Asymptotically exponential solutions in nonlinear integral and differential equations

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1 Asymptotically exponential solutions in nonlinear integral and differential equations István Győri and Ferenc Hartung Department of Mathematics, University of Pannonia, 821 Veszprém, P.O. Box 158, Hungary Abstract In this paper we investigate the growth/decay rate of solutions of an abstract integral equation which frequently arises in quasilinear differential equations applying a variation-of-constants formula. These results are applicable to some abstract equations which appear in the theory of age dependent population models and also to some quasilinear delay differential equations with bounded and unbounded delays. Examples are given to illustrate the sharpness of the results. Key words: exponential growth/decay, abstract integral equation, quasilinear differential equations, delay equations, mathematical biology 1 Introduction Structured population models have been studied at least from the sixties [3], and it is still an intensively studied area [1,2,12,13,16,23,29]. One important properties of age-structured population models is the so-called asynchronous exponential growth/decay property, i.e., when the age distribution tends to a limit independently of the initial age distribution see, e.g., [1 13,15,23,3 32]. In these papers the partial differential equation population model is transformed into an equivalent abstract linear inhomogeneous differential equation, so the solution is given by the variation-of-constant formula: xt = Ttu + Tt sfxsds, t, 1.1 This research was partially supported by Hungarian National Foundation for Scientific Research Grant No. K address: gyori@almos.uni-pannon.hu, hartung.ferenc@uni-pannon.hu István Győri and Ferenc Hartung. Preprint submitted to Elsevier 29 June 21

2 where u X, X is a Banach-space with norm, Tt is a strongly continuous semigroup of bounded operators in X. Gillenberg and Webb [15] and Webb [3] studied the asynchronous exponential growth in abstract differential equations originated from age-dependent population models, where the investigated abstract differential equation can be written in the form of 1.1. In [3] it has been shown that if lim t e αt t k Ttu exists for all u X for some α >, k 1, and Fx θ x, θs s lim s =, s θs is a monotone nonincreasing function on the interval,, then s lim t e αt t k xt exists, as well. Therefore the growth rate of the solutions of the homogeneous equation determines that of the solutions of the inhomogeneous equation. Motivated by this result, in this paper we study the asymptotic behavior of solutions of a nonlinear Volterra-type abstract integral equation xt = yt;ϕ + Tt sfs,x ds, t. 1.2 Here f is a Volterra-operator, i.e., ft,x = ft, x, if xs = xs, t 1 s t, where x, x : [t 1, X, t 1, and we associate the initial condition xs = ϕs, t 1 s 1.3 to 1.2. The class of Volterra integral equations of the form 1.2 contains the ordinary integral equation 1.1 as a special case using fs,x = Fxs, t 1 =, and in this case the initial condition 1.3 reduces to x = φ. The class of Eq. 1.2 also contains, e.g., functional integral equations of the form xt = Ttφ + Tt sfxs τsds, t. 1.4 In this paper we study the asymptotic behavior of solutions of a nonlinear Volterra-type abstract integral equation of the form 1.2 assuming the knowledge of an asymptotic formula for yt;ϕ. The function yt;ϕ in many applications is a solution of the linear part of a perturbed linear equation, although in our case it can be a nonlinear function of the initial function ϕ. In our main result Theorem 2.2 below we give sufficient conditions which imply that the asymptotic behavior of the linear part yt; ϕ is preserved for the solution of the nonlinear equation 1.2. In the case when yt;ϕ satisfies an exponential estimate of the form yt;ϕ m ϕe αt, we define a neighborhood of the zero initial function see Theorems 2.7 and 2.8 below such that solutions 2

3 starting from this neighborhood satisfy a similar exponential estimate with the same exponent. If the exponential growth/decay rate of yt;ϕ is known, then we give sufficient conditions under which the same growth/decay rate is preserved for the solutions of 1.2. In a special case see Corollary 2.11 below we give necessary and sufficient conditions for preserving this growth/decay rate for the solutions of 1.2. Our results applied for the ordinary integral equation 1.1 includes the result of Webb [3] under similar, or sometimes weaker condition see Theorem 2.12 below and they are applicable for the decaying case, as well. As an application of the main result, in Section 3 first we show the asynchronous exponential growth property of solutions of a nonlinear PDE model describing an age-dependent population with delayed birth process. Then we discuss asymptotic behavior of solutions of differential equations with bounded and unbounded delays. In this example yt;ϕ is a solution of an associated autonomous linear delay equation, where the asymptotic behavior is determined by the leading characteristic root of the equation. Our result can be applied in the case when the leading root is a complex number with multiplicity greater than 1. Illustrative examples are given for the pantograph and the sunflower equations. The study of asymptotic properties of different classes of integral and differential equations is an active research area, see, e.g., [6 8,12,14,15,17,24 26,28] and the references therein. Most of the work in this direction has been done for linear equations, and guarantees only pure exponential growth/decay of the solutions. Our method is applicable for nonlinear equations of the form 1.2 and for the case when yt;ϕ = e αt t k d ϕ cos βt + e ϕ sin βt + o1, as t. 2 Main results Let < t 1 < < be fixed, and C := C[t 1,,X denote the set of continuous functions mapping [t 1, into the Banach space X. Let C := C[t 1, ],X be the Banach space of continuous functions mapping [t 1, ] into X with the norm ϕ = max t 1 s ϕs, ϕ C, where denotes the norm in X. Any fixed norm on R n and its induced matrix norm on R n n are denoted by, as well. Let BX be the space of bounded linear operators mapping X into X, and R + = [,. A family T : R + BX of bounded linear operators is called strongly continuous if the map R + t Ttx X is continuous for any fixed x X. For any constant u R the corresponding constant function will be denoted by u, as well. 3

4 In this section we consider the Volterra-type integral equation xt = yt;ϕ + with initial condition Tt sfs,x ds, t 2.1 xs = ϕs, t 1 s, ϕ C. 2.2 We state the following hypotheses: H1 For all ϕ C the function y ;ϕ : [t 1, X is continuous, ys;ϕ = ϕs for s [t 1, ], and yt;ϕ m ϕe αt t + 1 k, t, 2.3 where α is a given constant, k is a nonnegative integer, and m : C R + is such that m ϕ as ϕ. H2 T : R + BX is a strongly continuous family of bounded linear operators on X, and c 1 := supe αt t + 1 k Tt <. t H3 f : [, C X is a Volterra-type functional, i.e., for all x C, the map [, t ft,x X is continuous, and for all t,x, t, x [, C, ft,x = ft, x, if xs = xs, t 1 s t. H4 For all t,z [, C, ft,e α t t k z ω t, max zs, 2.4 ζt s t where ζ : [, R satisfies t 1 ζt t, t, 2.5 and [, R + t,u ωt,u R + is a continuous function such that for any fixed t [,, the map R + u ωt,u R + is monotone nondecreasing, and for a positive constant v c 1 e αs ωs,v ds < v. 2.6 We define the constant { } m 1 ϕ := max max e αs t s t k ϕs, m ϕ t 1 s 2.7 4

