The stability and almost periodic solution for generalized logistic almost periodic system with delays (Q2888847)

This paper is concerned with the generalized logistic almost periodic system with infinite and discrete delays NEWLINE\[NEWLINE N'(t)=N(t)\left [a(t) -\sum_{j=1}^n c_j(t)N^j(t-\tau) - b(t)\int_0^\infty K(s)N(t-s)\,ds \right ], \tag{1} NEWLINE\]NEWLINE where \(K: [0,\infty) \to [0,\infty)\) is a piecewise-continuous function and satisfies \(\int_0^\infty K(s)\,ds=1\), \(\int_0^\infty sK(s)\,ds<\infty\), the coefficients \(a(t)\), \(b(t)\), and \(c_j(t)\), \(j=1,2,\dotsc,n\), are continuous almost periodic functions. By using the almost periodic function hull theory, under mild boundedness assumptions on the coefficient functions, it is proven that system (1) has a unique strictly positive almost periodic solution which is globally asymptotically stable. The boundedness conditions used in this paper are weaker than those in some previous papers by other authors. It also turns out that the discrete delay terms \(c_j(t)N^j(t-\tau)\), \(j=1,2,\dotsc,n\), in system (1) do not influence the existence and uniqueness of the strictly positive almost periodic solution and its asymptotic stability.NEWLINENEWLINEReviewer's remark: Note that the condition \(\int_0^\infty K(s)\,ds=1\) can be removed, since it can be obtained from the condition \(\int_0^\infty sK(s)\,ds<\infty\) and by an appropriate scaling.