A periodicity result for a nonlinear functional integral equation

A periodicity result for a nonlinear functional integral equation. Arino, O. & Mortabit, A. J. Math. Biol., 30(5):437–456, 1992.
abstract bibtex

This paper is a continuation of the considerations by the first author and \it M. Kimmel [ibid. 27, No. 3, 341-354 (1989; Zbl. 715.92011)]. The authors study the nonlinear functional integral equation $$n(t,x)=2\lambda\sigma[N(t-1)]\int\spa\sb 2\sba\sb 1f(x,ǎrphi(x))n(t-\psi(z),z)dz,$$ where $N(t)=\int\sp{a\sb 2}\sb{a\sb 1}\int\sp t\sb{t-\psi(u)}n(v,u)dv du$, for $t\ge 0$ and $x\in(a\sb 1,a\sb 2)$. Here, the functions $f(ḑot,ḑot),ǎrphi(ḑot),\psi(ḑot)$ and $\sigma(ḑot)$ have to satisfy suitable conditions.\par The existence and uniqueness of a nonnegative solution $n$ is proved. The asymptotic behavior of the solution is studied. For $\lambda$ sufficiently large slow oscillations take place. The existence of a nontrivial periodic solution is proved, too.

@Article{ArinoMortabit1992,
  author     = {Arino, Ovide and Mortabit, Abdessamad},
  title      = {A periodicity result for a nonlinear functional integral equation},
  journal    = {J. Math. Biol.},
  year       = {1992},
  volume     = {30},
  number     = {5},
  pages      = {437--456},
  issn       = {0303-6812},
  abstract   = {This paper is a continuation of the considerations
                  by the first author and {\it M. Kimmel} [ibid. 27,
                  No. 3, 341-354 (1989; Zbl. 715.92011)]. The authors
                  study the nonlinear functional integral equation
                  $$n(t,x)=2\lambda\sigma[N(t-1)]\int\sp{a\sb
                  2}\sb{a\sb 1}f(x,\varphi(x))n(t-\psi(z),z)dz,$$
                  where $N(t)=\int\sp{a\sb 2}\sb{a\sb 1}\int\sp
                  t\sb{t-\psi(u)}n(v,u)dv du$, for $t\ge 0$ and
                  $x\in(a\sb 1,a\sb 2)$. Here, the functions
                  $f(\cdot,\cdot),\varphi(\cdot),\psi(\cdot)$ and
                  $\sigma(\cdot)$ have to satisfy suitable
                  conditions.\par The existence and uniqueness of a
                  nonnegative solution $n$ is proved. The asymptotic
                  behavior of the solution is studied. For $\lambda$
                  sufficiently large slow oscillations take place. The
                  existence of a nontrivial periodic solution is
                  proved, too.},
  classmath  = {*45G10 Nonsingular nonlinear integral equations 45M05 Asymptotic theory of integral equations 45M15 Periodic solutions of integral equations 45M20 Positive solutions of integral equations 92C99 Physiological, cellular and medical topics  },
  coden      = {JMBLAJ},
  fjournal   = {Journal of Mathematical Biology},
  keywords   = {nonlinear integral equation with delay; ejective fixed point; cell proliferation; nonlinear functional integral equation; nonnegative solution; asymptotic behavior; slow oscillations; periodic solution},
  mrclass    = {34K15 (34K20 45M15 47N20 92C05)},
  mrnumber   = {93i:34126},
  mrreviewer = {Hans Engler},
  reviewer   = {W.Petry (Duesseldorf)},
}