5 and the function H : R + R, Hv = v c 1 e αs ωs,vds. 2.8 Assumption 2.6 yields Hv > for some v >, and therefore the constant ρ := sup {Hv: v > } 2.9 is well-defined, and it is either positive or +. The set U defined by U := { } ϕ C : m 1 ϕ < ρ 2.1 is not empty, since m ϕ, and therefore m 1 ϕ as ϕ. Hence the set { } Mϕ := m > : Hm > m 1 ϕ 2.11 is also not empty, and the constant mϕ := inf Mϕ 2.12 is a well-defined real number for all ϕ U. Definition 2.1 A function x is a solution of the initial value problem IVP if x C and it satisfies Eq. 2.1 on [, and initial condition 2.2 on [t 1, ]. In this paper we do not deal with the existence and uniqueness of solutions. We assume that some additional conditions are satisfied for f such that the solutions of the IVP exist locally on some interval [,t 1 ]. It will be shown in Theorem 2.2 that H1 H4 imply that any solution will exist globally on [,, as well. It is worth to note that the uniqueness of the solutions is not needed in our results. Any fixed solution of the IVP is denoted by x ;ϕ. In the first part of our main result, Theorem 2.2, we give an exponential upper bound for the solutions of the IVP , and in the second part of this theorem we give a limit relation based on the following three additional hypotheses: H5 There exist d, e : U X and β R such that lim e αt t t t k yt;ϕ d ϕ cos βt e ϕ sin βt =, t + for ϕ U. 5

6 H6 There exist P,Q BX for which lim e αt t + 1 k Tt P cosβt Q sin βt =. t + H7 There is an initial function ϕ U such that { } max d ϕ, e ϕ > P + Q e αs t ωs,mϕ ds Now, we are in a position to state and prove our main result. Theorem 2.2 Assume that H1-H4 are satisfied. i If ϕ U, then any solution x ;ϕ of the IVP exists on [t 1,, and satisfies xt;ϕ mϕe αt t t k, t t 1, 2.14 where mϕ is defined in ii If in addition H5-H6 hold, then for all ϕ U there are vectors dϕ and eϕ in X such that xt;ϕ = e αt t t t dϕ k cosβt + eϕ sin βt + o1, 2.15 as t +. Moreover, if H7 holds, then dϕ + eϕ =, where ϕ is given in Relations 2.14 and 2.15 can be reformulated in several forms. For example, the next result follows immediately from Theorem 2.2. Corollary 2.3 Assume that H1-H4 are satisfied. i If ϕ U, then any solution x ;ϕ of the IVP satisfies xt;ϕ mϕ t k e αt t + 1 k, t, 2.16 where mϕ is defined in ii If in addition H5-H6 hold, then for all ϕ U there are vectors dϕ and ẽϕ in X such that xt;ϕ = e αt t t k dϕ cos βt t +ẽϕ sin βt +o1, 2.17 as t +. Moreover, if H7 holds, then dϕ + ẽϕ =, where ϕ is given in

7 To prove Theorem 2.2 we need the following lemma which is interesting in its own right. Lemma 2.4 Assume that Ut,s, s t <, is a family of linear bounded operators on X that is jointly strongly continuous in t and s, moreover M 1 := sup Ut,s <, 2.18 s t< and there are strongly continuous operators P 1,Q 1 : R + BX and a constant β R such that lim Ut,s P 1s cos βt s Q 1 s sin βt s = 2.19 t + for any fixed s [,, and for some M 2 > sup P 1 s M 2, s sup Q 1 s M s Then for any continuous function g: [, X, relation gs ds < implies lim t + Ut,sgsds P 1 s cos βt s + Q 1 s sin βt s gs ds = Proof. From 2.2, we find Thus P 1 s cos βt s+q 1 s sin βt s gs ds 2M 2 gs ds <. δt := Ut,sgsds P 1 s cos βt s + Q 1 s sin βt s gs ds Ut,s P 1 s cosβt s Q 1 s sin βt s gs ds t 1 Ut,s gs ds t 1 P 1 s cos βt s + Q 1 s sin βt s gs ds for all t t 1. From 2.18 and 2.2, it follows 7

8 1 δt Ut,s P 1 s cosβt s Q 1 s sin βt s gs ds +M 1 + 2M 2 t 1 gs ds, for all t t 1, and hence 2.19 and the Lebesgue s Dominated Convergence Theorem imply lim sup δt M 1 + 2M 2 t + t 1 gs ds, t 1. This yields lim t + δt =, as t 1 +, and the proof of the lemma is complete. Remark 2.5 Lemma 2.4 is an essential generalization of a result in [4] which has been proved when Ut,s = at s, s t, is a scalar function, β = and P 1 s = p 1 is a constant. Proof. of Theorem 2.2 i Let ϕ U be an arbitrarily fixed initial function and x ;ϕ denote a noncontinuable solution of the corresponding IVP on [t 1,t 1 for some t 1 >. Define and zt = e αt t t k xt;ϕ, t [t 1,t 1 Ut,s = e αt s t t k Tt s, t s t Then it follows from 2.1 for t [,t 1 zt =e αt t t k yt;ϕ + Ut,se αs fs,e α t k z ds We obtain from assumptions H1 and H2, respectively e αt t t k yt;ϕ for t [,t 1, and = e αt t + 1 k yt;ϕ Ut,s = e αt s t s + 1 k Tt s k t t + 1 m ϕ t t k t s + 1 c 1 t t

