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)},
}

Downloads: 0