{
  "id": "1705.00898",
  "version": "v1",
  "published": "2017-05-02T10:40:58.000Z",
  "updated": "2017-05-02T10:40:58.000Z",
  "title": "Exponential stability for nonautonomous functional differential equations with state-dependent delay",
  "authors": [
    "Ismael Maroto",
    "Carmen Núñez",
    "Rafael Obaya"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "The properties of stability of compact set $\\mathcal{K}$ which is positively invariant for a semiflow $(\\Omega\\times W^{1,\\infty}([-r,0],\\mathbb{R}^n),\\Pi,\\mathbb{R}^+)$ 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 $\\mathcal{K}$ induce linear skew-product semiflows on the bundles $\\mathcal{K}\\times W^{1,\\infty}([-r,0],\\mathbb{R}^n)$ and $\\mathcal{K}\\times C([-r,0],\\mathbb{R}^n)$. 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 $\\mathcal{K}$ in $\\Omega\\times W^{1,\\infty}([-r,0],\\mathbb{R}^n)$ and also to the exponential stability of this minimal 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-05-02T10:40:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37B55",
      "34K20",
      "37B25",
      "34K14"
    ],
    "keywords": [
      "nonautonomous functional differential equations",
      "exponential stability",
      "state-dependent delay",
      "upper-lyapunov exponent",
      "induce linear skew-product semiflows"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}