9 for all s t <. Thus 2.23 together with 2.3 and 2.4 implies zt m ϕ + c 1 e αs t ω s, max zτ ds, t [,t ζt τ s Relation 2.7 and the definition of z yield zt max e αs s t k ϕs m1 ϕ, t 1 t. t 1 s 2.25 Combining 2.24, 2.25 and 2.7 together with the monotonicity of the righthand-side of 2.24 and the monotonicity of ω in its second argument we get wt m 1 ϕ + c 1 e αs ωs,wsds, t [,t 1, 2.26 where wt := max zτ, t [t 1,t 1. t 1 τ t On the other hand, 2.6 and the definitions of U and mϕ yield see 2.1 and 2.12 that there exists a decreasing sequence v n of nonnegative numbers such that v n > m 1 ϕ + c 1 Then for all n v n > m 1 ϕ + c 1 e αs ωs,v n ds and lim n v n = mϕ. e αs ωs,v n ds, t [,t Since v ωt,v is a monotone nondecreasing function on [, for any fixed t, a standard comparison result see, e.g., [22] yields from 2.26 and 2.27 that e αt t t k xt;ϕ wt < v n, t [,t 1 for all n. It follows from 2.12 that mϕ m 1 ϕ, therefore xt;ϕ < v n e αt t t k, t [t 1,t 1. Then t 1 = +, since otherwise lim t t1 xt;ϕ <. Statement i follows combining the previous inequality and 2.25, and taking the limit n. ii From 2.4 and 2.14 and the monotonicity of ω it follows that the function gt := e αt ft,x ;ϕ = e αt ft,e α t 1 +1 k z, t, satisfies gt e αt ωt,mϕ, t,

10 and 2.27 yields gt dt <. On the other hand, Ut,s defined in 2.22 satisfies Ut,s P cos βt s + Q sin βt s k e αt s t s + 1 k Tt s t s t t e αt s t s + 1 k Tt s P cosβt s Q sin βt s. Hence H2 and H6 yield lim Ut,s P cos βt s + Q sin βt s = t + for all fixed s [,. Since gt dt <, the integral It := exists, and in virtue of Lemma 2.4, we have lim t + P cosβt s + Q sin βt s gs ds Ut,sgsds It =. This, combined with 2.23, H5 and H6, yields zt =e αt t t t k yt;ϕ + Ut,sgsds =d ϕ cos βt + e ϕ sin βt + It + o1, as t +. Using the definition of gt and trigonometric identities we find It = d 1 ϕ cos βt + e 1 ϕ sin βt, t, where d 1 ϕ = P cos βs Q sin βse αs t fs,x ;ϕds and e 1 ϕ = P sin βs + Q cos βse αs t fs,x ;ϕds. Thus xt;ϕ satisfies 2.15 with the constants dϕ = d ϕ + d 1 ϕ and eϕ = e ϕ + e 1 ϕ. Now, we show that dϕ + eϕ for the initial function ϕ satisfying Indeed, 1

11 dϕ d ϕ d 1 ϕ d ϕ P + Q e αs fs,x ;ϕ ds, and eϕ e ϕ e 1 ϕ e ϕ P + Q Thus, from 2.13 and 2.28, it follows e αs fs,x ;ϕ ds. dϕ + eϕ max{ d ϕ, e ϕ } >. P + Q The proof of the theorem is complete. e αs fs,x ;ϕ ds If ωt,u is linear in u, then Theorem 2.2 yields easily the next result. Theorem 2.6 Assume H1-H3 are satisfied, and for all t,z [, C ft,e α t t k z e αt t at max zs, 2.29 ζt s t asds < 1, 2.3 where ζ satisfies 2.5, and a : [, R + is a continuous function such that c 1 where c 1 is defined in H2. Then i For all ϕ C the solution x ;ϕ of the IVP satisfies xt;ϕ m 2 ϕe αt t t k, t t 1, 2.31 where m 1 ϕ m 2 ϕ = 1 c 1 asds ii If H5 and H6 also hold, then for all ϕ C, there are dϕ, eϕ X such that 2.15 is satisfied. Moreover, if { } max d ϕ, e ϕ > m 2 ϕ P + Q asds, for some ϕ C, then dϕ + eϕ >. 11

12 Proof. The result is an easy consequence of Theorem 2.2 and hence its proof is omitted. In the next results we use the following conditions: H8 Suppose α, and there exists a continuous and monotone nondecreasing function b α : R + R + such that bu > for u >, and the inequality ft,e α t z b α e αt max zs, t,z [, C ζt s t holds, where ζ satisfies 2.5. H9 There exists ϕ 1 C such that lim t e αt yt;ϕ 1, 2.33 and yt;γϕ 1 = γyt;ϕ 1, t, γ > In the next result we study the case when k = in H1 and H2. In this case m 1 ϕ defined in 2.7 simplifies to { } m 1 ϕ := max e α t ϕ, m ϕ, 2.35 where m ϕ is defined in H1. Note that m 1 ϕ =, if and only if ϕ = and m ϕ =, and hence yt;ϕ =, t t 1. Theorem 2.7 Assume that H1, H2 and H3 are satisfied with k =, moreover H8 holds with α > and 1 b α u u 2 du < Then i for every ϕ C the equation m 1 ϕ + c 1 α m b α u du = m, m 2.37 m u 2 has at most two roots, and any solution x ;ϕ of the IVP satisfies xt;ϕ m α ϕe αt, t t 1, 2.38 where m α ϕ is the largest root of

13 ii If H5 and H6 are also satisfied with k =, then for every ϕ C there are d α ϕ, e α ϕ X such that xt;ϕ = e αt d α ϕ cos βt + e α ϕ sin βt + o1, t iii If, in addition, H9 holds, then there exists γ 1 > such that d α γϕ 1 + e α γϕ 1, γ γ 1. Proof. i Let ωt,u be defined by ωt,u = b α e αt u for all t and u. Then for m > e αt ωt,mdt = e αt b α e αt mdt = m α m b α u u 2 du <, where we used the substitution u = e αt m. Let H be defined by 2.8. Then Hm = mh 1 m, where H 1 m = 1 c 1 b α u du, α m u 2 and therefore ρ defined in 2.9 satisfies ρ = lim m + Hm = +, and U defined in 2.1 equals to C. Let ϕ C be fixed, and consider Mϕ := { } m > : Hm > m 1 ϕ. In view of Theorem 2.2, we have to show that m α ϕ := inf Mϕ satisfies the statement of the theorem. First note that H 1 is a monotone increasing function satisfying lim m + H 1 m = 1. Since H m = H 1 m+ c 1 b αm α, it follows that m H is monotone increasing for large enough m, and lim m + Hm = +. Consequently, Mϕ is not empty. We consider three cases. Case 1: Suppose b α u u 2 du α c 1. Then lim m + H 1 m R +, therefore H =, and H is monotone increasing on R +. In this case m α ϕ is the unique root of Case 2: Suppose α b α u < du <. c 1 u 2 Then H 1 m is negative for small m, and hence there exists m > such that H 1 m =, H 1 m < for m,m, and H 1 m > for m m,. 13

