Memory Dependent Growth in Sublinear Volterra Differential Equations
MMEMORY DEPENDENT GROWTH IN SUBLINEAR VOLTERRA DIFFERENTIALEQUATIONS
JOHN A. D. APPLEBY AND DENIS D. PATTERSON
Abstract.
We investigate memory dependent asymptotic growth in scalar Volterra equations withsublinear nonlinearity. To obtain precise results we utilise the powerful theory of regular variationextensively. By computing the growth rate in terms of a related ordinary differential equation we showthat when the memory effect is so strong that the kernel tends to infinity, the growth rate of solutionsdepends explicitly on the memory of the system. Finally, we employ a fixed point argument to determineanalogous results for a perturbed Volterra equation and show that, for a sufficiently large perturbation,the solution tracks the perturbation asymptotically, even when the forcing term is potentially highlynon-monotone. Introduction
We investigate explicit memory dependence in the asymptotic growth rates of positive solutions ofthe following scalar Volterra integro-differential equation x (cid:48) ( t ) = (cid:90) [0 ,t ] µ ( ds ) f ( x ( t − s )) , t > x (0) = ξ > , (1.1)where f is a positive sublinear function (i.e. lim x →∞ f ( x ) /x = 0) and µ is a non–negative Borel measure.The relevant existence and uniqueness theory regarding equations of the form (1.1) is well known andguarantees a unique solution x ∈ C ( R + ; (0 , ∞ )) in the framework of this article [10, Corollary 12.3.2],with the convention that R + := [0 , ∞ ). By defining the function M ( t ) := (cid:90) [0 ,t ] µ ( ds ) , t ≥ , (1.2)it follows that (1.1) is equivalent to x ( t ) = x (0) + (cid:90) t M ( t − s ) f ( x ( s )) ds, t ≥ , x (0) = ξ > . (1.3)We also study the asymptotic behaviour of the perturbed Volterra equation x (cid:48) ( t ) = (cid:90) [0 ,t ] µ ( ds ) f ( x ( t − s )) + h ( t ) , t > x (0) = ξ > . (1.4)As with the unperturbed equation, it is useful to consider an integral form of (1.4), and by defining H ( t ) := (cid:90) t h ( s ) ds, t ≥ , (1.5)it follows that (1.4) can be written in integral form as x ( t ) = x (0) + (cid:90) t M ( t − s ) f ( x ( s )) ds + H ( t ) , t ≥ x (0) = ξ > . (1.6) Date : November 7, 2018.2010
Mathematics Subject Classification.
Primary: 34K25; Secondary: 34K28.
Key words and phrases.
Volterra equations, asymptotics, subexponential growth, unbounded delay, regular variation.Denis Patterson is supported by the Government of Ireland Postgraduate Scholarship Scheme operated by the IrishResearch Council under the project GOIPG/2013/402. a r X i v : . [ m a t h . C A ] A p r JOHN A. D. APPLEBY AND DENIS D. PATTERSON
In [6], with µ a finite measure, we demonstrate that when f is sublinear and asymptotically increasing,the solution of (1.1) obeys lim t →∞ F ( x ( t )) /t = (cid:82) [0 , ∞ ) µ ( ds ) < ∞ , where F ( x ) := (cid:90) x f ( u ) du, x > . (1.7)In other words, the structure of the memory does not affect the asymptotic growth rate of the solution of(1.1) when the total measure is finite: indeed, the entire mass of µ could be concentrated at 0, because theordinary differential equation y (cid:48) ( t ) = (cid:82) [0 , ∞ ) µ ( ds ) · f ( y ( t )) for t ≥ F ( y ( t )) /t → (cid:82) [0 , ∞ ) µ ( ds )as t → ∞ . This is in contrast to the linear case where the growth rate depends crucially on thestructure of the memory (cf. [10, Theorem 7.2.3]). In [6] we also show that if lim t →∞ M ( t ) = ∞ , thenlim t →∞ F ( x ( t )) /t = ∞ . This result suggests that allowing the total measure to be infinite makes thelong run dynamics more sensitive to the memory but that comparison with a non-autonomous ordinarydifferential equation may be necessary in this case.To achieve precise asymptotic results for the solutions of (1.1) and (1.4) we employ the theory ofregular variation extensively. We record now for the reader’s convenience the definition of a regularlyvarying function (in the sense of Karamata) and allied notation. Definition 1.
Suppose a measurable function h : R → (0 , ∞ ) obeyslim x →∞ h ( λ x ) h ( x ) = λ ρ , for all λ > , some ρ ∈ R , then h is regularly varying at infinity with index ρ , or h ∈ RV ∞ ( ρ ).Regular variation provides a natural generalisation of the class of power functions and the applicationof the theory of regular variation to the study of qualitative properties of differential equations is anactive area of investigation. Recent research themes in this direction are recorded in reviews such as [12]and [14] and all properties of regularly varying functions employed can be found in the classic text [8].The authors of the present paper give a highly abridged list of the properties we have found useful inthe introduction to our work [5], which concerns ordinary differential equations.Many of the applications of regular variation in the asymptotic theory of linear Volterra equationsdeal with the situation in which it is desired to capture slow decay in the memory, as captured by ameasure or kernel, or a singularity. Of course, slowly fading memory can be described in other ways,using for instance the theory of L weighted spaces (see e.g.,[16] and for stochastic equations, [7]). Whenthe kernel is integrable, it is often possible to obtain precise rates of decay in L ∞ by means of a largerclass of kernels (such as the subexponential class studied in [3], of which regularly varying kernels are asubclass). However, for singular equations, or equations with non–integrable kernels, the full power ofthe theory of regular variation is often needed: in particular, for linear equations, transform methodsand the Abelian and Tauberian theorems for regular variation are exploited (see e.g.,[4, 17]). It shouldbe stressed, though, that such methods are of greatest utility for linear equations: indeed, there doesnot seem to be especial benefit gained in this work in applying such a transform approach. Moreover,in this paper, the equation is intrinsically non–linear: f ( x ) is not of linear order as x → ∞ , and regularvariation arises both in the slow decay of µ and in the sublinear growth of f . Also, it is a general themeof the works cited above that the slow decay in the memory, combined with an appropriate type ofstability, give rise to convergence at a certain rate to equilibrium. By contrast in this paper, solutionsgrow, rather than decay.With a view to applications, we believe the most interesting subclass of equations will retain theproperty that the asymptotic contribution to the growth rate from a moving interval of any fixed duration( τ >
0, say) is negligible, in the sense thatlim t →∞ (cid:90) [ t,t + τ ) µ ( ds ) = 0 , for each τ > . (1.8)It should be noted that our proofs do not require this stipulation, but we mention it in order to motivateshortly a stronger hypothesis on M . EMORY DEPENDENT GROWTH IN SUBLINEAR VOLTERRA DIFFERENTIAL EQUATIONS 3
