Global attractivity for a delay logistic equation with instantaneous terms
Let \(g:[0,\infty)\rightarrow {\mathbb R}\) be a non-decreasing continuous function such that \(g(t)\leq t\) for all \(t\geq 0\) and \(g(t)\rightarrow \infty\) as \(t\rightarrow \infty\). Let \(BC\) be the Banach space consisting of bounded right-hand continuous functions \(\phi:(-\infty,0]\rightarrow {\mathbb R}\) with the sup norm \(\| \phi\| =\sup_{s\in (-\infty,0]}| \phi(s)| \). If \(a>0, \psi\in C([g(0),a],{\mathbb R})\) and \(0\leq t\leq a\), then \(\psi_t\in BC\) is defined by \(\psi_t(s)=\psi(t+s)\), for \(g(t)-t\leq s\leq 0\) and \(\psi_t(s)=0\), for \(-\infty<s<g(t)-t.\) In this paper, the author is concerned with the following one-dimensional delayed logistic differential equation \[ x^{\prime}(t)=-r(t)[1+x(t)][cx(t)+F(t,x_t)],\quad t\geq 0,\tag{1} \] with initial condition \[ x(t)=\phi(t),\text{ for } g(0)\leq t\leq 0,\quad \phi\in C([g(0),0],[-1,\infty)),\quad \phi(0)>-1, \] where \(c\geq 0, r(t)\in C([0,\infty),(0,\infty))\), \(F:[0,\infty)\times BC\rightarrow {\mathbb R}\) and for any \(\psi\in C([g(0),\infty),{\mathbb R})\) with \(\psi_t\in BC\) the function \(t\mapsto F(t,\psi_t)\) is continuous on \([0,\infty)\), and satisfies \(-M_t(-\phi)\leq F(t,\phi)\leq M_t(\phi)\) for \(t\geq 0\), \(\phi\in BC\), where \(M_t(\phi)=\max\{0,\sup_{s\in [g(t)-t,0]}\phi(s)\}\). By using the inequality technique, sufficient conditions are derived for the global attractivity of the zero solution to equation (1). Existing related results in the literature are extended and improved.
scientific article; zbMATH DE number 2117571
scientific article