On stability of equations with an infinite distributed delay

For scalar equations of population dynamics with an infinite distributed delay x′(t)=r(t)[∫−∞tf(x(s))dsR(t,s)−x(t)], x(t)=φ(t),  t⩽t0, where f is the delayed production function, we consider asymptotic stability of the zero and a positive equilibrium K. It is assumed that the initial distribution is an arbitrary continuous function. Introducing conditions on the memory decay, we characterize functions f such that any solution with nonnegative nontrivial initial conditions tends to a positive equilibrium. The differences between finite and infinite delays are outlined, in particular, we present an example when the weak Allee effect (meaning that f′(0)=1 together with f(x)>x , x∈(0,K) ) which has no effect in the finite delay case (all solutions are persistent) can lead to extinction in the case of an infinite delay.


Abstract
For scalar equations of population dynamics with an infinite distributed delay where f is the delayed production function, we consider asymptotic stability of the zero and a positive equilibrium K.It is assumed that the initial distribution is an arbitrary continuous function.Introducing conditions on the memory decay, we characterize functions f such that any solution with nonnegative nontrivial initial conditions tends to a positive equilibrium.The differences between finite and infinite delays are outlined, in particular, we present an example when the weak Allee effect (meaning that f ′ (0) = 1 together with f(x) > x, x ∈ (0, K)) which has no effect in the finite delay case (all solutions are persistent) can lead to extinction in the case of an infinite delay.
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Introduction and preliminaries
The purpose of the current paper is to combine the idea of the connection between difference equations and delay differential models with infinite delays.We consider a scalar equation incorporating an infinite distributed delay with the initial condition Denote by χ u (t) the characteristic function of (u, ∞) and recall that for R(t, s) = ∑ m k=1 a k (t)χ h k (t) (s), where ∑ m k=1 a k (t) = 1 for any t, equation (1.1) takes the form ( We have an unbounded delay if max Also, if R(t, •) is continuously differentiable with ∂R (t, s) ∂s = K (t, s) then (1.1) is an integro-differential equation Without an assumption on some memory decay in (1.1) (and in particular for (1.4)) it is impossible to predict asymptotic behaviour of solutions from the form of f.Example 1.Consider the problem for t 0 = 1 with, generally, unbounded delay The positive equilibrium is K = 1 + ln 2. If h satisfies lim t→∞ h(t) = ∞, a solution of (1.6) (and any solution of (1.6) and (1.2) with a nonnegative initial function and a positive initial value) tends to K [2,9].However, if the delayed argument is constant for t ⩾ 1, in particular, h(t) ≡ a ∈ [0, 1], t ⩾ 1, the solution of (1.6) behaves differently.If a = 0, the solution x(t) = e −(t−1) of x ′ (t) = −x(t), x(1) = 1 tends to zero, for any a ∈ (0, 1] we get x ′ (t) = 2ae 1−a − x (t) , x (1) = 1.
Its solution x(t) = (1 − 2ae 1−a )e −(t−1) + 2ae 1−a tends to 2ae 1−a , it can take any value from (0, 2] which can be less or greater than K. Introduction of infinite delays is closely connected to distributed delays with decaying (usually exponentially) memory.The literature on this topic is extensive, we refer to the monograph [18] and some recent publications [12, 13, 19-22, 33, 35, 36, 38, 40-44].Most of the papers refer to autonomous integro-differential equations, there are only few publications where a general approach to equations with infinite delays is developed.To the best of our knowledge, [13] is one of the first papers in this direction.
The idea to connect global behaviour of delay differential models and dynamics of discrete maps was initiated in [31,32].The method to deduce global asymptotic stability of an equation with a delayed production term and a non-delay negative term from stability of a difference equation x n+1 = f(x n ) was further developed in [15,16] and later in [24][25][26].In population dynamics models, production can include maturation, gestation and many other common delays, and a negative non-delay term which describes instantaneous mortality.The above ideas were combined with efficient global stability conditions due to [37] using the notion of the Schwarzian derivative of a three times continuously differentiable function f The Schwarzian derivative vanishes for linear fractional functions and plays a crucial role in identifying global stability of maps [37].This notion can be significantly used in the theory of delay equations, in particular, by applying the following attractivity condition [25].