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The authors study the nonlinear functional integral equation $$n(t,x)=2\\lambda\\sigma[N(t-1)]\\int\\spa\\sb 2\\sba\\sb 1f(x,ǎrphi(x))n(t-\\psi(z),z)dz,$$ where $N(t)=\\int\\sp{a\\sb 2}\\sb{a\\sb 1}\\int\\sp t\\sb{t-\\psi(u)}n(v,u)dv du$, for $t\\ge 0$ and $x\\in(a\\sb 1,a\\sb 2)$. Here, the functions $f(ḑot,ḑot),ǎrphi(ḑot),\\psi(ḑot)$ and $\\sigma(ḑot)$ have to satisfy suitable conditions.\\par The existence and uniqueness of a nonnegative solution $n$ is proved. The asymptotic behavior of the solution is studied. For $\\lambda$ sufficiently large slow oscillations take place. 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Biol.},\r\n year = {1992},\r\n volume = {30},\r\n number = {5},\r\n pages = {437--456},\r\n issn = {0303-6812},\r\n abstract = {This paper is a continuation of the considerations\r\n by the first author and {\\it M. Kimmel} [ibid. 27,\r\n No. 3, 341-354 (1989; Zbl. 715.92011)]. The authors\r\n study the nonlinear functional integral equation\r\n $$n(t,x)=2\\lambda\\sigma[N(t-1)]\\int\\sp{a\\sb\r\n 2}\\sb{a\\sb 1}f(x,\\varphi(x))n(t-\\psi(z),z)dz,$$\r\n where $N(t)=\\int\\sp{a\\sb 2}\\sb{a\\sb 1}\\int\\sp\r\n t\\sb{t-\\psi(u)}n(v,u)dv du$, for $t\\ge 0$ and\r\n $x\\in(a\\sb 1,a\\sb 2)$. Here, the functions\r\n $f(\\cdot,\\cdot),\\varphi(\\cdot),\\psi(\\cdot)$ and\r\n $\\sigma(\\cdot)$ have to satisfy suitable\r\n conditions.\\par The existence and uniqueness of a\r\n nonnegative solution $n$ is proved. The asymptotic\r\n behavior of the solution is studied. For $\\lambda$\r\n sufficiently large slow oscillations take place. The\r\n existence of a nontrivial periodic solution is\r\n proved, too.},\r\n classmath = {*45G10 Nonsingular nonlinear integral equations 45M05 Asymptotic theory of integral equations 45M15 Periodic solutions of integral equations 45M20 Positive solutions of integral equations 92C99 Physiological, cellular and medical topics \u0003 },\r\n coden = {JMBLAJ},\r\n fjournal = {Journal of Mathematical Biology},\r\n keywords = {nonlinear integral equation with delay; ejective fixed point; cell proliferation; nonlinear functional integral equation; nonnegative solution; asymptotic behavior; slow oscillations; periodic solution},\r\n mrclass = {34K15 (34K20 45M15 47N20 92C05)},\r\n mrnumber = {93i:34126},\r\n mrreviewer = {Hans Engler},\r\n reviewer = {W.Petry (Duesseldorf)},\r\n}\r\n\r\n","author_short":["Arino, O.","Mortabit, A."],"key":"ArinoMortabit1992","id":"ArinoMortabit1992","bibbaseid":"arino-mortabit-aperiodicityresultforanonlinearfunctionalintegralequation-1992","role":"author","urls":{},"keyword":["nonlinear integral equation with delay; ejective fixed point; cell proliferation; nonlinear functional integral equation; nonnegative solution; asymptotic behavior; slow oscillations; periodic solution"],"downloads":0},"bibtype":"article","biburl":"https://server.math.umanitoba.ca/~jarino/ovide/papers/BiblioOvide.bib","creationDate":"2019-12-20T14:16:41.214Z","downloads":0,"keywords":["nonlinear integral equation with delay; ejective fixed point; cell proliferation; nonlinear functional integral equation; nonnegative solution; asymptotic behavior; slow oscillations; periodic solution"],"search_terms":["periodicity","result","nonlinear","functional","integral","equation","arino","mortabit"],"title":"A periodicity result for a nonlinear functional integral equation","year":1992,"dataSources":["DJzjnMX7p3giiS766"]}