14 Moreover, H is monotone increasing on m,. Then 2.37 has a unique root on m, for any m 1 ϕ >, and 2.37 has two roots, m = and m = m for m 1 ϕ = i.e., for ϕ =. Case 3: Suppose Then L Hospital s Rule yields b α u u 2 du =. lim m b α u du = lim m + m u 2 m + d dm m b αu u 2 d 1 dm m du = b α, therefore lim Hm = c 1 m + α b α. Then there exists m such that Hm < for m [,m, Hm =, and Hm > for m > m. Then 2.37 has a unique root on [m, for any m 1 ϕ. Thus statement i is a consequence of statement i of Theorem 2.2 with k =. Note that in all cases above we get m α ϕ as m 1 ϕ. ii In virtue of Theorem 2.2, it is also clear that 2.39 is satisfied with some suitable d α ϕ and e α ϕ for all ϕ C. iii It follows from 2.34 that m γϕ 1 = γm ϕ 1 for γ >, and so m 1 γϕ 1 = γ m 1 ϕ 1 for γ >. The proof of the theorem is complete if we show that H7 holds with ϕ = γϕ 1 for any large enough γ >. First note that d γ 1 ϕ 1 + e γ 1 ϕ 1 =γa for γ >, where A:= d ϕ 1 + e ϕ 1. Then inequality 2.13 for ϕ = γϕ 1 has the form γa > P + Q P + Q = m α γϕ 1 α e αt b α m αγϕ 1 e αt t m α γϕ 1 dt b α u u 2 du. 2.4 Therefore relations γ m 1 ϕ 1 = m 1 γϕ 1 = m α γϕ 1 H 1 m α γϕ 1 yield an equivalent form of 2.4: A > P + Q m 1ϕ 1 αh 1 m α γϕ 1 m αγϕ 1 b α u u 2 which holds for γ γ 1 for some γ 1 >, since m α γϕ 1 as γ. The proof of the theorem is complete. du, 14

15 The next result deals with the case α <. Theorem 2.8 Assume that H1, H2 and H3 are satisfied with k =, moreover H8 holds with α < Let m 1 ϕ be defined in 2.35, and 1 b α u u 2 du <. U α := {ϕ C : m 1 ϕ < ρ α }, 2.41 where Then ρ α := sup <v { v 1 c 1 α v b α u u 2 du }. i for all initial function ϕ U α any corresponding solution x ;ϕ of the IVP satisfies 2.38, where m = m α ϕ is the smallest root of the function Hm := m c m 1 α m b α u du m u 2 1 ϕ, 2.42 where the function H is monotone increasing. ii If H5 and H6 are also satisfied with k =, then for every ϕ U α there exist d α ϕ, e α ϕ X such that 2.39 is satisfied. iii If, in addition, H9 holds, then there exists γ 2 > such that d α γϕ 1 + e α γϕ 1, < γ γ 2. Proof. i Let ωt,u be defined by ωt,u = b α e αt u for all t and u. Then = m m b α u du α u 2 t e αt t ωt,mdt = e αt t b α e αt t m dt <, where we used the substitution u = e αt m. Thus ρ defined in 2.9 is equal to ρ α. We rewrite H in the form Hm = mh 2 m m 1 ϕ, H 2 m = 1 c m 1 b α u du. α u 2 Now H 2 = 1, and there exists m > such that 1/2 H 2 m 1 for m [,m ]. Therefore it is easy to see that m α ϕ as ϕ. 15

16 Statement i is an easy consequence of statement i of Theorem 2.2 with k =. ii The asymptotic formula 2.39 is an immediate consequence of the assumptions and part ii of Theorem 2.2 for all ϕ U α. iii Similarly to the proof of Theorem 2.7 we can argue that condition H7 with ϕ = γϕ 1 is equivalent to the inequality A > P + Q m 1ϕ 1 α H 2 m α γϕ 1 mαγϕ 1 b α u u 2 which holds for < γ γ 2 for some γ 2 >, since m α γϕ 1 as γ. The proof of the theorem is complete. du, Proposition 2.9 We have ρ α = and U α = C in Theorem 2.8, if and only if b α u du < α 2.43 u 2 c 1 or b α u u 2 du = α c 1 and lim m + b αm = Proof. If 2.43 holds, then H 2 m ε for m > for some ε >, and hence Hm εm, m >, therefore ρ α =. If b α u du > α, u 2 c 1 then there exists m > such that H 2 m = and H 2 m < for m > m. This yields ρ α <. If then H 2 m lim Hm = lim m + m + 1 m which completes the proof. b α u u 2 du = α c 1, H = lim 2m m + 1 m 2 c 1 = lim m + α b αm, The sharpness of Theorems 2.7 and 2.8 is analyzed for the following scalar equation xt = V t ϕ + V t sgs,x ds, t,

17 with initial condition Here xs = ϕs, t 1 s, ϕ C[t 1, ], R. A1 V : R +, is a continuous function, and there exists a real number α such that the limit P = lim t + e αt V t is a positive number. A2 g : [, C[t 1,, R R is a continuous Volterra-type functional and there exists a continuous function γ α : [, R + R + such that the map R + u γ α t,u R + is monotone nonincreasing for any fixed t [,, moreover for any t,y [, C[t 1,, R +, g t,e α t y γ α t, min ys, 2.46 ξt s t where ξ: [, [t 1, is a continuous function satisfying t 1 ξt t for t, and ξt + as t +. Theorem 2.1 Assume that A1 and A2 are satisfied, and there exists an initial function ϕ C[t 1, ], R + such that ϕ > and Eq has a solution x ;ϕ : [t 1, R for which the limit c := lim t + e αt xt;ϕ exists and is positive. Then there exist v > and T > satisfying the inequality P +T e αt γ α t,v dt < v. Proof. First we show that the function x = x ;ϕ is positive on [,. Otherwise there exists a t 1 > such that Then from 2.45 it follows x t >, t < t 1, and x t 1 =. x t 1 V t 1 ϕ >, which is a contradiction. Thus x t >, t, and hence the function yt = e αt t x t, t t 1, satisfies 17

