Wright type delay differential equations with negative Schwarzian

We prove that the well-known 3/2 stability condition established for the Wright equation (WE) still holds if the nonlinearity $p(\exp(-x)-1)$ in WE is replaced by a decreasing or unimodal smooth function f with $f'(0)<0$ satisfying the standard negative feedback and below boundedness conditions and having everywhere negative Schwarz derivative.

(H2) f is bounded below and has at most one critical point x * ∈ R which is a local extremum.
(H3) (Sf )(x) < 0 for all The negative feedback condition (H1) and boundedness condition (H2) are very typical in the theory of (1.1); the first one causes solutions to tend to oscillate about zero, while both of them guarantee the existence of the global compact attractor to Eq. (1.1) (e.g. see [9]). On the other hand, the Schwarzian negativity condition (H3) is rather common in the theory of one-dimensional dynamical systems (see [12,16]) while it is not very usual for the studies of delay-differential equations. Here, we introduce (H3) in hope of obtaining an analogue of the Singer global stability result for an one-dimensional map f : I → I; this result (see its complete formulation below) states that the local stability of a unique fixed point e ∈ I plus an appropriate (monotone or unimodal) form of the map f imply the global attractivity of the equilibrium e ∈ I [12]. The negative Schwarzian condition is not artificial at all, it appears naturally also in some other contexts of the theory of delay differential equations, see e.g. [10,, where it is explicitely used, and [3,Theorem 7.2,p. 388], where the condition Sf < 0 is implicitely required; moreover, many nonlinear delay differential equations used in mathematical modelling in biology (e.g. Mackey-Glass, Lasota-Wazewska, Nicholson, Goodwin equations) have their right-hand sides satisfying the hypothesis (H3). Take also, for example, the celebrated Wright equation which was used to describe the distribution of prime numbers or to model population dynamics of a single species: x ′ (t) = −px(t − 1)[1 + x(t)], p > 0. (1.2) For x(t) > −1, Eq. (1.2) is reduced to (1.1) after applying the transformation y = − ln (1 + x). In this case f (x) = (exp(−x) − 1) and, by abuse of notation, we will again refer to the transformed system x ′ (t) = p(exp(−x(t − 1)) − 1), p > 0, (1.3) as the Wright equation. In this case, f is strictly decreasing and has no inflexion points; both these facts simplify considerably the investigation of (1.3). Below, we will present two other important examples, with nonlinearities which may have an inflexion point (some "food-limitation" models) or even a local extremum (population model exhibiting the Allee effect).
Eq. (1.1) has been considered before by several authors but only assuming conditions (H1) and (H2), see [9,11,22,23] and references therein. In particular, the Morse decomposition of its compact attractor has been described in detail in [9]. Moreover, it was proved also that the Poincaré-Bendixson theorem holds for (1.1) with the decreasing nonlinearity f so that the asymptotic periodicity is the "most complicated" type of behavior in (1.3) [9,11,23]. It should be noted also that the above information has essentially a "qualitative" character. So that adding (H3), we can hope to obtain some additional information of analytic nature about possible bifurcations in parametrized families of (1.1). The following variational equation along its unique steady state x = 0 plays a very important role in the study of such bifurcations: As is well-known, this equation is unstable when −f ′ (0) > π/2, and this instability implies the existence of slowly oscillating periodic solutions to (1.1) (see e.g. [22]). Surprisingly, the dynamically simpler case −f ′ (0) < π/2 = 1.571... has not been studied thoroughly before, and, in particular, it seems that the following Wright conjecture has not been solved: the inequality p < π/2 is sufficient for the global stability in (1.3). On the other hand, the sufficiency of the stronger condition p < 3/2 for the global stability of Eq. (1.3) was proved in 'a very difficult theorem of Wright [26]' (see [2, page 64]), where also the sharper conditions p < 37/24 = 1.5416... and p < 1.567... were announced. It should be noted that proofs of the 3/2 stability condition for Eq. (1.3) have strongly used the specific exponential form of the nonlinearity f (x) = p(exp(−x) − 1) and, in particular, the monotonicity properties of such f . This fact explains why any analogue of this Wright result has not been proved for other, essentially different (nonexponential), right-hand sides in Eq. (1.1) (even for monotone f , the general situation being considerably more complicated). An important step in solving the Wright conjecture was made in Theorem 3 from [22] which provides us with some examples of Eq. (1.1) which satisfy (H1), (H2) and have slowly oscillating periodic solutions, even when the corresponding linearized equation (1.4) is exponentially stable. This means that the local exponential stability of the steady state in (1.1) (or, what is the same, exponential stability in (1.4)) with f having only these standard and usual properties (H1), (H2) in general does not imply the global asymptotic stability in (1.1). Moreover, as a simple consequence of an elegant approach towards stable periodic orbits for scalar equations of the form x ′ (t) = −µx(t) + f (x(t − 1)), µ ≥ 0 with Lipschitz nonlinearities proposed recently in [24] (see also [25]), we get the following This result is of special importance for us, since it shows clearly that the strong correlation between local (at zero) and global properties of Eq. (1.3) can not be explained only with the concepts presented in (H1), (H2).
