Convergence of a class of discrete-time semiflows with application to neutral delay differential equations

A New Generalization of the Haddock Conjecture

“…Since the asymptotic behavior of NFDE with time-varying delays has not been touched in [1,3,6,8,9,10,11,12,13,14,15], one can find that the analysis method of the dynamical systems in the above references cannot be applied to prove Theorem 1. Noting that F (x) = x 1 3 satisfies (1.5), it follows from Theorem 1 that the Haddock conjecture is true when for the delays are time-varying functions.…”

“…Later on, variants of the above equations are used to model some population growth and spread of epidemics, and hence they have received considerable attention (see, for example, [1,9,10,11,12,14] and the references therein). In particular, the asymptotic behavior of the following non-autonomous NFDE [x(t) − px(t − r)] = H(t, x(t), x(t − r)) (1.3) has been deeply studied in [3,8,13]. Here, H ∈ C(R 3 , R) is periodic in its first argument, and locally Lipschitz continuous in its second argument.…”