With (1.8) still in force, if µ is absolutely continuous and admits a non–negative and continuousdensity k , such that µ ( ds ) = k ( s ) ds , we see that k (cid:54)∈ L (0 , ∞ ) because M ( t ) → ∞ as t → ∞ . Inparticular the property (1.8) is implied by k ( t ) → t → ∞ . Therefore, it is perfectly possible for k tolie in another L p space, for some p >
1. As an example, suppose that k ( t ) ∼ t − θ as t → ∞ for θ ∈ (0 , p > /θ > k ∈ L p (0 , ∞ ), while k (cid:54)∈ L (0 , ∞ ). In this sense, our work shares concerns withexisting results in the literature in which the Volterra equation does not possess an integrable kernel (seee.g., [16, 11]).The type of fading memory property (1.8) we suggested was of interest motivates a stronger assumptionon M . First, we see that (1.8) implies1 nτ M ( nτ ) = 1 τ n n − (cid:88) j =0 (cid:90) [ jτ,jτ + τ ) µ ( ds ) → n → ∞ and so the non–negativity of µ implies that M ( t ) /t → t → ∞ . Since M ( t ) → ∞ as t → ∞ , M isnon–decreasing, and M ( t ) /t → t → ∞ , it is reasonable to suppose that M ∈ RV ∞ ( θ ) for θ ∈ [0 , θ = 1 in the parameter range does not lead to any problems in the analysis,and indeed it transpires that our arguments are valid for all θ ≥ f , is a positive and asymptotically increasing function such that f ( x ) →∞ and f ( x ) /x → x → ∞ ; hence it is natural to assume that f ∈ RV ∞ ( β ) for β ∈ [0 , β rapidly: if β > f ( x ) /x → ∞ as x → ∞ , and if β < f isasymptotic to a decreasing function. When β = 0 we append the hypotheses of asymptotic monotonicityand increase to infinity on f , as these are not necessarily satisfied by functions in RV ∞ (0), but otherwisethe analysis is essentially the same as when β ∈ (0 , β = 1 is largely ontechnical grounds: informally, when β = 1, the inverse of the increasing function F defined by (1.7) is no longer regularly varying ; F − now belongs to the class of rapidly varying functions (which we definebelow). It also can be seen from the nature of our results that the asymptotic behaviour of solutionsmust be of a different form from those that hold when β <
1. For β <
1, no such technical problemarises, and indeed F − is regularly varying with index 1 / (1 − β ).In some situations, we will consider very rapidly growing forcing terms H in the perturbed equation(1.6) which are not regularly varying. We sometimes consider forcing terms from the class of rapidlyvarying functions , and a definition of this class follows. Definition 2.
Suppose a measurable function h : R → (0 , ∞ ) obeys for λ > x →∞ h ( λ x ) h ( x ) = , λ < , , λ = 1 , + ∞ , λ > . Then h is rapidly varying at infinity , or h ∈ RV ∞ ( ∞ ). If on the other hand, h : R → (0 , ∞ ) obeys for λ >
0: lim x →∞ h ( λ x ) h ( x ) = + ∞ , λ < , , λ = 1 , , λ > . Then we write h ∈ RV ∞ ( −∞ ).The proof of our main result for (1.1), Theorem 4, relies principally upon comparison methods,properties of regularly varying functions and a time change argument for delay differential equations.We first use constructive comparison methods, similar in spirit to those employed by Appleby andBuckwar [2] for linear equations, to establish “crude” upper and lower bounds on the solution of (1.1).The more challenging construction is that of the lower bound and is completed by comparing solutionsof (1.1) with those of a related nonlinear pantograph equation using time change arguments inspired byBrunner and Maset [9]. Finally, we prove a convolution lemma for regularly varying functions (cf. [1,Theorem 3.4]) which is then used, in conjunction with straightforward comparison methods, to sharpenthe aforementioned “crude” upper and lower bounds, and show that they coincide. Another paper which JOHN A. D. APPLEBY AND DENIS D. PATTERSON uses similar iterative methods to sharpen estimates in the growth of solutions of nonlinear convolutionVolterra equations is Schneider [15].With ¯ M ( t ) := (cid:82) t M ( s ) ds , we obtain lim t →∞ F ( x ( t )) / ¯ M ( t ) = Λ( β, θ ), or that the growth rate ofsolutions of (1.1) depend explicitly on both indices of regular variation, and therefore the memory ofthe system (Theorem 4). The value of the parameter–dependent limit Λ can be determined explicitly interms of the Gamma function. This result is only valid for β ∈ [0 ,
1) and hence may not hold if f is onlyassumed to be sublinear (i.e. lim x →∞ f ( x ) /x = 0). In this sense, it appears that the imposition of thehypothesis of regular variation on f and M is intrinsic to the form of the asymptotic behaviour deduced,rather than a being a purely technical contrivance, and the restriction to β (cid:54) = 1 also seems justified bygrounds other than the complexity of the analysis needed to prove a sharp result.The results and methods outlined above for (1.1) can also be used to yield sharp asymptotics for theperturbed equation (1.4). If H is positive, solutions to (1.4) will be positive and exhibit unboundedgrowth; therefore there is no need to assume pointwise positivity of h . Hence solutions of (1.4) are nolonger necessarily non–decreasing and more delicate comparison techniques are required to treat thisadditional difficulty.When H is of the same order of magnitude as the solution of (2.4), we establish non-trivial upperand lower bounds on the solution and then employ a simple fixed point iteration argument to calculatethe exact asymptotic growth rate of the solution in terms of a characteristic equation (Theorem 5).Moreover, the converse also holds: growth in the solution of (1.4) at a rate proportional to that of thesolution of (2.4) is possible only when H is of the same order as that solution. In these results, theparameter θ characterises the dependence of the growth rate on the degree of memory in the system.When the perturbation term grows sufficiently quickly, the solution tracks H asymptotically, in the sensethat lim t →∞ x ( t ) /H ( t ) = 1, even when H is allowed to be highly non-monotone. Indeed, under certainrestrictions we can show that our characterisation of rapid growth in the perturbation is necessary inorder for lim t →∞ x ( t ) /H ( t ) = 1 to prevail.2. Main Results and Discussion
The following equivalence relation on the space of positive continuous functions and shorthand areused throughout.
Definition 3.
Suppose a, b ∈ C ( R + ; R + ). a and b are asymptotically equivalent if lim t →∞ a ( t ) /b ( t ) = 1;we often write a ( t ) ∼ b ( t ) as t → ∞ for short. µ is a non-negative Borel measure on R + with infinite total variation; more precisely µ ( E ) ≥ E ∈ B ( R + ) , (cid:90) [0 , ∞ ) µ ( ds ) = lim t →∞ M ( t ) = ∞ , (2.1)where M is defined as in (1.2). Our first result gives precise information on the asymptotic growth rateof the solution to (1.1). We state our result before carefully analysing the conclusion. We defer the proofto Section 5. Theorem 4.
