Bifurcation properties for a sequence of approximation of delay equations. Arino, O. & Benkhalti, R. J. Math. Anal. Appl., 171(2):377–388, 1992.
abstract   bibtex   
The purpose of the paper is to investigate properties of the differential equations $\dot x=λf(x(t),$ $x(t-1))$ and its approximation by difference equations. In section II the authors show that in any segment $[ǎrphi,\psi]$ of initial values (where $ǎrphi<0<\psi)$ there is at least one initial value of an oscillating solution which is the limit of a sequence of initial values for the approximating equations. The results in section III contain information about bifurcation of Rabinowitz type.
@Article{ArinoBenkhalti1992,
  author     = {Arino, Ovide and Benkhalti, Rachid},
  title      = {Bifurcation properties for a sequence of approximation of delay equations},
  journal    = {J. Math. Anal. Appl.},
  year       = {1992},
  volume     = {171},
  number     = {2},
  pages      = {377--388},
  issn       = {0022-247X},
  abstract   = {The purpose of the paper is to investigate
                  properties of the differential equations $\dot
                  x=\lambda f(x(t),$ $x(t-1))$ and its approximation
                  by difference equations. In section II the authors
                  show that in any segment $[\varphi,\psi]$ of initial
                  values (where $\varphi<0<\psi)$ there is at least
                  one initial value of an oscillating solution which
                  is the limit of a sequence of initial values for the
                  approximating equations. The results in section III
                  contain information about bifurcation of Rabinowitz
                  type.},
  classmath  = {*34K15 Qualitative theory of solutions of functional-differential equations 34C23 Bifurcation (periodic solutions) 34A45 Theoretical approximation of solutions of ODE 34A47 Bifurcation 39A10 Difference equations  },
  coden      = {JMANAK},
  fjournal   = {Journal of Mathematical Analysis and Applications},
  keywords   = {delay differential equations; approximation by difference equations; oscillating solution; bifurcation of Rabinowitz type},
  mrclass    = {34K15},
  mrnumber   = {93j:34098},
  mrreviewer = {Xiao Biao Lin},
  reviewer   = {G.Karakostas (Ioannina)},
}

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