18 yt = e αt t V t ϕ + e αt s V t se αs t g s,e α t y ds e αt V t ϕ + e αt s V t se αs γ α s, min yτ ds ξs τ s for t. But yt c > and ξt +, as t +, and hence for an arbitrarily fixed ε, 1, there exists a T = T ε > such that e αt V t 1 εp and 1 + εc yt min ys 1 εc, t + T. ξt s t This yields the inequality 1 + εc 1 εpϕ + 1 εp +T e αs γ α s, 1 εc ds. Since Pϕ >, there is a constant ε, 1 that satisfies and hence v > P 1 + ε c 1 ε Pϕ < 1 ε 2 c, +T e αs γ α s,v ds, where v = 1 ε c. This completes the proof of the theorem. Next we consider a special case of Eq xt = V t ϕ + V t sbxs τsds, t with the initial condition xs = ϕs, r s, ϕ C[ r, ], R. In the next corollary for α defined in A1, S α denotes the set of all initial functions ϕ C[ r, ], R such that the limit c := lim t + e αt xt;ϕ 2.48 exists and is not zero for a solution xt;φ of Eq corresponding to ϕ. It is easy to see that if b is an odd function and ϕ S α, then ϕ S α, as well. Now we state and prove the next sharp result. 18

19 Corollary 2.11 Assume that V : R +, satisfies condition A1, b : R R is a continuous, monotone nondecreasing and odd function satisfying bu > for u >, τ : [, [,r] is a continuous function, and r >. Then i If S α is not empty, then α. ii If α <, then S α is not empty if and only if 1 iii If α >, then S α is not empty if and only if 1 bu u 2 du <. bu u 2 du <. Proof. It is easy to check that the assumptions imply H1 H3 and H5 H6 with X = R, t 1 = r, Tt = V t, yt;ϕ = V t ϕ, ft,x = bxt τt, ζt = t τt, k =, c 1 = sup t e αt V t, m ϕ = c 1 ϕ, β =, Q =, d ϕ = Pe α ϕ, e ϕ =, and H8 with b α u = bω α u, where ω α = sup t e ατt. i Suppose ϕ S α, and let xt = xt;ϕ be a fixed solution of 2.47 satisfying Without the loss of generality we assume that ϕ is such that c defined in 2.48 is positive. For the sake of contradiction we assume that α =. But in that case there exists a t 1 > such that xt τt > c/2, t +t 1, and hence c c t t 1 V t sbxs τsds b V t sds = b V udu 2 +t 1 2 for t + t 1. It follows from A1 with α = that lim t + V t = P >, and hence from 2.47 we obtain lim t + xt = +. This contradicts to the definition of S α, and consequently α. ii In virtue of Theorem 2.8, it is clear that 1 bu du < implies that S u 2 α is not empty. For the necessary part, assume ϕ S α. Define the functions g: [, C[t 1,, R R and γ α : [, R + R + by gt,x = bxt τt, t,x [, C[t 1,, R, and γ α t,u = be αt δ α u, t,u [, R +, where δ α = inf t e ατt. Then for any t,y [, C[t 1,, R, relation 2.46 is satisfied, where ht = t τt, t. Thus by Theorem 2.1 we have constants v > and T > such that v > P +T e αt γ α t,v dt = P +T e αt bδ α e αt v dt. It can be easily seen by using the substitution u = δ α e αt t v that the above inequality implies 1 bu du <. u 2 19

20 Part iii can be argued similarly to the proof of ii. Closing this section we consider Eq. 2.1 in the case when t 1 =, i.e., consider xt = yt;ϕ + Tt sfs,xsds, t, 2.49 and the initial condition x = ϕ. 2.5 We will need the following assumptions. H8* Suppose α, and there exists a continuous and monotone nondecreasing function θ: R + R + such that θu > for u >, and ft,e αt t t +1 k z θ e αt t t +1 k z, t,z [, X. H9* There exists ϕ 1 C such that and 2.34 holds. lim t e αt t + 1 k yt;ϕ 1, 2.51 Theorem 2.12 Suppose H1-H3 hold with t 1 =, there exists η 1 such that the function,,, u θu u η is monotone nonincreasing, moreover H8* holds with α > and 1 θu u 2 log ukη du < Then i for every ϕ C, any solution x ;ϕ of the IVP satisfies 2.14, where m = mϕ is the largest root of the equation m 1 ϕ + c 1 α kη+1m m θu log u kη u 2 m + 1 du = m ii If H5 and H6 are also satisfied with t 1 =, then for every ϕ X there are d α ϕ, e α ϕ X such that 2.15 holds. 2

21 iii If, in addition, H9* holds, then there exists γ 1 > such that d α γϕ 1 + e α γϕ 1, γ γ 1. Proof. Let ωt,u be defined by ωt,u = θe αt t + 1 k u for all t and u. Then for m e α e αt ωt,mdt = e αt θeαt t t + 1 k m e αt t + 1 k m η eαt t + 1 k m η dt e αt θeαt t m e αt m η eαt t + 1 k m η dt θe αt t m = m e αt m t + 1 kη dt = m θu 1 α m u 2 α log u kη m + 1 du m θu α kη+1 m u log 2 ukη du <, where we used the substitution u = e αt m and the fact that t + 1 = 1 α log u m + 1 = 1 α log u log m + α 1 α log u for log m α. Therefore ρ defined in 2.9 is + and U defined in 2.1 equals to C. The rest of the proof is identical to that of Theorem 2.7. In the case α < we have e αt ωt,mdt m α kη+1 m θu log m kη u 2 u α du <. Therefore, analogously to Theorem 2.8 and Theorem 2.12, we can formulate a result for the case α < using condition instead of θu log 1 kη u 2 u α du < We remark that Theorem 2.12 includes the result of Webb [3] using η = 1, = and yt;ϕ = Ttϕ. 21

22 3 Applications In this section we apply our main results to a class of age-dependent population models with delayed birth process, to a certain class of differential equations with bounded and unbounded delays, and also to the pantograph and sunflower differential equations. 3.1 An age-dependent population with delayed birth process In [23] the following age-structured population model has been studied: u t t,a = ut,a µaut,a, t, a, 3.1 a ut, = τ βσ,aut + σ,adσ da, t, 3.2 us,a =F s,a, s [ τ,, a >, 3.3 u,a =f a, a. 3.4 Here ut,a denotes the density of the population at time t and age a, the death rate of the individuals is described by µa. τ > is a constant denoting the maximal delay and βσ, a represents the probability that an individual of age a reproduces after a time lag σ starting from conception. We introduce the Banach-spaces X = L 1 R +, R +, E = L 1 [ τ, ],X = L 1 [ τ, ] R +, R and Z = E X with the product norm F = F E + f X, f Z and define the delay operator We assume Φ: E R, ΦF = τ βσ,afσ,adσ da. B1 β L [ τ, ] R +, R and µ L R +, R are bounded, nonnegative functions, and lim a µa >, F E, f X. We use the notations ut = ut, and u t : [ τ, ] X, u t s = ut+s, and define the function z: R + Z, zt = u t. ut 22