Suppose the measure µ obeys (2.1) with M ∈ RV ∞ ( θ ) , θ ≥ and that f ∈ RV ∞ ( β ) , β ∈ [0 , . When β = 0 let f be asymptotically increasing and obey lim x →∞ f ( x ) = ∞ . Then the solution, x ,of (1.1) satisfies x ∈ RV ∞ ((1 + θ ) / (1 − β )) and lim t →∞ F ( x ( t ))¯ M ( t ) = Γ( θ + 1)Γ (cid:16) βθ − β (cid:17) Γ (cid:16) θ − β (cid:17) =: Λ( β, θ ) , (2.2) where Γ( x ) := (cid:82) ∞ t x − e − t dt and ¯ M ( t ) := (cid:82) t M ( s ) ds . EMORY DEPENDENT GROWTH IN SUBLINEAR VOLTERRA DIFFERENTIAL EQUATIONS 5
By Karamata’s Theorem (cf. [8, Theorem 1.5.11]), lim t →∞ ¯ M ( t ) /tM ( t ) = 1 / (1 + θ ). Hence theconclusion of Theorem 4 is equivalent tolim t →∞ F ( x ( t )) t M ( t ) = (1 + θ ) Γ( θ + 1)Γ (cid:16) βθ − β (cid:17) Γ (cid:16) θ − β (cid:17) = 11 − β B (cid:18) θ + 1 , θβ − β (cid:19) , where B denotes the Beta function, which is defined by B ( x, y ) := (cid:82) λ x − (1 − λ ) y − dλ (cf. [13, p.142]).Furthermore, since F − ∈ RV ∞ (1 / (1 − β )), (2.2) is also equivalent tolim t →∞ x ( t ) F − ( t M ( t )) = (cid:26) − β B (cid:18) θ + 1 , θβ − β (cid:19)(cid:27) − β . (2.3)Theorem 4 expresses the leading order asymptotics of the solution in terms of the functions F and ¯ M .The dependence of Λ on β and θ is known explicitly and this can be used to gain some insight into secondorder effects of the nonlinearity and the memory on the growth rate. The following proposition recordssome properties of the function Λ( β, θ ) that are useful when interpreting the conclusion of Theorem 4.The proofs of the forthcoming claims are deferred to Section 6. Proposition 1.
Suppose Λ( β, θ ) is defined by (2.2) with β ∈ [0 , and θ ∈ [0 , ∞ ) . Then(i.) Λ(0 , θ ) = 1 for fixed θ ∈ (0 , ∞ ) and Λ( β,
0) = 1 for fixed β ∈ (0 , ,(ii.) lim β ↑ Λ( β, θ ) = 0 for fixed θ ∈ (0 , ∞ ) and lim θ →∞ Λ( β, θ ) = 0 for fixed β ∈ (0 , ,(iii.) β (cid:55)→ Λ( β, θ ) is decreasing, β ∈ (0 , , θ (fixed) ∈ (0 , ∞ ) ,(iv.) θ (cid:55)→ Λ( β, θ ) is decreasing, θ ∈ (0 , ∞ ) , β (fixed) ∈ (0 , ,(v.) Λ( β, θ ) ∈ (0 , for β ∈ (0 , and θ ∈ (0 , ∞ ) . Figure 1.
Plot of the surface Λ( β, θ ) with β ∈ [0 ,
1) and θ ∈ [0 , β ∈ (0 , θ = 0 in Theorem 4 yields lim t →∞ F ( x ( t )) / ¯ M ( t ) = 1. The authorshave previously obtained this conclusion for sublinear equations of the form (1.3) without regular vari-ation but with lim t →∞ M ( t ) = M ∈ (0 , ∞ ). Therefore Theorem 4 can be thought of as a continuousextension of our previous results for (1.1) with sublinear nonlinearities and finite measures (see [6] forfurther details).For a fixed β ∈ (0 , θ represents an increase in the rate of decay of themeasure µ . This can be made precise by supposing that the measure µ is absolutely continuous, andspecifically that µ ( ds ) = m ( s ) ds for continuous m ∈ RV ∞ ( θ − , θ ∈ (0 , JOHN A. D. APPLEBY AND DENIS D. PATTERSON value of θ gives more weight to values of the solution in the past (more memory) and we expect thegrowth rate of solutions of (1.1) to be slower than that of the related ordinary differential equation y (cid:48) ( t ) = M ( t ) f ( y ( t )) , t > y (0) = ξ > . (2.4)The equation (2.4), in contrast, places the entire weight M ( t ) at the present time, when the solution islargest. Hence, increasing the value of θ (putting more weight further into the past) slows the growthrate and it is intuitive that Λ( β, θ ) is decreasing in θ . Using this comparison with (2.4) once more, it isclear that Proposition 1 ( v. ) must hold since solutions of (1.1) can never grow faster than those of (2.4)(if f is strictly increasing this can be seen by inspection).For a fixed θ ∈ (0 , ∞ ), one might expect an increase in β to lead to a faster rate of growth ofthe solution of (1.1). Therefore, it may initially be surprising that Λ( β, θ ) is decreasing in β . Thiscounter-intuitive result is best understood by explaining the error introduced in the approximation ofthe right-hand side of (1.1). From (1.1) x (cid:48) ( t ) = (cid:90) [0 ,t ] µ ( ds ) f ( x ( t − s )) = (cid:90) [0 ,t ] µ ( ds ) f ( x ( t − s )) f ( x ( t )) f ( x ( t )) , t > . The error of our upper bound on the solution is proportional to the ratio f ( x ( t − s )) /f ( x ( t )) for s ∈ (0 , t ),or f ( x ( λt )) /f ( x ( t )) for λ ∈ (0 , f ◦ x ∈ RV ∞ ( β (1 + θ ) / (1 − β ))lim t →∞ f ( x ( λt )) f ( x ( t )) = λ β (1+ θ )1 − β =: γ ( β ) . When γ ( β ) is close to one, the solution of (1.1) is close to that of (2.4) and hence our estimate is sharp.However, γ ( β ) is decreasing and lim β ↑ γ ( β ) = 0. Thus the zero limit as β ↑ ii. )represents the fact that the solution of (2.4) increases much faster in β than the solution to (1.1), for afixed value of θ . 3. Results for Perturbed Volterra Equations
We now present a result which illustrates how our precise understanding of the asymptotics of solutionsof (1.1) can be applied to perturbed versions of the equation, such as (1.4). This result applies toperturbations of (1.1) that are of the same, or smaller, order of magnitude as solutions of the ordinarydifferential equation (2.4). Our assumptions on H guarantee that lim t →∞ x ( t ) = ∞ but this limit isno longer necessarily achieved monotonically and this is reflected in the added complexity of certaintechnical aspects of the proofs. The proofs of the results in this section are largely deferred to Section 5. Theorem 5.