Theorem A ([25, Proposition 3.3]).Assume that f : (0, ∞) → (0, ∞) is three times continuously differentiable function that has a unique fixed point K such that f(x) > x for x ∈ (0, K), f(x) < x for x > K, and f is bounded on (0, K].If in addition f has a unique critical point (maximum) and a negative Schwarzian derivative (Sf)(x) < 0 everywhere but at the critical point and |f ′ (K)| ⩽ 1 then all the solutions of the equation x n+1 = f(x n ) with x 0 > 0 tend to K as n → ∞.
The fact that a unique equilibrium is globally stable if and only if f has no 2-cycles goes back to 1950s [10], see also [11].We refer the reader to the equivalence result in [26].
The connection between stability of (1.1), where a finite lower bound h(t) in the integral is used instead of −∞, to the fact that K is a global attractor of the map f, has a long history, see e.g.[10].Generally, if K is not a global attractor on [0, ∞), meaning that there exists a (stable or unstable) 2-cycle, for a delay equation K can still be globally asymptotically stable if delays are small; however, there exist delays such that there are solutions with sustainable oscillations around K [9].When generalizing results for scalar equations to systems, a notion of a strong attractor plays a key role [26], which was earlier introduced in more general settings than onedimensional functions [31].Note that for systems the fact that all solutions of a relevant discrete system converge to a unique equilibrium is not sufficient for global asymptotic stability of a corresponding delay differential equations, see an example in [27].
Also, in future we will apply the notion of the strong attractor adapted from [26].The point K > 0 is a strong attractor of a continuous function f on (0, ∞) if for every compact subset D ⊂ (0, ∞) such that its interior contains K, there exists a family of compact enclosed intervals J n ⊂ D such that f(J n−1 ) belongs to J n which is in the interior of J n−1 , and K is in the interior of all J n and is the only intersection point of these sets.Without loss of generality, we notice that it is equivalent to the fact that for any J 0 = [a 0 , b 0 ] ⊂ (0, ∞) such that a 0 < K < b 0 , there exists a sequence of enclosed intervals J n ⊂ (0, ∞) such that The notion of a strong attractor appears in much more general settings than one-dimensional functions [31], see also [26].
However, for a continuous scalar function f : [0, ∞) → [0, ∞) such that f(x) > x for 0 < x < K and 0 < f(x) < x for x > K the following statements are equivalent [26]: Thus, in the framework of a scalar model considered in (1.1), everywhere further we just need K to be globally attractive for the difference equation Theorem A combined with [26, Proposition A.1] leads to the following result.
Theorem B. Let f : (0, ∞) → (0, ∞) be a three times continuously differentiable function that has a unique fixed point K such that f(x) > x for x ∈ (0, K) and f(x) < x for x > K, and f be bounded on (0, K].If in addition f has a unique critical point (maximum) and a negative Schwarzian derivative (Sf)(x) < 0 everywhere but at the critical point and |f Typical examples of unimodal functions satisfying for certain values of the parameters all the assumptions of theorem B are the Ricker map for βe p > 1 and the generalized Beverton-Holt maps Analysis of the maps f 1 , f 2 is classical, see, for example, [23] and references therein, as well as [5,9,14].
If β ⩽ e −p then zero is the only nonnegative equilibrium and is a global attractor on [0, ∞) of x n+1 = f 1 (x n ), where f 1 is defined in (1.9).Then K = (β − 1) 1/γ is a strong attractor of f 2 .
These and some other examples are considered later in the context of non-autonomous models with an infinite delay.
The main results of the present paper can be summarized as follows.
