An affirmative answer to the extended Gopalsamy and Liu’s conjecture on the global asymptotic stability in a population model

We study the global asymptotic stability of the positive equilibrium in a population model with a piecewise constant argument. Gopalsamy and Liu conjectured that the positive equilibrium N∗=1a+b is globally asymptotically stable if and only if the following inequality holds, r≤r̄ˆ(α)≡1+ααln1+α1−α which has been solved by Muroya and Kato (2005) [2], Li and Yuan (2008) [1] for α≔ab∈[0,1). But, for α∈(−1,0)α∈(−1,0), is the above inequality the necessary and sufficient condition for the global asymptotic stability of the positive equilibrium 1a+b? In this paper, we will give an affirmative answer to the extended Gopalsamy and Liu’s conjecture.