Suppose the measure µ obeys (2.1) with M ∈ RV ∞ ( θ ) , θ ≥ and that f ∈ RV ∞ ( β ) , β ∈ [0 , . When β = 0 let f be asymptotically increasing and obey lim x →∞ f ( x ) = ∞ . Let x denote thesolution of (1.4) and suppose H ∈ C ((0 , ∞ ); (0 , ∞ )) . Then the following are equivalent: ( i. ) lim t →∞ x ( t ) F − ( t M ( t )) = ζ ∈ [ L, ∞ ) , ( ii. ) lim t →∞ H ( t ) F − ( t M ( t )) = λ ∈ [0 , ∞ ) , (3.1) where L = (cid:110) B (cid:16) θ, θβ − β (cid:17) / (1 − β ) (cid:111) / (1 − β ) , and moreover ζ = ζ β − β B (cid:16) θ, θβ − β (cid:17) + λ. (3.2)We notice that, when there is a sufficiently slowly growing forcing term H , λ = 0, and we recoverfrom (3.2) exactly the asymptotic behaviour of the unperturbed equation, given by (2.3). Also, in thelimit as λ → + , the rate of the unperturbed equation is recovered.Condition ( ii. ) on H in Theorem 5 does not cover the case when H is of larger magnitude than thesolution of the unperturbed equation (1.1) (or that of (2.4)). To deal with this case, we would like toknow the growth rate of the solution when lim t →∞ H ( t ) /F − ( t M ( t )) = ∞ . Insight into what happenscan be gained by sending λ → ∞ in Theorem 5. For λ >
0, from Theorem 5, we havelim t →∞ x ( t ) H ( t ) = ζ ( λ ) λ =: η ( λ ) , EMORY DEPENDENT GROWTH IN SUBLINEAR VOLTERRA DIFFERENTIAL EQUATIONS 7 where ζ depends on λ through (3.2). Since ζ = ζ ( λ ) is the unique positive solution of (3.2), η = η ( λ ) isthe unique positive solution of η = 1 + Kη β λ β − where K > λ –independent positive quantity K = 11 − β B (cid:16) θ, θβ − β (cid:17) . Clearly η ( λ ) > λ (cid:55)→ η ( λ ) is in C , by the implicit function theorem. Moreover, by implicitdifferentiation, η (cid:48) ( λ ) obeys η (cid:48) ( λ ) (cid:26) − β η ( λ ) − η ( λ ) (cid:27) = K ( β − η ( λ ) β λ β − . Therefore, as the bracket on the left–hand side is positive, λ (cid:55)→ η ( λ ) is decreasing. Hence for λ >
1, wehave η ( λ ) < Kη (1) β λ β − , so lim sup λ →∞ η ( λ ) ≤
1, and so η ( λ ) → λ → ∞ .In view of this discussion, one might expect that lim t →∞ H ( t ) /F − ( t M ( t )) = ∞ implies x ( t ) ∼ H ( t )as t → ∞ , or less precisely that sufficiently rapid growth in H forces x ( t ) to grow at the rate H ( t ).Therefore, it is natural to ask under what conditions we would have x ( t ) ∼ H ( t ) as t → ∞ . It isstraightforward to show that a necessary condition for lim t →∞ x ( t ) /H ( t ) = 1 is that lim t →∞ (cid:82) t M ( t − s ) f ( H ( s )) ds/H ( t ) = 0. This motivates the hypothesislim t →∞ M ( t ) (cid:82) t f ( H ( s )) dsH ( t ) = 0 , (3.3)and the following result. This result requires no monotonicity in H and as such allows for H to undergoconsiderable fluctuation, a point we will illustrate further in Section 4. Theorem 6.
Suppose the measure µ obeys (2.1) with M ∈ RV ∞ ( θ ) , θ ≥ and that f ∈ RV ∞ ( β ) , β ∈ [0 , . When β = 0 let f be asymptotically increasing and obey lim x →∞ f ( x ) = ∞ . Let H be a functionin C ((0 , ∞ ); (0 , ∞ )) satisfying (3.3) . Then the solution, x , of (1.1) obeys lim t →∞ x ( t ) /H ( t ) = 1 . When H is regularly varying at infinity the hypotheses of Theorems 5 and 6 align to give a completeclassification of the asymptotics (Corollary 1). However, assuming regular variation of H imposes con-siderable regularity constraints. In particular, H is then asymptotic to an increasing function and thisrestricts potential applications of Theorem 6 to stochastic functional differential equations. Corollary 1.
Let M ∈ RV ∞ ( θ ) , θ ≥ with lim t →∞ M ( t ) = ∞ . Suppose that f ∈ RV ∞ ( β ) , β ∈ [0 , .When β = 0 let f be asymptotically increasing and obey lim x →∞ f ( x ) = ∞ . If H ∈ RV ∞ ( α ) , α > , thenthe following are equivalent:(i.) lim t →∞ M ( t ) (cid:82) t f ( H ( s )) ds/H ( t ) = 0 ,(ii.) lim t →∞ H ( t ) /F − ( tM ( t )) = ∞ ,(iii.) lim t →∞ (cid:82) t M ( t − s ) f ( H ( s )) ds/H ( t ) = 0 . We exclude the case α = 0, because it is covered by Theorem 5 with λ = 0. Proof of Corollary 1.
By hypothesis f ◦ H ∈ RV ∞ ( αβ ) and M ∈ RV ∞ ( θ ). Hence (cid:82) t f ( H ( s )) ds ∈ RV ∞ (1 + αβ ) and Karamata’s Theorem yields (cid:82) t f ( H ( s )) ds ∼ t f ( H ( t )) / (1 + αβ ), as t → ∞ . Thus M ( t ) (cid:82) t f ( H ( s )) dsH ( t ) ∼ M ( t ) t f ( H ( t ))(1 + αβ ) H ( t ) , as t → ∞ . (3.4)Lemma 1 yields (cid:90) t M ( t − s ) f ( H ( s )) ds ∼ B (1 + αβ, θ ) t M ( t ) f ( H ( t )) , as t → ∞ . (3.5)Therefore (3.4) and (3.5) together yield (cid:82) t M ( t − s ) f ( H ( s )) dsH ( t ) ∼ (1 + αβ ) B (1 + αβ, θ ) M ( t ) (cid:82) t f ( H ( s )) dsH ( t ) , as t → ∞ . JOHN A. D. APPLEBY AND DENIS D. PATTERSON
Hence ( i. ) and ( iii. ) are equivalent. By Karamata’s Theorem, F ( H ( t )) ∼ H ( t ) / ((1 − β ) f ( H ( t ))) or f ( H ( t )) /H ( t ) ∼ / (1 − β ) F ( H ( t )), as t → ∞ . Hence (3.4) can be restated as M ( t ) (cid:82) t f ( H ( s )) dsH ( t ) ∼ M ( t ) t (1 + αβ )(1 − β ) F ( H ( t )) , as t → ∞ . Thus if ( i. ) holds, then lim t →∞ M ( t ) t/F ( H ( t )) = 0. This implies lim t →∞ F ( H ( t )) /M ( t ) t = ∞ and hencethat ( iii. ) holds, by the regular variation of F − . The reverse implications are all also true and ( i. ) and( ii. ) are equivalent. (cid:3) We state without proof a partial converse to Theorem 6 with H ∈ RV ∞ ( α ) , α >
0. The proof followsfrom Corollary 1 and estimation arguments similar to those used throughout this paper.
Theorem 7.