23 It was shown in [23] that the PDE model is equivalent to the abstract Cauchy-problem z t = A zt, t, z = z := where the linear operator A on DA Z is defined by F f A F d σ,a = Fσ dσ, f f a µafa, 3.5 where DA = F f W 1,1 [ τ, ],X W 1,1 R +, R: F f = f ΦF. The following result is a consequence of Theorems 3.3, 4.6 and 4.7 of [23]. Proposition 3.1 Assume B1. Then A generates a strongly continuous positive semigroup T t on Z, and the population model is well-posed. Moreover, if τ βσ,ae a µs ds dσ da > 1, 3.6 then α := ω A = sa > is an eigenvalue of A with a one dimensional spectral projection Π and there exist constants M,δ > such that e α t T t Π Me δt, for all t. Here ω A denotes the growth bound of T t, i.e., ω A = lim t + log T t t, and sa is the spectral bound of A, i.e., sa = sup{re λ: λ σa}, where σa is the spectrum of A. Now we consider a nonlinear version of 3.1 u t,a = u t a t,a µa + Gut +,a ut,a, t, a, 3.7 where the nonlinear functional G: E R + accounts for the loss of individuals due to crowding. Similar nonlinearity was considered, e.g., in [1] and [12]. It is easy to see that 3.7 associated with boundary and initial conditions is equivalent to the Cauchy-problem z t = A zt + Gzt, t, z = z,

24 where G F σ,a =. f GFσfa Then the mild solution of 3.8 is the solution of zt = T tz + We assume on the nonlinearity that T t sgzsds, t. B2 G: E R + is such that i there exist constants M and 1 < η < satisfying GF M F η E, F E, ii G is locally Lipschitz-continuous, i.e., for every K > there exists L = LK such that GF G F L F F E, F E, F E K. The next result shows that the asynchronous exponential growth of the linear equation 3.5 is preserved for the mild solution of the nonlinear equation 3.8. Theorem 3.2 Assume B1, B2 and 3.6. Then 3.8 has a unique mild solution zt on [,, and there exists a one dimensional projection Π such that the solution satisfies lim t + e α t zt Πz Z =, z Z. Proof. Assumption B2 yields easily that G is locally Lipschitz-continuous, therefore a standard argument see, e.g., Section 4.3 of [5] shows the local existence and also the uniqueness of the mild solution of 3.8. It follows from Proposition 3.1 that H1 H3, H5, H6 and H9 are satisfied with = t 1 = and k =, and 3.6 yields α >. Let z = F Z. Then assumption B2 i yields f G e α t z Z = Ge α t Fe α t f X = Ge α t Fe α t f X Me 1+ηα t F η E f X Me 1+ηα t z 1+η Z t,z Z. Therefore H8 holds with b α u = Mu 1+η, which satisfies 2.36, as well. Therefore, Theorem 2.7 yields the statement of the theorem. 24

25 3.2 A system of linear delay differential equations Consider the system of linear delay equations We assume N M ẋt = A i xt τ i + B j txt σ j t, t. 3.9 i= j= i, ii τ i R + and A i R n n, i N, and iii B j : R + R n n and σ j : R + R + are continuous functions, lim t + t σ j t = +, j M. Let τ = max i N τ i and t 1 := min We associate the initial condition { } τ, min {inf {t σ jt}}. j M t xt = ϕt, t 1 t 3.1 to Eq In this section we will consider the initial time as a parameter, so the solution of the IVP will be denoted by x ;,ϕ. The asymptotic property of the solutions of Eq. 3.9 is given by using our main results and certain asymptotic properties of the solutions of the autonomous system N ẏt = A i yt τ i, t, 3.11 i= and the associated initial condition yt = ϕt, τ t By definition, the fundamental solution Tt to 3.11 is the n n matrix valued function satisfying Tt = N A i Tt τ i, t, and T = I, Ts = for τ s <. i= Here I and denote the n n identity and zero matrices, respectively. The characteristic equation associated to 3.11 is N λ := det λi A i e λτ i = i= 25

26 A complex number λ is called an eigenvalue of Eq if it is a solution of Eq Definition 3.3 λ = α + β i i = 1, is called a dominant eigenvalue of 3.11 if λ = and Reλ > Re λ for λ C satisfying λ =, λ λ and λ λ. The ascent of λ is the order of λ as a pole of 1 λ see [9], [18]. It is known [9], [18] that the ascent of an eigenvalue λ is less or equal to the algebraic multiplicity of λ. We assume C λ = α + β i is a dominant eigenvalue of Eq. 3.11, and let k + 1 be its ascent. The following result follows from the general theory of the series representation of the solutions of Eq see, e.g., [4], [18]. Proposition 3.4 Assume C. Then the following statements hold. i For all and ϕ C[ τ, ], R n there exist d,ϕ, e,ϕ R n such that the solution y ;,ϕ of Eq through,ϕ satisfies yt;,ϕ = e αt t d k,ϕ cos β t + e,ϕ sin β t + o1, as t. ii There exist constant matrices P,Q R n n for which Tt = e α t t + 1 k P cos β t + Q sin β t + o1, as t +. In the proof of the next theorem we will need the following estimate which can be proved using Gronwall s inequality see, e.g., [17] for a proof of a similar result. Lemma 3.5 The solution x ; ;ϕ of the IVP satisfies N i= xt;,ϕ e A M t i t + j= B j s ds max ϕs, t. t 1 s Now, we are in a position to state and prove the following result. 26

