Existence results for initial value problems for neutral functional-differential equations. Arino, O., Benkhalti, R., & Ezzinbi, K. J. Differential Equations, 138(1):188–193, 1997.
abstract   bibtex   
The authors give a counterexample to show that theorem 3.1 from \it S. K. Ntouyas, Y. G. Sficas and \it P. Ch. Tsamatos [J. Differ. Equations 114, No. 2, 527-537 (1994; Zbl 810.34061)] fails. This theorem is concerned with an existence theorem for initial value problems for neutral functional differential equations of the form $$d\over dt[x(t)-λg(t,x\sb t)]=λf(t,x\sb t),\quad t\in[0,T],\tag 1$$ $$x\sb 0=ǎrphi∈C([-r,0],\bbfR\sp n),\tag 2$$ where $x\sb t(s)=x(t+s)$ ($s\in[-r,0]$), and $f$, $g$ are continuous functions from $[0,T]\times C([-r,0],{\bbfR}\sp n)$ into ${\bbfR}\sp n$, $\lambda\in[0,1]$. \par The given correction of this theorem is as follows. Let $f$ and $g$ be completely continuous and for any bounded $\Lambda∈C([-r,T],{\bbfR}\sp n)$ the set $\{t↦g(t,x\sb t): x\in\Lambda\}$ is equicontinuous in $C([0,T],{\bbfR}\sp n)$. Let, moreover, exist a constant $K$ such that $‖x‖\sb{C([-r,T],{\bbfR}\sp n)}≤K$ for each solution of (1)–(2) and any $\lambda\in[0,1]$. Then the problem (1)–(2) for $\lambda=1$ has at least one solution in $[-r,T]$. The equations with infinite delay and the problem of periodic solutions are also discussed.
@Article{ArinoBenkhaltiEzzinbi1997,
  author    = {Arino, Ovide and Benkhalti, R. and Ezzinbi, K.},
  title     = {Existence results for initial value problems for neutral functional-differential equations},
  journal   = {J. Differential Equations},
  year      = {1997},
  volume    = {138},
  number    = {1},
  pages     = {188--193},
  issn      = {0022-0396},
  abstract  = {The authors give a counterexample to show that
                  theorem 3.1 from {\it S. K. Ntouyas, Y. G. Sficas}
                  and {\it P. Ch. Tsamatos} [J. Differ. Equations 114,
                  No. 2, 527-537 (1994; Zbl 810.34061)] fails. This
                  theorem is concerned with an existence theorem for
                  initial value problems for neutral functional
                  differential equations of the form $${d\over
                  dt}[x(t)-\lambda g(t,x\sb t)]=\lambda f(t,x\sb
                  t),\quad t\in[0,T],\tag 1$$ $$x\sb 0=\varphi\in
                  C([-r,0],{\bbfR}\sp n),\tag 2$$ where $x\sb
                  t(s)=x(t+s)$ ($s\in[-r,0]$), and $f$, $g$ are
                  continuous functions from $[0,T]\times
                  C([-r,0],{\bbfR}\sp n)$ into ${\bbfR}\sp n$,
                  $\lambda\in[0,1]$. \par The given correction of this
                  theorem is as follows. Let $f$ and $g$ be completely
                  continuous and for any bounded $\Lambda\in
                  C([-r,T],{\bbfR}\sp n)$ the set $\{t\mapsto g(t,x\sb
                  t): x\in\Lambda\}$ is equicontinuous in
                  $C([0,T],{\bbfR}\sp n)$. Let, moreover, exist a
                  constant $K$ such that $\Vert x\Vert
                  \sb{C([-r,T],{\bbfR}\sp n)}\leq K$ for each solution
                  of (1)--(2) and any $\lambda\in[0,1]$. Then the
                  problem (1)--(2) for $\lambda=1$ has at least one
                  solution in $[-r,T]$. The equations with infinite
                  delay and the problem of periodic solutions are also
                  discussed.},
  classmath = {*34K40 Neutral equations 34K05 General theory of functional-differential equations 47H09 Mappings defined by shrinking properties },
  coden     = {JDEQAK},
  fjournal  = {Journal of Differential Equations},
  keywords  = {neutral functional differential equations; initial value problem; existence theorem; infinite delay; periodic solutions},
  mrclass   = {34K05 (34K40)},
  mrnumber  = {98d:34099},
  pdf       = {ArinoBenkhaltiEzzinbi-1997-JDE138.pdf},
  reviewer  = {R.R.Akhmerov (Novosibirsk)},
}
Downloads: 0
About
Documentation
Keywords
Search
Users
Terms and Conditions of Use
Privacy Policy
Sitemap
© BibBase.org 2021