Suppose the measure µ obeys (2.1) with M ∈ RV ∞ ( θ ) , θ ≥ and that f ∈ RV ∞ ( β ) , β ∈ [0 , . When β = 0 let f be asymptotically increasing and obey lim x →∞ f ( x ) = ∞ . Let x denote thesolution of (1.4) , H ∈ C ((0 , ∞ ); (0 , ∞ )) ∩ RV ∞ ( α ) with α > . Then the following are equivalent: ( i. ) lim t →∞ M ( t ) (cid:82) t f ( H ( s )) dsH ( t ) = 0 , ( ii. ) lim t →∞ x ( t ) H ( t ) = 1 . While discussing the hypothesis that lim t →∞ H ( t ) /F − ( tM ( t )) = ∞ in the context of regular variationit is worth remarking that this hypothesis is also satisfied for H ∈ RV ∞ ( ∞ ), the so-called rapidly varyingfunctions (see [8, p.83]). If H ∈ RV ∞ ( ∞ ), then (3.3) holds and Theorem 6 can be applied; this fact isrecorded in the following corollary. Corollary 2.
Suppose the measure µ obeys (2.1) with M ∈ RV ∞ ( θ ) , θ ≥ and that f ∈ RV ∞ ( β ) , β ∈ [0 , . When β = 0 let f be asymptotically increasing and obey lim x →∞ f ( x ) = ∞ . Let x ( t ) denotethe solution of (1.4) and suppose H ∈ C ((0 , ∞ ); (0 , ∞ )) ∩ RV ∞ ( ∞ ) is asymptotically increasing. Then lim t →∞ x ( t ) /H ( t ) = 1 .Proof of Corollary 2. Define the function a by M ( t ) t f ( H ( t )) H ( t ) =: M ( t ) t a ( H ( t )) . Then a ∈ RV ∞ ( β −
1) and a ◦ H ∈ RV ∞ ( −∞ ). Hence lim t →∞ M ( t ) t a ( t ) = 0 and because f ◦ H isasymptotically increasing (3.3) holds. Applying Theorem 6 then yields the desired conclusion. (cid:3) Corollary 2 will also hold if H ∈ MR ∞ ( ∞ ), a sub-class of RV ∞ ( ∞ ) (see [8, p.68] for the definition ofMR ∞ ( ∞ )), because this guarantees that H is asymptotic to an increasing function (see [8, p.83]).4. Examples
The requisite proofs and justifications supporting the discussion in this section are deferred to Section6.4.1.
Application of Theorem 4.
The main attraction of Theorem 4 is that it largely reduces theasymptotic analysis of solutions of (1.1) to the computation, or asymptotic analysis, of the function F − . This is because under the appropriate hypotheses, Theorem 4 yields x ( t ) ∼ F − ( t M ( t )) (cid:26) − β B (cid:18) θ + 1 , θβ + 11 − β (cid:19)(cid:27) − β , as t → ∞ . In general, exact computation of F − in closed form is not possible. The following result provides theasymptotics of F − for a large class of f ∈ RV ∞ ( β ) for β ∈ [0 ,
1) using some classic results from thetheory of regular variation. It’s principal appeal is that it can be applied by calculating the limit of areadily–computed function which can be found directly in terms of f , without need for integration. EMORY DEPENDENT GROWTH IN SUBLINEAR VOLTERRA DIFFERENTIAL EQUATIONS 9
Proposition 2.
Suppose f ∈ RV ∞ ( β ) , β ∈ [0 , is continuous and that (cid:96) ( x ) := (cid:0) f ( x ) /x β (cid:1) − β obeys lim x →∞ (cid:96) ( x (cid:96) ( x )) (cid:96) ( x ) = 1 . (4.1) Then F ( x ) ∼ − β xf ( x ) , F − ( x ) ∼ (1 − β ) − β (cid:96) (cid:16) x − β (cid:17) x − β , as x → ∞ . The following examples illustrate the convenience of Proposition 2 in practice.
Example 8.
Suppose f ( x ) ∼ a x β log log( x α ) as x → ∞ with β ∈ [0 , a > α >
0. In this case (cid:96) ( x ) ∼ ( a log log ( x α )) − β , as x → ∞ . Hence¯ L ( x ) := (cid:96) ( x (cid:96) ( x )) (cid:96) ( x ) = ( a log log ( x α (cid:96) α ( x ))) − β ( a log log ( x α )) − β = (cid:16) log log (cid:16) x α a α − β { log log( x α ) } α − β (cid:17)(cid:17) − β (log log ( x α )) − β , x > . Let log log( x α ) = u to obtain¯ L (exp exp( u ) /α ) = (cid:32) log log( x α a α − β u α − β ) u (cid:33) − β = (cid:32) log( e u + log( a α − β ) + log( u α − β )) u (cid:33) − β = (cid:32) log( e u ) /u + log (cid:32) a α − β ) + log( u α − β ) e u (cid:33) /u (cid:33) − β =: (1 + G ( u )) − β , where lim u →∞ G ( u ) = 0. Therefore lim x →∞ ¯ L ( x ) := (cid:96) ( x (cid:96) ( x )) /(cid:96) ( x ) = 1 and applying Lemma 2 yields F − ( x ) ∼ (1 − β ) − β (cid:110) a log log (cid:16) x α − β (cid:17)(cid:111) − β x − β , as x → ∞ . This example is also valid with log log( x ) replaced by (cid:81) ni =1 log i − ( x ), where log i ( x ) = log log i − ( x ). Theproof in this case is essentially the same but the resulting formulae are rather convoluted. Example 9.
Suppose f ( x ) ∼ x β (2 + sin(log log( x ))) as x → ∞ , with β ∈ (0 , (cid:96) ( x ) ∼ (2 + sin(log log( x ))) − β , as x → ∞ . Hence ¯ L ( x ) := (cid:18) x (cid:96) ( x )))2 + sin(log log( x )) (cid:19) − β . Once more make the substitution log log( x ) = u to obtain¯ L (exp exp( u )) = u ) { u ) } − β ))2 + sin( u ) − β = u ) + log { u ) } − β ))2 + sin( u ) − β = (cid:18) u + log[1 + log { u ) } − β /e u ] (cid:19) u ) − β . Letting u → ∞ in the above yields lim u →∞ ¯ L (exp exp( u )) = 1 and hence Proposition 2 applies. Therefore F − ( x ) ∼ (1 − β ) − β (cid:110) (cid:16) log log( x − β ) (cid:17)(cid:111) − β x − β , as x → ∞ . Discrete Measures.