27 Theorem 3.6 Assume C, and t k M j= B j t e α σ j t dt < Then for every there exists M such that for every ϕ C[t 1, ], R n the solution x ;,ϕ of Eq. 3.9 through,ϕ satisfies xt;,ϕ M e α t t t 1 +1 k max ϕs, t t 1. t 1 s Moreover, there are vectors d,ϕ and ẽ,ϕ in R n such that 3.15 xt;,ϕ = e αt t k d,ϕ cos β t + ẽ,ϕ sin β t + o1, as t +, where d,ϕ + ẽ,ϕ for some ϕ C[t 1, ], R n. Proof. From part c of Proposition 3.4, it follows that c 1 := supt + 1 k e αt Tt <. t Assumption 3.14 yields there exists S > such that { } A := max c 1, P + Q t t k M S Since t 1 t 1 for any, it follows j= 3.16 B j t e α σ j t dt < 1. { } M max c 1, P + Q t t k B j t e α σ j t dt A < 1 j= 3.17 for all S. Let S be fixed, and f : [, C[t 1,, R n R n be defined by M ft,y = B j tyt σ j t. j= Then, by using the variation of constants formula see, e.g., [18], we obtain that the solution x ;,ϕ of Eq. 3.9 through,ϕ satisfies xt;,ϕ = yt;,ϕ + Tt sfs,x ;,ϕds, t, where y ;,ϕ denotes the solution of Eq through,ϕ. Moreover, for all t,z [, C[t 1, ], R n R n 27

28 f t,e α t k z M = B j te α t σ j t t σ j t t k zt σ j t j= e α t at max ζt s t zs, where ζt = min{t σ j t: j =,...,M}, and M at = t t k B j t e α σ j t, t. j= Thus it follows from 3.17 that all conditions of Theorem 2.6 hold, therefore x ;,ϕ satisfies 2.31, i.e., where xt;,ϕ m 2,ϕe α t t t k, t S, 3.18 m 2,ϕ = max { max t 1 s e α s s t k ϕs, m ϕ } 1 c 1 asds Since Eq is autonomous and linear, there exists a constant K 1 such that yt;,ϕ Ke αt max ϕs, t. t 1 s Therefore and so m ϕ = K max ϕs, t 1 s m 2,ϕ ˆm 2 max ϕs, S, ϕ C[t 1, ], R n, t 1 s where ˆm 2 = max{e α t 1, K} 1 A, α >, K, α 1 A Now, consider an arbitrarily fixed [,S and an initial function ϕ C[t 1, ], R n. Lemma 3.5 yields there exists a constant m 3 1 such that xt;, ϕ m 3 max ϕs, t S. 3.2 t 1 s On the other hand, the solution x ;, ϕ of Eq. 3.9 is unique on [, and xt;, ϕ = xt;s,ϕ, t S, 28

29 where ϕs = xs;, ϕ, t 1 S s S. Consequently, 3.18, 3.19 and 3.2 yield for t S xt;, ϕ = xt;s,ϕ ˆm 2 Se α t S t t 1 S + 1 k max ϕs t 1 S s S ˆm 2 Se α S e α t m 3 t t k max ϕs. t 1 s Therefore 3.15 is satisfied with ˆm 2 Sm 3 e α S, < S, M = ˆm 2, S. The rest of the proof of this theorem is an easy consequence of Theorem 2.6 and the ideas used above, therefore it is omitted. We get immediately from the proof: Remark 3.7 If α or t 1 is bounded for, then M in 3.16 is independent of. Now, we give the following definition of stability: Definition 3.8 We say that the zero solution of Eq. 3.9 is equistable on R + if for all and ϕ C[t 1, ], R n the solution x ;,ϕ of Eq. 3.9 through,ϕ is bounded on [,. If for all and ϕ C[t 1, ], R n the solution x ;,ϕ tends to zero as t +, then we say that the zero solution of Eq. 3.9 is equi-asymptotically stable. The next stability result is an easy consequence of Theorem 3.6, therefore we state it without proof. Theorem 3.9 Assume C. If 3.14 holds, then i The zero solution of Eq. 3.9 is equistable on R + if and only if either α < or α = and k =. ii The zero solution of Eq. 3.9 is equi-asymptotically stable if and only if α <. We close this subsection with the following corollary of Theorem 3.6, which shows the importance of the factor e α σ j t in condition We apply our 29

30 results to the pantograph equation see, e.g., [19]. Corollary 3.1 Consider the delay differential equation N M ẋt = A i xt τ i + i= j= B j xγ j t 3.21 where τ i R +, A i R n n, i n, and γ j, 1, B j R n n, j M. If C holds and α >, then the statements of Theorem 3.6 are valid for any solution x ;,ϕ of Eq through,ϕ where, t 1 = min{ τ, min j M γ j }, and ϕ C[t 1, ], R n. Proof. Let σ j : R + R + and B j : R + R n n be defined by σ j t = 1 γ j t and B j t B j, t, j M, and ζt = min{γ j t: j =,...,M}. Then t k M j= B j t e α σ j t dt = M j= B j t k e α 1 γ j t dt <, and hence by Theorem 3.6 the statement of the corollary follows. Remark 3.11 Corollary 3.1 shows that if the zero solution of Eq is unstable then the zero solution of Eq is also unstable. 3.3 The sunflower equation In this section we consider the second-order delay differential equation where ẍt + Aẋt + Bgxt r =, t, 3.22 A,B R, r R +, g CR, R is odd satisfying gu > for u > When gx = sin x, Eq is the so-called sunflower equation, which was investigated extensively see, e.g., [2], [21] and [27] and the references therein. Consider an associated linear equation ÿt + Aẏt + Byt r =, t,

31 and its characteristic equation λ 2 + Aλ + Be λr = Theorem 3.12 Assume 3.23, let λ = α +β i be a simple dominant eigenvalue of Eq and α > and 1 1 u max gs s du < s u Then for all ϕ C[ r, ], R the solution x ;,ϕ of Eq through, ϕ satisfies xt;,ϕ = e α t d 1 ϕ cosβ t + γ 1 ϕ + o1, t +, 3.27 where d 1 ϕ, γ 1 ϕ R and d 1 ϕ for some ϕ C[ r, ], R. Proof. Set =, t 1 = r, and ft,x = B gxt r xt r, t,x [, C[t 1,, R. Moreover, for all ϕ C[t 1, ], R the solution x ;,ϕ of Eq through,ϕ is the unique solution of ẍt + Aẋt + Bxt r = ft,x, t. Thus by the variation of constants formula we find xt;,ϕ = yt;,ϕ + Tt sfs,x ;,ϕds, t, 3.28 where y ;,ϕ is the solution of Eq through,ϕ and Tt is the unique function satisfying Tt + A Tt + BTt r =, a.e. t, and T = 1 and Ts =, r s. Since λ = α + β i is a simple dominant eigenvalue of Eq. 3.25, from the series representation of the solutions of Eq see Proposition 3.4, we have yt;,ϕ = e α t d ϕ cosβ t + e ϕ sin β t + o1, t +, where d,e : C[ r, ], R R are such that d ϕ + e ϕ for some ϕ C[ r, ], R, and Tt = e α t P cos β t + Q sin β t + o1, t +, 31