(i) For equations with an infinite distributed delay of quite a general nature in the production term and instantaneous linear mortality, we state conditions on the memory decay and the production function guaranteeing global asymptotic stability for any nontrivial nonnegative initial function.Note that limitations on the memory fading are minimal and sharp.They can be compared to the assumption h(t) → ∞ for t → ∞ in the case of a single concentrated delay, without it asymptotic behaviour is unpredictable, see example 1.However, the restrictions on the production function f in (1.1) are strong, in particular, all solutions of x n+1 = f(x n ) with a positive x 0 should converge to the unique positive equilibrium.(ii) In addition to the fact that the equilibrium of the map is globally stable, either the derivative of f at zero should exceed one, or the initial function should be separated from zero.Without this condition, even persistence of solutions is not guaranteed, as example 2 illustrates.In biological models, such type of a production function can be connected to a weak Allee effect.It was well known previously that the strong Allee effect [1] can lead to extinction in delayed models.For finite delays, weak Allee effect in models of type (1.1) generally has no negative influence on survival.However, the situation changes when infinite delays are involved, even with a fading memory.(iii) Various applied examples, such as Nicholson's blowflies and Mackey-Glass equations, illustrate that the developed scheme works for several population dynamics models, under some assumptions on the parameters.
We start with boundedness and persistence (eventual separation from zero) properties in section 2. The main part of the paper is section 3 where global asymptotic stability results are obtained.General stability results are applied to population dynamics models in section 4, while final section 5 involves some discussion.

Existence and permanence of solutions
We consider problem (1.1) and (1.2) when some of the following assumptions hold.
) is the right-side limit of a function u at point t, and for any t 1 ⩾ t 0 and ε ∈ (0, 1) there exists The first four conditions (a1)-(a4) apply to any of (1.4) and (1.5).For (1.4), assumption (a5) is equivalent to: k=1 a k (t) ≡ 1 for any t ⩾ t 0 and for any t 1 ⩾ 0 and ε ∈ (0, 1) there exists As mentioned in (a5), R(t, t + ) = 1 is a normalization condition describing the probability density function R(t, s) that the reference delayed point at t is (t − s) units in the past.For (1.4), it corresponds to the condition that the sum of the coefficients is equal to 1 at each point.The idea of the equilibrium is that, without delay, population decreases for values above the equilibrium and increases if the values are less than the carrying capacity.This equilibrium is assumed to be the only positive fixed point of the production function f, which in (a1) is set to be time-independent.
Let us fix t 1 > t 0 .The delays h k either tend to infinity for t → ∞ or not.In the former case, lim and this term is not included in the sum in condition (a5A) where h k (t) ⩽ t 1 participate in the sum.Overall, all the terms with lim t→∞ h k (t) = ∞ and in particular, terms with bounded delay t − h k (t) ⩽ δ, are excluded for t 2 large enough.In the latter case when h k (t) < t 1 for some t ⩾ t 2 , the non-negative coefficient at such t should not exceed ε.
For example, condition (a5A) holds if the arguments of x are of two types: tending to +∞ as t → ∞ for terms with positive coefficients not exceeding one and those with h k (t) ⩽ t * for some t * with positive coefficients that tend to zero.In particular, For (1.5), assumption (a5B) becomes: (a5B) K(t, s) ⩾ 0 are measurable integrable in s with ∫ t −∞ K(t, s) ds ≡ 1, where for any t 1 ⩾ 0 and ε ∈ (0, 1), there exists Quite common kernels in the case of infinite delays , but exponential decay of the kernel is not required.The kernel K(t, s) = 1 (1+t−s) 2 also satisfies (a5B) though does not decay exponentially with memory.