It may appear that our inclusion of a general measure µ in (1.1) and thehypothesis that the integral of µ is regularly varying are only compatible when µ is an absolutelycontinuous measure. The following proposition allows us to easily construct examples to show ourresults also cover a variety of equations involving discrete measures. Proposition 3.
Let x ≥ and δ x be the Dirac measure at x on ( R + , B ( R + ) . Suppose that θ ∈ (0 , and that µ ∈ RV ∞ ( θ − . Let τ > and µ ( ds ) = (cid:98) t/τ (cid:99) (cid:88) j =0 µ ( jτ ) δ jτ ( ds ) . (4.2) Hence M ( t ) = (cid:90) [0 ,t ] µ ( ds ) = (cid:98) t/τ (cid:99) (cid:88) j =0 µ ( jτ ) , (4.3) and M ∈ RV ∞ ( θ ) . Furthermore, M ( t ) ∼ ˜ M ( t ) := (cid:82) t ˜ µ ( s ) ds as t → ∞ , where ˜ µ ∈ RV ∞ ( θ − is any C , decreasing function such that µ ( s ) ∼ ˜ µ ( s ) as s → ∞ . The following examples illustrate, using Proposition 3, the application of some of our results toequations involving discrete measures.
Example 10.
Using the notation of Proposition 3, suppose that x (cid:48) ( t ) = (cid:98) t/τ (cid:99) (cid:88) j =0 µ ( jτ ) f ( x ( t − jτ )) + (cid:90) t µ ( s ) f ( x ( t − s )) ds, t > , (4.4)where m given by m ( E ) = (cid:82) E µ ( s ) ds for any Borel set E ⊂ [0 , ∞ ) is an absolutely continuous measure.Therefore µ ( ds ) = ∞ (cid:88) j =0 µ ( jτ ) δ jτ ( ds ) + µ ( s ) ds. (4.5)If µ ∈ RV ∞ ( θ −
1) and µ ∈ RV ∞ ( α ), then M ∈ RV ∞ (max( θ, α + 1)). Suppose that θ > α + 1 for thepurposes of this example. Thus M ( t ) ∼ (cid:80) (cid:98) t/τ (cid:99) j =0 µ ( jτ ) as t → ∞ and choose µ ( x ) ∼ log( x + 1) / (1 + x ) − θ =: ˜ µ ( x ) , θ ∈ (0 , , where the asymptotic relation holds as x → ∞ . Hence ˜ µ ∈ RV ∞ ( θ −
1) and it is straightforward to showthat ˜ M ( t ) = 1 θ ( t + 1) θ log( t + 1) − θ (( t + 1) θ − ∼ t θ θ log( t ) , as t → ∞ . Combining these facts, and Proposition 3, with Examples 8 and 9 we can provide the exact asymptoticsfor particular classes of solutions to (1.1).From Example 8 and (2.3), when µ ( ds ) is given by (4.5) and f ( x ) ∼ a x β log log( x α ) as x → ∞ , x ( t ) ∼ (cid:40) a t θ +1 log( t ) θ log log (cid:32) t α (1+ θ )1 − β log( t ) α − β θ α − β (cid:33) B (cid:18) θ + 1 , θβ − β (cid:19)(cid:41) − β ∼ (cid:26) a t θ +1 log( t ) θ log log (cid:16) t α (1+ θ )1 − β (cid:17) B (cid:18) θ + 1 , θβ − β (cid:19)(cid:27) − β , as t → ∞ . EMORY DEPENDENT GROWTH IN SUBLINEAR VOLTERRA DIFFERENTIAL EQUATIONS 11
Similarly, using Example 9, with f ( x ) ∼ x β (2 + sin(log log( x ))) as x → ∞ , x ( t ) ∼ (cid:40) t θ +1 log( t ) θ (cid:34) (cid:32) log log (cid:32) t θ − β log( t ) − β θ − β (cid:33)(cid:33)(cid:35) B (cid:18) θ + 1 , θβ − β (cid:19)(cid:41) − β ∼ (cid:26) t θ +1 log( t ) θ (cid:104) (cid:16) log log (cid:16) t θ − β (cid:17)(cid:17)(cid:105) B (cid:18) θ + 1 , θβ − β (cid:19)(cid:27) − β , as t → ∞ . Perturbed Equations and Application of Theorems 5 and 6.
Using a parametrized examplewe illustrate how the asymptotic behaviour of solutions of (1.4) can be classified using the range ofpossibilities covered by the results in Section 3.
Example 11.
For ease of exposition suppose that β ∈ (0 ,
1) and let f ( x ) = x β , x ≥ M ( t ) = (1 + t ) θ − , t ≥ H ( t ) = (1 + t ) α e γt − , t ≥ , with θ > α ∈ R , and γ ≥
0. Hence (1.4) becomes x ( t ) = x (0) + (cid:90) t ((1 + t − s ) θ − x ( s ) β ds + (1 + t ) α e γt − , t ≥ , with x (0) >
0. Therefore F − ( t M ( t )) ∼ (1 − β ) − β t θ +11 − β , as t → ∞ . Case ( i. ) : γ = 0 . In this case H ∈ RV ∞ ( α ) and H ( t ) F − ( t M ( t )) ∼ (1 − β ) β − t α − θ +11 − β , as t → ∞ . If α < ( θ + 1) / (1 − β ), then lim t →∞ H ( t ) /F − ( t M ( t )) = 0 and Theorem 5 yields the limitlim t →∞ x ( t ) /F − ( t M ( t )) = L, where L = (cid:110) B (cid:16) θ, θβ − β (cid:17) / (1 − β ) (cid:111) / (1 − β ) . If α = ( θ + 1) / (1 − β ), then lim t →∞ H ( t ) /F − ( t M ( t )) = (1 − β ) / ( β − =: λ and Theorem 5 giveslim t →∞ x ( t ) /F − ( t M ( t )) = ζ , where ζ satisfies (3.2).Finally, if α > ( θ + 1) / (1 − β ), then lim t →∞ H ( t ) /F − ( t M ( t )) = ∞ . Then, by Corollary 1, (3.3) holdsand Theorem 6 yields lim t →∞ x ( t ) /H ( t ) = 1. Case ( ii. ) : γ > . In this case H ∈ RV ∞ ( ∞ ) and Corollary 2 immediately gives lim t →∞ x ( t ) /H ( t ) = 1for all α ∈ R , β ∈ (0 ,
1) and θ > H is required to have some form monotonicity in the results of Section 3. When λ = 0 in Theorem 5 there is no monotonicity requirement on H but λ > H asymptoticto the monotone increasing function F − , modulo a constant. By contrast, Theorem 6 allows for large“fluctuations”, or irregular behaviour, in H ; the following examples illustrate this point. Example 12.
