Dynamical properties of nonautonomous functional differential equations with state-dependent delay
| Autor: | Maroto Camarena, Ismael, Núñez Jiménez, María del Carmen, Obaya García, Rafael |
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| Rok vydání: | 2017 |
| Zdroj: | UVaDOC. Repositorio Documental de la Universidad de Valladolid instname |
| Popis: | The properties of stability of a compact set $K$ which is positively invariant for a semiflow $(\W\times W^\infty([-r,0],\mathbb{R}^n),\Pi,\mathbb{R}^+)$ determined by a family of nonautonomous FDEs with state-dependent delay taking values in $[0,r]$ are analyzed. The solutions of the variational equation through the orbits of $K$ induce linear skew-product semiflows on the bundles $K\times W^\infty([-r,0],\R^n)$ and $K\times C([-r,0],\R^n)$. The coincidence of the upper-Lyapunov exponents for both semiflows is checked, and it is a fundamental tool to prove that the strictly negative character of this upper-Lyapunov exponent is equivalent to the exponential stability of $\mK$ in $\W\times W^\infty([-r,0],\R^n)$ and also to the exponential stability of this compact set when the supremum norm is taken in $W^\infty([-r,0],\R^n)$. In particular, the existence of a uniformly exponentially stable solution of a uniformly almost periodic FDE ensures the existence of exponentially stable almost periodic solutions. Ministerio de Economía, Industria y Competitividad (MTM2015-66330-P) |
| Databáze: | OpenAIRE |
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