On the other hand, Walther's result from [22] cannot be applied to Eq.  We remark that this conjecture is very close to the Hal Smith conjecture [17] to the effect that the local and global asymptotic stabilities for Nicholson's blowflies equation are equivalent. Observe that this equation also has a unique positive steady state and nonlinearity satisfying the negative feedback and the negative Schwarzian conditions (see [1,8,18] for further discussions).
Furthermore, due to recent results of Krisztin [5], now we can indicate some class of symmetric and monotone nonlinearities (e.g. f (x) = −p tanh x, f (x) = −p arctan x, p > 0) for which the above conjecture is true. Although for both the mentioned functions condition Sf < 0 holds, in general the additional convexity condition imposed on f in [5] (see also [6]) is different from (H3): evidently, the requirement of the negative Schwarzian is not the unique way to approach the problem (the same situation that we have in the theory of one-dimensional maps). In fact, to prove our main result (Theorem 1.3), we only need some geometric consequences of the inequality Sf < 0 for the graph of f . For instance, if f ′′ (0) = 0 then this geometric consequence is given by (f (x) − f ′ (0)x)x > 0 for x = 0 (that necessarily holds also under the above mentioned convexity, symmetry and monotonicity assumptions from [5,6]). This geometric approach was developed further in [8], where a generalization of the Yorke condition [7, Section 4.5] was proposed instead of (H3).
In this paper we carry out the first step towards Conjecture 1. To prove Theorem 1.3, we will essentially use an idea from [4], which allows us to construct some one-dimensional map inheriting some properties of Eq. (1.1). Roughly speaking, we consider maps F k = F k (z) : R → R, F k (0) = 0, which give the values of the k-th consecutive extremum of the oscillating solutions x(t, z), z = 0, satisfying x(s, z) ≡ z, s ∈ [−1, 0]. Then we investigate some relations existing between the global attractivity properties of F k and (1.1), trying to deduce in this way the global asymptotical stability of (1.1) from the corresponding property of the discrete dynamical system generated by F k . Since the computation becomes more and more complicated with the growth of k, we only consider the simplest case k = 1 here. Computer experiments show that, increasing k, we obtain better approximations to the condition −f ′ (0) ≤ π/2 given in Conjecture 1.2 (for example, for k = 2 we get −f ′ (0) ≤ 37/24 and so on). However, due to the technical complications, this way to approach the above conjecture could be used only for very special cases.
Curiously, the note [4] devoted to the study of the Yorke type functional differential equations with sublinear nonlinearity (see [27]) and, in particular, the Yorke 3/2 stability criterion, still can be extended to the class of nonlinear Wright's type delay differential equations. We consider the special nature of the number 3/2 (which was found almost simultaneously by A.D. Myshkis [14] and by E.M. Wright [26]) as an invariant of such a prolongation. Moreover, there exists an interesting interplay between both these types of functional differential equations if we consider a variable coefficient p(t) instead of the constant p in Eq. (1.3) (see [19] for details).