We can also consider a combination of (1.4) and (1.5) Here condition (a5) should be replaced by (a5C) a k (t) : [t 0 , ∞) → [0, ∞) are Lebesgue measurable, K(t, s) ⩾ 0 are measurable integrable in s, where ∑ m k=1 a k (t) + ∫ t −∞ K(t, s) ds ≡ 1 for any t ⩾ t 0 , and also for any t 1 ⩾ 0 and ε ∈ (0, 1) there exists We start with stating under (a1)-(a5) boundedness of all solutions.Proof.First, let us notice that, as long as all x(t) ⩾ 0, t ⩽ t 1 , the solution satisfies since by (a4), the initial function and thus f(x(s)), s ⩽ t 0 , is nonnegative.Extending the estimate x(t) ⩾ φ(t 0 ) exp{− ∫ t t0 r(s) ds} to all the line, we get positivity (and the lower estimate) of x(t) for t ⩾ t 0 .
To get the upper bound, introduce where by (a4) the value of M is positive and finite.Let us take any ε > 0 and prove that x(t) < M + ε for any t ⩾ t 0 .By definition, x(t 0 ) ⩽ M. Assume the contrary that there exists t such that x(t) ⩾ M + ε.Denote and remark that which is a contradiction.Therefore x(t) < M + ε for any t ⩾ t 0 .Since ε is arbitrary, we get x(t) ⩽ M for any t ⩾ t 0 , which concludes the proof.
Consider now equation (1.1) with the unique fixed point x * = 0. Note that in the assumptions of lemma 2.1, positivity of K is not used.This allows to consider the case K = 0.
Remark 2.3.Note that in corollary 2.2 memory decay in (a5) is not really used.If we omit this assumption, solutions not necessarily tend to zero, as the solution illustrates.Had an additional condition lim t→∞ h(t) = ∞ been introduced, the solution would tend to zero as t → ∞ [2].a5C) is satisfied then all solutions of either (1.4) or (1.5) or (2.1), respectively, with initial conditions (1.2) are bounded.
We recall that a solution x is called persistent if there is ε > 0 such that x(t) ⩾ ε, t ⩾ t 0 and permanent if in addition it is bounded on [t 0 , ∞).Lemma 2.5.Suppose that (a1)-(a5) hold and either (a6) there exists α > 1 and δ ∈ (0, K) such that f(x) Then the solution of (1.1) and (1.2) is persistent.
Proof.First, let (a6) hold.If x(t) ⩾ δ for any t ⩾ t 0 , the solution of (1.1) and (1.2) is persistent.Let x(t 1 ) < δ for some t 1 ⩾ t 0 .Since α > 1, we can choose By (a5), there exists where ε ∈ (0, 1) is defined in (2.2).Note that, as x ′ (t) ⩾ −r(t)x(t), we get the estimate (2.4) Also, by lemma 2.1, there is M > 0 such that x(t) ⩽ M for any t ∈ (−∞, ∞).Denote and notice that by x(t 1 ) < δ we have m 0 < δ.Also, by (2.4), . Therefore, we conclude that also 2) and βm 0 < m 0 , for any t ⩾ t 2 , we get persistence.Otherwise, let On the one hand, by definition, Here in the first inequality we used the fact that x(t) < δ for t ⩾ [t we can denote the solution is non-decreasing, which contradicts the assumption This concludes the proof.
Next, we discuss the sharpness of the assumptions of lemma 2.5.Let us illustrate the fact that if both (a6) and (a7) fail, persistence is not guaranteed.
Note that even with (a6) not satisfied for (2.6), once the initial function is separated from zero, i.e. condition (a7) holds, persistence is observed.Consider instead of (2.5) another problem with the same f defined in (2.6).Then the initial function is not less than 1 2 , f(x(−1)) = 5 8 , the derivative at zero is −3/8 and further is also negative, as the solution of (2.7) satisfies The growth function in (2.6) experiences a so called weak Allee effect [1].Unlike the strong Allee effect [8], there is no domain (0, ε) where f(x) < x, leading to extinction in the scenario of bounded domains with insufficient (for survival) population density [3] but the growth rate, if differentiable, does not satisfy f ′ (0 + ) > 1.In the case of the weak Allee effect and the unique positive equilibrium, whenever the delayed argument(s) tend to infinity as time tends to infinity, all solutions are permanent [2] (i.e. both persistent and bounded).Infinite delays outline the danger of the weak Allee effect as in example 2.