Suppose f ∈ RV ∞ ( β ) , β ∈ (0 , M ∈ RV ∞ ( θ ) , θ ≥ H ( t ) = (1+ t ) α (2 + sin( t )) − α >
0. From Karamata’s Theoremlim sup t →∞ M ( t ) (cid:82) t f ( H ( s )) dsH ( t ) = lim sup t →∞ M ( t ) (cid:82) t f ((1 + s ) α (2 + sin( s ) − ds (1 + t ) α (2 + sin( t )) − ≤ lim sup t →∞ (1 + (cid:15) ) M ( t ) (cid:82) t φ (3 s α ) dst α . Since M ( t ) (cid:82) t φ (3 s α ) dst α ∼ M ( t ) t f (3 t α )(1 + αβ ) t α , as t → ∞ , a sufficient condition for (3.3) to hold, and hence for Theorem 6 to apply, is α > (1 + θ ) / (1 − β ). Evenmore rapid variation in H is permitted; for example let H ( t ) = e t (2 + sin( t )) −
2. In this case asymptoticmonotonicity of f and the rapid variation of e t yieldlim sup t →∞ M ( t ) (cid:82) t f ( H ( s )) dsH ( t ) ≤ lim sup t →∞ M ( t ) t f (3 e t ) e t = 0 , and once more Theorem 6 applies to yield x ( t ) ∼ H ( t ) as t → ∞ , where x is the solution to (1.6). Byfixing f ( x ) = x β , we can immediately see that it is possible to capture more general types of exponentiallyfast oscillation in Theorem 6. Choose H ( t ) = e σ ( t ) t , where σ ( t ) obeys 0 < σ − ≤ σ ( t ) ≤ σ + < ∞ for all t ≥
0, for some constants σ − and σ + . Checking condition (3.3) yieldslim sup t →∞ M ( t ) (cid:82) t f ( H ( s )) dsH ( t ) ≤ lim sup t →∞ M ( t ) t e βσ + t e σ − t . The limit of the right hand side will be zero if σ − > βσ + .Finally we present an example of a H for which condition (3.3) fails to hold. This example illustratesthe limitations of our results by showing that when the exogenous perturbation exhibits rapid, irregulargrowth we are unable to capture the dynamics of the solution. This example is constructed by consideringan extremely ill-behaved perturbation with periodic fluctuations of exponential order. Example 13.
Choose f ( x ) = x β and H ( t ) ∼ e t (1+ αp ( t )) := H ∗ ( t ), as t → ∞ , with α ∈ (0 , β ∈ (0 , p a continuous 1 − periodic function such that max t ∈ [0 , p ( t ) = 1 and min t ∈ [0 , p ( t ) = −
1. Let t > n ( t ) ∈ N such that n ( t ) ≤ t < n ( t ) + 1. Then S ( t ) := (cid:90) t f ( H ∗ ( s )) ds = (cid:90) t e βs (1+ αp ( s )) ds = n ( t ) − (cid:88) j =0 (cid:90) j +1 j e βs (1+ αp ( s )) ds + (cid:90) tn ( t ) e βs (1+ αp ( s )) ds = n ( t ) − (cid:88) j =0 (cid:90) e β ( u + j )(1+ αp ( u )) du + (cid:90) t − n ( t )0 e β ( u + n ( t ))(1+ αp ( u )) du. Let I j := (cid:82) e β ( u + j )(1+ αp ( u )) du and S n := (cid:80) n − j =0 I j . Then S ( t ) = S n ( t ) = (cid:82) t − n ( t )0 e β ( u + n ( t ))(1+ αp ( u )) du. Hence S ( t ) ≤ S n ( t ) = (cid:82) e β ( u + n ( t ))(1+ αp ( u )) du = S n ( t )+1 . Thus S n ( t ) ≤ S ( t ) ≤ S n ( t )+1 . Now estimate I j as j → ∞ as follows. Letting c ( u ) = e u (1+ αp ( u )) β and d ( u ) = β (1 + αp ( u )) we have I j = (cid:82) c ( u ) e jd ( u ) du . By hypothesis 0 < c = min u ∈ [0 , c ( u ) ≤ max u ∈ [0 , c ( u ) ≤ c < ∞ . Hence c (cid:90) e jd ( u ) du ≤ I j ≤ c (cid:90) e jd ( u ) du, j ≥ . Therefore, for j ≥ c j (cid:18)(cid:90) e jd ( u ) du (cid:19) j ≤ I jj ≤ c j (cid:18)(cid:90) e jd ( u ) du (cid:19) j . For any continuous, non-negative function a : [0 , (cid:55)→ (0 , ∞ ), lim j →∞ (cid:16)(cid:82) a ( u ) j du (cid:17) /j = max u ∈ [0 , a ( u )and thus lim j →∞ I /jj = max u ∈ [0 , e d ( u ) . Thereforelim j →∞ j log I j = max u ∈ [0 , d ( u ) = β max u ∈ [0 , (1 + αp ( u )) = β (1 + α ) > . Since S n = (cid:80) n − j =0 I j , this gives us lim n →∞ log S n /n = β (1 + α ). Hencelim inf t →∞ t log S ( t ) ≥ lim inf t →∞ t log S n ( t ) = lim inf t →∞ n ( t ) log S n ( t ) n ( t ) t = β (1 + α ) . EMORY DEPENDENT GROWTH IN SUBLINEAR VOLTERRA DIFFERENTIAL EQUATIONS 13
An analogous calculation for the limit superior then yields lim t →∞ log S ( t ) /t = λ . Therefore, aslog H ∗ ( t ) /t = 1 + αp ( t ),lim sup t →∞ t log (cid:18) H ∗ ( t ) S ( t ) (cid:19) = lim sup t →∞ (1 + αp ( t )) − β (1 + α ) = (1 + α )(1 − β ) > . Hence lim sup t →∞ H ∗ ( t ) / (cid:82) t f ( H ∗ ( s )) ds = ∞ , and because H ( t ) ∼ H ∗ ( t ) as t → ∞ and f ∈ RV ∞ ( β ),we have lim sup t →∞ H ( t ) / (cid:82) t f ( H ( s )) ds = ∞ . Similarlylim inf t →∞ t log (cid:18) H ∗ ( t ) S ( t ) (cid:19) = lim inf t →∞ (1 + αp ( t )) − β (1 + α ) = 1 − β − α (1 + β ) . Choose α > (1 − β ) / (1+ β ) > − β − α (1+ β ) < t →∞ H ∗ ( t ) / (cid:82) t f ( H ∗ ( s )) ds =0. As above, this gives lim inf t →∞ H ( t ) / (cid:82) t f ( H ( s )) ds = 0. We remark that because the function t (cid:55)→ H ( t ) / (cid:82) t f ( H ( s )) ds is of exponential order, (3.3) is violated for any M ∈ RV ∞ ( θ ) , θ ∈ [0 , ∞ ).5. Proofs of Results
In the proofs that follow we will often choose to work with a monotone function approximating f .This monotone approximation will be denoted by φ . If f is regularly varying with a positive index thenThere exists φ ∈ C ((0 , ∞ ); (0 , ∞ )) ∩ C ( R + , (0 , ∞ )) such that f ( x ) ∼ φ ( x ) and φ (cid:48) ( x ) > x > . (5.1)by [8, Theorem 1.3.1 and Theorem 1.5.13]. It is immediate that if f is regularly varying and asymptoticto φ , then φ is also regularly varying with the same index. If f ∈ RV ∞ (0) we assume that a φ satisfying(5.1) exists since only a smooth, but not necessarily monotone, approximation is guaranteed in this case.The function F ( x ) is approximated by Φ( x ) := (cid:82) x du/φ ( u ) and Φ − is the inverse function of Φ. If f ( x ) ∼ φ ( x ) as x → ∞ it follows trivially that F ( x ) ∼ Φ( x ) and F − ( x ) ∼ Φ − ( x ), as x → ∞ .The proof of Theorem 4 is decomposed into the following lemmata, the first of which provides a preciseestimate on the asymptotics of the convolution of two regularly varying functions. Lemma 1.
