{
  "id": "1108.3392",
  "version": "v1",
  "published": "2011-08-17T04:16:53.000Z",
  "updated": "2011-08-17T04:16:53.000Z",
  "title": "Existence of bounded uniformly continuous mild solutions on $\\Bbb{R}$ of evolution equations and some applications",
  "authors": [
    "Bolis Basit",
    "Hans Günzler"
  ],
  "comment": "16 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "We prove that there is $x_{\\phi}\\in X$ for which (*)$\\frac{d u(t)}{dt}= A u(t) + \\phi (t) $, $u(0)=x$ has on $\\r$ a mild solution $u\\in C_{ub} (\\r,X)$ (that is bounded and uniformly continuous) with $u(0)=x_{\\phi}$, where $A$ is the generator of a holomorphic $C_0$-semigroup $(T(t))_{t\\ge 0}$ on ${X}$ with sup $_{t\\ge 0} \\,||T(t)|| < \\infty$, $\\phi\\in L^{\\infty} (\\r,{X})$ and $i\\,sp (\\phi)\\cap \\sigma (A)=\\emptyset$. As a consequence it is shown that if $\\n$ is the space of almost periodic $AP$, almost automorphic $AA$, bounded Levitan almost periodic $LAP_b$, certain classes of recurrent functions $REC_b$ and $\\phi \\in L^{\\infty} (\\r,{X})$ such that $M_h \\phi:=(1/h)\\int_0^h \\phi (\\cdot+s)\\, ds \\in \\n$ for each $h >0$, then $u\\in \\n\\cap C_{ub}$. These results seem new and generalize and strengthen several recent Theorems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-08-17T04:16:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47D06",
      "43A60",
      "43A99",
      "47A10"
    ],
    "keywords": [
      "bounded uniformly continuous mild solutions",
      "evolution equations",
      "applications",
      "recurrent functions",
      "bounded levitan"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1108.3392B"
    }
  }
}