Another growth function refers to the Mackey-Glass equation [28][29][30] x (2.9) Some other equations, such as both with a prescribed equilibrium K, also satisfy (a1).Thus, once (a2)-(a4) and (a5b) are satisfied, a solution of any of the above equations is bounded and persistent, by lemma 2.5.

Global asymptotic stability
We will use a regular definition of global asymptotic stability meaning that K is stable and all solutions with the initial conditions satisfying (a4) tend to K as t → ∞.
If (a1) holds and f is a three times differentiable function with a unique critical point which is a local maximum and a negative Schwarzian derivative at all positive x but a critical point, once K is a local attractor, it is also a global attractor.Sufficient conditions are outlined in theorem B and propositions 1.1 and 1.2.Proof.By (a4), the initial function is bounded: φ(t) ⩽ M 0 < ∞.The assumption f(x) < x for any x > 0 implies that The sequence We prove by induction validating that there exists a sequence of t n such that x(t) ⩽ M n for t ⩾ t n .Let us justify the first induction step that there is a time moment t 1 > t 0 such that x(t) ⩽ M 1 for any t ⩾ t 1 .
First, let us notice that for any M ∈ (M 1 , M 0 ), the solution is decreasing as long as moreover, by divergence of the integral of r(t) there is t such that x( t) ⩽ M and x(t) ⩽ M for any t ⩾ t.This also implies x(t) ⩽ M 0 for any t ∈ (−∞, ∞) and thus f(x(t)) ⩽ M 1 for any t including t < t 0 , due to the bound of the initial function.
Since f(x) < x, the value and for any x * ∈ (M 1 , x * ) we have In particular, taking above M = x * ∈ (M 1 , M 0 ), we get that x(t) ⩽ x * for t ⩾ t leading to x(t) ⩽ M < M 1 and thus sup Then for t ⩾ t and as long as x(t) ⩾ M 1 , ) .
Here the second inequality uses (a5), while in the third one we use the fact that f(x(t)) ⩽ M 1 for any t and max s∈[t,t] f (x(s)) ⩽ M due to the choice of t ⩾ t and sup Since by (3.2), the difference M − M 1 is a negative constant and also the integral of r diverges, we conclude that there is t 1 such that x(t) ⩽ M 1 for some t ⩾ t 1 , moreover, the solution stays below M 1 .
The next induction steps are similar.Here x(t) ⩽ M 0 as an upper bound is substituted with as M n < M < M 1 .By (a5), we can find t ⩾ t n−1 such that R(t, t) ⩽ ε for t ⩾ t.For t ⩾ t, as long as x(t) ⩾ M, where M n − M is a negative constant.By the divergence of the integral of r, x(t) < M for t large enough.The proof that in fact x(t) < M n for such t follows the same scheme as above.Thus lim t→∞ x(t) = 0, which concludes the proof.
Proof.As mentioned in the introduction, in the scalar case, any global attractor is a strong attractor, i.e. there are sequences a n , b n such that (1.8) holds.By lemmas 2.1 and 2.5, the solution has lower and upper positive bounds.Denote by a 0 the lower bound and by b 0 the upper bound and notice that Let us prove that there exists for any t ⩾ t 0 .Further, let us apply (a5) and find t1 such that R(t, t 0 ) ⩽ ε 1 for t ⩾ t1 , where Then, for any x(t) ⩽ A, Here R(t, t 0 ) ⩽ ε 1 implies which justifies the second inequality, and the choice of ε 1 gives the third one.Thus there is t * > t1 such that x(t * ) ⩾ A. Moreover, as f(x(t)) ⩾ a 1 for any t ⩾ t 0 , the derivative is positive for any x(t) ∈ [a 0 , A], and x(t) ⩾ A for any t ⩾ t * .Then, by (1.8), there is ā > a 1 such that f(x) ⩾ ā for x ∈ [A, ā].Similarly to above, we prove that x(t * ) = a 1 for some t * > t * and that x(t) ⩾ a 1 for all larger t ⩾ t * .
