2026-04-27T22:00:27Zhttps://uvadoc.uva.es/oai/request

oai:uvadoc.uva.es:10324/257592025-02-20T13:54:34Zcom_10324_1176com_10324_931com_10324_894col_10324_1359

Exponential stability for nonautonomous functional differential equations with state-dependent delay Maroto Camarena, Ismael Núñez Jiménez, María del Carmen Obaya, Rafael The properties of stability of a compact set $K$ which is positively invariant for a semiflow $(\W\times W^{1,\infty}([-r,0],\mathbb{R}^n),\Pi,\mathbb{R}^+)$ 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 $K\times W^{1,\infty}([-r,0],\mathbb{R}^n)$ and $\mK\times C([-r,0],\mathbb{R}^n)$. 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 $K$ in $\W\times W^{1,\infty}([-r,0],\mathbb{R}^n)$ and also to the exponential stability of this compact set when the supremum norm is taken in $W^{1,\infty}([-r,0],\mathbb{R}^n)$. 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. 2017-09-19T18:47:34Z 2017-09-19T18:47:34Z 2017 info:eu-repo/semantics/article Discrete and Continuous Dynamical Systems, Series B 22 (8) 2017, 3167-3197 1531-3492 http://uvadoc.uva.es/handle/10324/25759 10.3934/dcdsb.2017169 eng info:eu-repo/grantAgreement/EC/H2020/643073 info:eu-repo/semantics/openAccess American Institute of Mathematical Sciences SI