and the solution $u$. Then we give similar results for the second-order differential system: $u'' = \nabla \Phi (u) + e(t)$.

#### Article information

Source
Differential Integral Equations, Volume 18, Number 4 (2005), 361-378.

Dates
First available in Project Euclid: 21 December 2012

https://projecteuclid.org/euclid.die/1356060192

Mathematical Reviews number (MathSciNet)
MR2122704

Zentralblatt MATH identifier
1212.34112

Subjects
Primary: 34C11: Growth, boundedness
Secondary: 34C12: Monotone systems 34C27: Almost and pseudo-almost periodic solutions

#### Citation

Cieutat, Philippe. Necessary and sufficient conditions for existence and uniqueness of bounded or almost-periodic solutions for differential systems with convex potential. Differential Integral Equations 18 (2005), no. 4, 361--378. https://projecteuclid.org/euclid.die/1356060192