Globally asymptotic behavior of solutions for an integro-differential equation with infinite delay (Q2765932)

The author considers two integro-differential equations with infinite delay, by the form NEWLINE\[NEWLINEdN(t)/dt=-\gamma N(t)+ \alpha\int^\infty_0 K(s)F \bigl(N(t-s) \bigr)ds,\;t\geq 0NEWLINE\]NEWLINE and NEWLINE\[NEWLINEdN(t)/dt= -\gamma N(t)-\beta F\bigl(N (t)\bigr)+ \alpha\int^\infty_0 K(s)F\bigl(N(t-s) \bigr)ds,\;t\geq 0,NEWLINE\]NEWLINE generalizing two models of population dynamics in hemetology. They are functional-differential equations with infinite delay. In the above equations \(F(u)= u/(1+u^n)\), \(n\in(0,1]\); \(\alpha,\beta, \gamma\) are strictly positive parameters, \(n\in(0,1]\); \(K:[0,\infty)\to[0,\infty)\) has the property \(\int^\infty_0 K(s)ds=1\); \(N(t)\) is the density of mature stem cells in the blood circulatory procedure.NEWLINENEWLINENEWLINEThe above equations were discussed by other authors respecting the existence, the uniqueness, the continuous dependence and continuations of the solutions, in an admissible space. One of them, quoted in the references list, gave a criterion for globally asymptotic behaviour of an autonomous functional-differential equation.NEWLINENEWLINENEWLINEIntroducing some function spaces and conditions for the functions and parameters entering the above equations, the author shows that they satisfy the conditions in this criterion; under these conditions, the equilibria of these equations are globally asymptotically stable. In this manner, he renounces to the admissible space notion, on which the previous studies are based; it isn't uniquely determined.