32 where P,Q R and P + Q. In that case, it can be easily seen that Eq satisfies all of the conditions of Theorem 2.7, whenever b α u = max s u gs s for all u. Therefore in virtue of Theorem 2.7, we obtain the following representation of x ;,ϕ: xt;,ϕ = e α t d α ϕ cos β t + e α ϕ sin β t + o1, t +, 3.29 where d α,e α : C[ r, ], R R are such that d α ϕ + e α ϕ, for some ϕ C[ r, ], R. Let and γ 1 ϕ be such that d 1 ϕ = d α ϕ 2 + e α ϕ 2 d 1 ϕ cos γ 1 ϕ = d α ϕ and d 1 ϕ sin γ 1 ϕ = e α ϕ. Then d 1 ϕ and 3.29 yields 3.27, and this completes the proof of the theorem. Similarly to Theorem 3.12, Theorem 2.8 yields the next result. Theorem 3.13 Assume 3.23, let λ = α +β i be a simple dominant eigenvalue of Eq and α < and 1 1 u max gs s du < s u Then there exists a δ = δα >, such that for all ϕ C[ r, ], R, ϕ = max ϕs < δ, r s the solution x ;,ϕ of Eq through,ϕ satisfies 3.27, where d 1 ϕ, γ 1 ϕ R and d 1 ϕ for some ϕ C[ r, ], R, and ϕ < δ. When gx = sinx, i.e., Eq is the so-called sunflower equation then u max gs s du = 1 2 s u u max sin s s du <, 2 s u and therefore Theorem 3.13 is applicable. 32

33 References [1] O. Arino, A. Bertuzzi, A. Gandolfi, E. Sánchez, C. Sinisgalli, The asynchronous exponential growth property in a model for the kinetic heterogeneity of tumour cell populations, J. Math. Anal. Appl [2] O. Arino, E. Sánchez, G. Webb, Necessary and sufficient conditions for asynchronous exponential growth in age structured cell populations with quiescence, J. Math. Anal. Appl [3] G.I. Bell, E.C. Anderson, Cell growth and division I. A mathematical model with applications to cell volume distribution in mammalian suspension cultures, Biophys. J [4] R. Bellman, K.L. Cooke, Differential Difference Equations, Academic Press, New York, [5] T. Cazenave, A. Haraux, An Introduction to Semilinear Evolution Equations, Clarendon Press, Oxford, [6] J. Cermak, A change of variables in the asymptotic theory of differential equations with unbounded delays, J. Comput. Appl. Math [7] J.G. Dix, C.G. Philos, I.K. Punaras, An asymptotic property of solutions to linear nonautonomous delay differential equations, Electron. J. Diff. Equ [8] J.G. Dix, C.G. Philos, I.K. Punaras, Asymptotic properties of solutions to linear non-autonomous neutral differential equations, J. Math. Anal. Appl [9] O. Diekmann, S.A. van Gils, S.M. Verduyn Lunel, H.-O. Walther, Delay Equations: Functional-, Complex-, and Nonlinear Analysis, Springer-Verlag, New York, Applied Mathematical Sciences Vol. 11, [1] J. Dyson, R. Villella-Bressan, G.F. Webb, A nonlinear age and maturity structured model of population dynamics, J. Math. Anal. Appl., [11] J. Dyson, R. Villella-Bressan, G.F. Webb, Asynchronous exponential growth in an age structured population of proliferating and quiescent cells, Mathematical Biosciences, [12] J. Dyson, R. Villella-Bressan, G.F. Webb, Asymptotic behaviour of solutions to abstract logistic equations, Math. Biosc., [13] J.Z. Farkas, Note on asynchronous exponential growth for structured population models, Nonlin. Anal [14] J. Gallardo, M. Pinto, Asymptotic integration of nonautonomous delaydifferential systems, J. Math. Anal. Appl., [15] M. Gillenberg, G.F. Webb, Asynchronous exponential growth of semigroups of nonlinear operators, J. Math. Anal. Appl

34 [16] M.E. Gurtin, R.C. MacCamy, Non-linear age-dependent population dynamics, Arch. Rational Mech. Anal., [17] I. Győri, F. Hartung, Exponential stability of a state-dependent delay system, Discrete Contin. Dynam. Systems [18] J.K. Hale, S.M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, [19] A. Iserles, On the generalized pantograph functional-differential equations, European J. Appl. Math [2] D. Israelsson, A. Johnsson, A theory of circumnutations of Helianthus annus, Physiol. Plant [21] M.R.S. Kulenovic, G. Ladas, Oscillations of the sunflower equation, Quart. Appl. Math [22] V. Laksmikantham, S. Leela, Differential and Intregral Inequalities, vol. 1. Academic Press, New York, [23] S. Piazzera, L. Tonetto, Asynchronous exponential growth for an age dependent population equation with delayed birth process, J. Evol. Equ., [24] M. Pinto, Nonlinear delay-differential equations with small lag, Internat. J. Math. Math. Sci., [25] M. Pituk, A Perron type theorem for functional differential equations, J. Math. Anal. Appl., [26] M. Pituk, Asymptotic behavior and oscillation of functional differential equations, J. Math. Anal. Appl [27] A.S. Somolinos, Periodic solutions of the sunflower equation: ẍ + a/rẋ + b/r sin xt r =, Quart. Appl. Math [28] H.R. Thieme, Balanced exponential growth of operator semigroups, J. Math. Anal. Appl [29] G.F. Webb, Theory of Nonlinear Age Dependent Population Dynamics, Marcel Dekker, New York, [3] G.F. Webb, Growth bounds of solutions of abstract nonlinear differential equations, Differential Integral Equations [31] G.F. Webb, Asynchronous exponential growth in differential equations with inhomogeneous nonlinearities, Differential Equations in Banach Spaces, Lecture Notes in Pure and Applied Mathematics Series, Vol. 148, Marcel Dekker, New York, 1993, [32] G.F. Webb, Asynchronous exponential growth in differential equations with asymptotically homogeneous nonlinearities, Adv. Math. Sci. Appl., /