A New Generalization of the Haddock Conjecture

In this paper, we establish some new criteria on the boundedness and asymptotic constancy of solutions for a class of scalar neutral functional differential equations with time-varying delays via differential inequality techniques. Our results are an improvement of existing ones and generalization of the Haddock conjecture.

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. Note that x 1 3 is not locally Lipschitz continuous on R, and the Haddock conjecture equations (1.1) and (1.2) can not be contained in (1.3). On the other hand, the delays in differential equations of population and ecology problems are usually time-varying in the real world, then equation (1.1) can be naturally generalized to the following non-autonomous NFDE with a time-varying delay: where F ∈ C(R, R) is strictly increasing, σ, b ∈ C(R, R) are bounded, and Obviously, (1.1) and (1.2) are particular cases of (1.4) with b(t) = 1 and σ − = σ + . It is well-known that a non-autonomous NFDE with a time-varying delay generally does not generate a semiflow and hence the methods for NFDE with constant delays in [1,9,10,11,12,14] are not suitable for (1.4). Moreover, it should be mentioned that Propositions 4 and 5 of [6] played a crucial role in the discussions of its main results, but they are not true as demonstrated by a counterexample in [15]. Most recently, Liu [4,5] corrected these two propositions, and successfully used them to analyze the asymptotic behavior of some functional differential equations (see Propositions 4 * and 5 * in [5]). Therefore, the proof in [1] needs further improvement.
Thus a natural question arises: whether we can use the improved Proposition 4 * and Proposition 5 * of [5] to correct the proof of the main result of [7]. This is the main purpose of this paper.
Throughout this paper, we assume that The remaining part of this paper is organized as follows. In Section 2, we show the global existence of every solution for (1.4) with the initial data x t0 = ϕ ∈ C. Based on the preparation in Section 2, we state and prove our main result in Section 3.

Preliminaries
In the following, assume that G : R → R is continuous and strictly increasing, and for all x ∈ R \ {0}. Then, from Lemma 1, Proposition 4 * and Proposition 5 * in [5], we have Proposition 1. Consider the differential equation, where c = 0 is a given constant, ε is a parameter satisfying 0 ≤ ε ≤ |c| 2 . Moreover, assume the initial condition (

2.2)
Let u = u(t; t 0 , u 0 ) be the solution of the initial value problems (2.1) and (2.2), andᾱ > 0 be a given constant. Then there exists a positive real number µ independent of t 0 and ε such that Proposition 2. Consider the differential equation, where c = 0 is a given constant, ε is a parameter satisfying 0 ≤ ε ≤ |c| 2 . Moreover, assume the initial condition Lemma 1. (see [4]). Let t 0 ∈ R, T > 0,h ∈ C([t 0 , t 0 + T ] × R, R), and h be nonincreasing with respect to the second variable. Then the initial value Consider the solution y(t) of the following initial value problem, By Lemma 1 , y(t) exists and is unique on [t 0 , t 0 +σ − ]. Denote x(t) = y(t)+a(t).
By the proof of Lemma 3 in [7], we can take the following lemma: Assume that x ∈ C((−σ + , +∞), R) is a bounded, and there exist constantsδ,p ∈ R such that |p| < 1 and lim 3 Boundedness and asymptotic constancy Theorem 1. Let ϕ ∈ C. Then x(t; t 0 , ϕ) is bounded and tends to a constant as t → +∞.
Now, we distinguish two cases to prove that D + z(t) ≤ 0 for all t ≥ t 0 .
From the continuity of x(·), we can choose a positive constant δ < σ + such that We obtain Case 2. When t ∈ Q. We first claim that can not occur. Otherwise,

S. Xiao
which contradicts that So (3.1) is true, and thus we only consider the case that .
Secondly, assume that Clearly, (3.2) gives us This, together with the fact that t − σ + < s 1 − σ + ≤ t + 1 2 σ − − σ + < t < s 1 , implies that We claim that Otherwise, Then t <T < s 1 and It follows that which contradicts the facts that

Hence, (3.3) holds, and
Analogously, we can get From the above results, we know that x(t; t 0 , ϕ) is bounded on [−σ + , +∞), and Finally, we prove that x(t; t 0 , ϕ) tends to a constant as t → +∞. Let It suffices to show that α − = α. By way of contradiction, suppose that α − < α. Then α − and α are not zero simultaneously. Without loss of generality, we assume that α = 0 since the proof for the case of α − = 0 is quite similar. For Let w(t) = x(t)−px(t−σ(t)) for all t ≥ t 0 . It is clear that, for t ∈ [r m , r m +2σ + ], and This implies Now, we prove In fact, if x(t) < x(t − σ(t)) and t ∈ [r m , r m + 2σ + ], then

Conclusions
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