For the upper bound we choose ε 2 ∈ ( 0, b0−b1 4b0 ) ∈ (0, 1) as 0 < b 1 < b 0 , and find t2 such that R(t, t 0 ) ⩽ ε 2 for t ⩾ t2 .Then, for any all solutions tend to K, so K is the global attractor.The proof also implies stability, so K is globally asymptotically stable.

Examples and applied models
We start with stability conditions for Nicholson's blowflies model (2.8).
By proposition 1.To illustrate (a5), let us present some examples.We can consider an infinite number of concentrated delays where s n is an increasing positive sequence, the delay is infinite if lim s→∞ s n = +∞.Not only a standard exponential-type kernel in the integro-differential equation will satisfy (a5) but also a kernel with a slower decay Next, consider the Mackey-Glass equation describing white blood cells production (2.9).Corollary 1.2, theorems 3.1 and 3.2 lead to persistence and stability results in this case.Proposition 4.4.Suppose that r, φ and R satisfy (a2),(a4) and (a5), β > 0, γ > 0.
If β ⩽ 1 then any solution of (2.9) and (1.2) converges to zero.If either 0 < γ ⩽ 2 or γ > 2 and 1 < β < γ γ − 2 then all solutions of (2.9), (1.2) Consider in addition two other models [39] with a modification of the Ricker map for and the combination of models by Hassel and Maynard Smith for We use the following result from [23, example 3.1] for (4.4).
If a ⩽ 1 then all solutions of (4.7) and (1.2) tend to zero.
If a > 1 and ζγ ⩽ 2a 1/ζ a 1/ζ − 1 then all solutions of (4.7) and (1.2) Sharp stability conditions [23, example 3.2] are simpler for the normalized form of f 4 defined in (4.5) when the equation is scaled to . We get the following statement from theorem 3.2.
then any solution of problem (4.8) tends to one as t → ∞.

Conclusions, discussion, open problems
In contrast to most publications on infinite distributed delays, equation (1.1) is nonautonomous with minimal requirements on the delay distribution and the set of initial functions.Continuity and boundedness of initial functions are required to provide that the integral is well-defined for any R(t, •) and can be relaxed for integro-differential models.To the best of our knowledge, equations with multiple concentrated delays such as (1.4) for possible m = ∞ also have not been considered at this level of generality.Note that exponential memory decay is not required for our results, see, for example, (4.3).The results of the present paper can be summarized as follows.
(i) For quite general assumptions on continuous f with a unique positive fixed point, positivity and boundedness of all solutions is verified.If, in addition, either f ′ (0) > 1 or the initial function is separated from zero, we get persistence of solutions.(ii) If under the above conditions, the unique positive equilibrium attracts all the orbits of x n+1 = f(x n ) with x 0 > 0, this equilibrium is also a global attractor for all the solutions of (1.1) with nonnegative nontrivial initial conditions.If all solutions of the difference equation are attracted to zero, so are the solutions of (1.1).(iii) For a one-dimensional map, conditions of global stability for a unique positive equilibrium are well known including non-existence of a two-cycle and explicit tests for a unimodal f with a negative Schwarzian derivative.These conditions are specified for many applied equations, which allowed to analyze population dynamics models with infinite delay.