In any case, it should be noted that the Yorke and the Wright type equations have rather different structures (see [4,19,20] for more comments).
Completing our discussion, we consider briefly two other Wright type equations studied recently by several authors: [19, p. 456] or, what is basically the same, the Michaelis-Menten single species growth equation with one delay (see [7, p. 132]): and (Sf )(y) = −1/2 < 0 for all y. By Theorem 1.3, the inequality rh ≤ (3/2)(1 + cr) implies the global stability of the zero solution to Eq. (1.5) (compare with [7] and [19]). We also point out that Eq. (1.5) with c = 0 coincides with the Wright equation (1.2), so that Wright's 3/2 stability theorem is a very special case of our result. Example 1.5. Consider now a population model described by This equation has a unique positive equilibrium e * and the change of variables . Note that f has negative S-derivative as a composition of a quadratic polynomial and the real exponential function. If b ≤ 0 then f is strictly decreasing, and if b > 0, then f has exactly one critical point (minimum). In the latter case, the population model exhibits the so-called Allee effect [13]. Applying Theorem 1.3, we see that e * attracts all positive solutions of (1.6) once (2ce * − b)he * ≤ 1.5 (compare e.g. with [7, pp. 143-146], where also other references can be found).
The paper is organized as follows. In Section 2 we define several auxiliary scalar functions and study their properties as well as relations connecting them. Finally, in the last section, we use these functions to prove Theorem 1.3 (notice that, in contrast with [1], we may not use the above formulated Singer's result for this purpose).

Auxiliary functions
To prove our main result, we will proceed in analogy to [4], so that the construction of one-dimensional maps inheriting attractivity properties of the dynamical system generated by Eq. (1.1) is the main tool here. In this section, we introduce several such scalar maps and study their properties as well as the relations existing among them.
First we note that we can only have eight different possibilities for the maps satisfying hypotheses (H), depending on the situation of the eventual critical point and the inflexion points. A graphic representation of all these cases is given in Fig.  1 below, where x * denotes the critical point and c 1 , c 2 are the inflexion points. We recall that a real function has at most one inflexion point in any interval in which the Schwarz derivative is well defined and is negative (see [16]).
In the sequel, up to the proof of Theorem 1.3 and with the unique exception made for Corollary 2.2, we will always assume that f satisfies (H) and f ′′ (0) > 0. This situation corresponds to the pictures (a)-(e) in Fig. 1.
Next, for a < 0, b > 0, we introduce the set Also, for every a < 0 and every b > 0, consider the rational function r(x, a, b) = a 2 x/(a − bx) defined over (ab −1 , +∞). Let K + a,b (respectively, K − a,b ) be the set of restrictions of elements of K a,b to [0, +∞) (respectively, to (ab −1 , 0]). We will denote by r + and r − the restrictions of r(·, a, b) to the intervals [0, +∞) and (ab −1 , 0] respectively. The following properties are elementary:  Futhermore, it can be proved that r + and r − are respectively the minimal element of K + a,b and the maximal element of K − a,b with respect to the usual order. The following slightly different result will play a key role in the sequel: Lemma 2.1. For all y ∈ K a,b with (Sy)(x) < 0 for x ∈ {w : y ′ (w) = 0}, we have r(x, a, b) < y(x) for all x > 0 and also r(x, a, b) > y(x) for all x ∈ (ab −1 , 0).
The proof for (b) and (e) is slightly different if the inflexion point c 1 of y belongs to the interval (ab −1 , 0). In this case, we can use the same arguments only to show that r ′ (x) < y ′ (x) and r(x) > y(x) for x ∈ [c 1 , 0). Next, by convexity arguments, for all x ∈ (ab −1 , c 1 ).
Finally, case (c) can be studied analogously taking into account that r is strictly decreasing on (ab −1 , 0), whereas y reaches its maximum at x * .
In the following lemmas we establish some relations between m and M which are needed in the proof of Theorem 1.3.
Let fix now this solution z = z(t).