Everywhere in the current manuscript, we assumed the only positive equilibrium K to be time-independent.If K(t) is variable, two approaches are possible: • consider a periodic K(t), say, with seasonal changes, as a variable attractor; • in the most general case, when, for example, overall population size can increase with time, we can fix K 0 > 0 and scale N to the variable K, considering N(t) K(t) K 0 , keeping a scaled equilibrium K 0 constant and referring the variable part to r(t).
Let us dwell a little bit on the second approach for a varriable carrying capacity.Then it is natural to assume that the production function dependent on the value of x in the past, also refers to the carrying capacity at the same moment of time.We can also describe the dynamics for the ratio of the population size to the carrying capacity.With the Beverton-Holt growth function, we get a model ] , r 1 (t) = r (t) K (t) , α ∈ (0, 1) , h (t) ⩽ t, t > 0, its convergence to the constant equilibrium K 0 ≡ 1 can be deduced using the results of the present paper (cerrtainly, with more general delays, such as infinite delays).Nevertheless, the current research is just the beginning of exploration for (1.1).Let us mention some extensions immediately arising from the results of the present paper.
(i) As mentioned above, persistence is guaranteed for f ′ (0) > 1 or when the initial function is separated from zero.For systems with f ′ (0) = 1, what are sufficient conditions on memory decay dependent on the higher derivatives of f at zero?The answer can shed light on the influence of the weak Allee effect on survival and extinction.(ii) Development of delay-dependent conditions for equations with infinite distributed delays is an important problem.For finite distributed delays, stability for non-autonomous equations in some cases was explored, see, for example, [9] and references therein.The relation of autonomous equations with distributed and concentrated delays was a focus of [6,7].In particular, the following hypothesis was eventually verified: if an equation with a constant concentrated delay is asymptotically stable then an equation with a distributed delay with the same 'center of mass' as the above concentrated delay is also asymptotically stable.Extension of this result to non-autonomous equations, including equations with an infinite delay, is an interesting area of investigation.In particular, consider nonautonomous (2.8) for β > e and deduce delay-dependent global attractivity conditions.(iii) In view of the previous item, is it possible to construct delay-dependent asymptotic stability conditions for (1.1) with prescribed exponential memory fading R(t, s) > e −λ(t−s) ?Also, instead of global attractivity of a positive equilibrium, stability of a unique positive periodic solution can be explored.(iv) As a generalization of equation (1.1), we can consider a model with several production functions and two partial cases of (5.1) : x Is it possible to generalize the notion of a strong attractor to the case of a family of functions {f j } to establish global asymptotic stability of (5.1)-( 5. 3)?Is it equivalent to global attractivity, as the case is still one-dimensional?(v) A natural extension of the main results of the present paper would be proceeding to the vector analogue of equation (1.1) in the spirit of [4].
will lead to the existence of x * > 0 (all M n are positive) such that f(x * ) ⩾ M n−1 ⩾ x * > 0 which contradicts the assumption f(x) < x for any x > 0. As a monotone decreasing positive sequence, {M n } should have a limit d ⩾ 0. Let d > 0.Then, getting a limit in (3.1), by continuity of f, we conclude that d = max x∈[0,d] f(x) > 0. Thus by continuity of f this maximum is attained at x * ∈ (0, d] and f(x * ) = d ⩾ x * , which contradicts f(x) < x for any x > 0. Therefore the solution decreases.Therefore, there is t > t2 such that x( t) ⩽ B, moreover, x(t) ⩽ B, t ⩾ t.Again, by(1.8),there is b < b 1 such that f(x) ⩽ b for x ∈ [ b, B].So we conclude that x(t) ⩽ b 1 for t > t, for some t > t.Choosing t 1 = max{t * , t}, we conclude thatx (t) ∈ [a 1 , b 1 ] , t ⩾ t 1 .The induction step from [a n , b n ] to [a n+1 , b n+1 ] is similar.Since x (t)∈ [a n , b n ] , t ⩾ t n and lim 1, K = 1 + 1 p ln β is a strong attractor of the function f 1 (x) = βxe p(K−x) if e −p < β < e.