Suppose a ∈ RV ∞ ( ρ ) and b ∈ RV ∞ ( σ ) , where ρ ≥ and σ ≥ , and lim t →∞ a ( t ) = ∞ . If σ = 0 let b be asymptotically increasing and obey lim t →∞ b ( t ) = ∞ . Then lim t →∞ (cid:82) t a ( s ) b ( t − s ) dst a ( t ) b ( t ) = (cid:90) λ ρ (1 − λ ) σ dλ =: B ( ρ + 1 , σ + 1) , where B denotes the Beta function.Proof. Let (cid:15), η ∈ (0 , ) be arbitrary. Define I ( t ) := (cid:90) t a ( s ) b ( t − s ) ds = (cid:90) (cid:15)t a ( s ) b ( t − s ) ds + (cid:90) (1 − η ) t(cid:15)t a ( s ) b ( t − s ) ds + (cid:90) t (1 − η ) t a ( s ) b ( t − s ) ds =: I ( t ) + I ( t ) + I ( t ) . (5.2)By making the substitution s = λtI ( t ) t a ( t ) b ( t ) = (cid:82) (1 − η ) t(cid:15)t a ( s ) b ( t − s ) dst a ( t ) b ( t ) = (cid:90) − η(cid:15) a ( λt ) a ( t ) b ( t (1 − λ )) b ( t ) dλ. By the Uniform Convergence Theorem for Regularly Varying Functions (see [8, Theorem 1.5.2]) it followsthat lim t →∞ I ( t ) t a ( t ) b ( t ) = (cid:90) − η(cid:15) λ ρ (1 − λ ) σ dλ. (5.3)Since both a and b are positive functions it is clear that I ( t ) ≥ I ( t ) and hencelim inf t →∞ I ( t ) t a ( t ) b ( t ) ≥ (cid:90) − η(cid:15) λ ρ (1 − λ ) σ dλ. Letting η and (cid:15) → + then yieldslim inf t →∞ I ( t ) t a ( t ) b ( t ) ≥ (cid:90) λ ρ (1 − λ ) σ dλ. (5.4)By hypothesis there exists an increasing, C function β such that b ( t ) /β ( t ) → t → ∞ . It followsthat there exists T > t ≥ T implies b ( t ) /β ( t ) ≤
2. Therefore with (cid:15) ∈ (0 , ), t ≥ T wehave that (1 − (cid:15) ) t ≥ T . Suppose t ≥ T and estimate as follows I ( t ) = (cid:90) (cid:15)t a ( s ) b ( t − s ) ds ≤ β ( t ) (cid:90) (cid:15)t a ( s ) ds = 2 β ( t ) (cid:15) t a ( (cid:15)t ) (cid:82) (cid:15)t a ( s ) ds(cid:15) t a ( (cid:15)t ) . Hence, for t ≥ T , I ( t ) t a ( t ) b ( t ) ≤ (cid:15) β ( t ) b ( t ) a ( (cid:15)t ) a ( t ) (cid:82) (cid:15)t a ( s ) ds(cid:15) t a ( (cid:15)t ) .a ∈ RV ∞ ( ρ ) implies that lim t →∞ a ( (cid:15)t ) /a ( t ) = (cid:15) ρ and similarly, by Karamata’s Theorem,lim t →∞ (cid:82) (cid:15)t a ( s ) ds/(cid:15) t a ( (cid:15)t ) = 1 / (1 + ρ ). Thuslim sup t →∞ I ( t ) t a ( t ) b ( t ) ≤ (cid:15) ρ +1 ρ . (5.5)Finally, consider I ( t ). By construction t ≥ T implies b ( t ) /β ( t ) ≤ b, β are continuous andpositive, with β bounded away from zero, sup ≤ t ≤ T b ( t ) /β ( t ) = max ≤ t ≤ T b ( t ) /β ( t ) := B < ∞ . Thusthere exists B > b ( t ) ≤ B β ( t ) for all t ≥
0. Therefore I ( t ) = (cid:90) t (1 − η ) t a ( s ) b ( t − s ) ds ≤ B (cid:90) t (1 − η ) t a ( s ) β ( t − s ) ds ≤ B β ( ηt ) (cid:90) t (1 − η ) t a ( s ) ds. Hencelim sup t →∞ I ( t ) t a ( t ) b ( t ) ≤ B lim sup t →∞ β ( ηt ) b ( t ) lim sup t →∞ (cid:82) t (1 − η ) t a ( s ) dst a ( t ) = B η σ lim sup t →∞ (cid:82) t (1 − η ) t a ( s ) dst a ( t ) . (5.6)The final limit on the right-hand side of (5.6) is calculated by once more calling upon the UniformConvergence Theorem for Regularly Varying Functionslim t →∞ (cid:82) t (1 − η ) t a ( s ) dst a ( t ) = lim t →∞ (cid:90) − η a ( λt ) a ( t ) dλ = (cid:90) − η λ ρ dλ. Returning to (5.6)lim sup t →∞ I ( t ) t a ( t ) b ( t ) ≤ B η σ (cid:90) − η λ ρ dλ = B η σ (cid:18) ρ + 1 − (1 − η ) ρ +1 ρ + 1 (cid:19) . (5.7)Therefore, combining (5.3), (5.5) and (5.7), we obtainlim sup t →∞ I ( t ) t a ( t ) b ( t ) ≤ (cid:15) ρ +1
11 + ρ + (cid:90) − η(cid:15) λ ρ (1 − λ ) σ dλ + B η σ (cid:90) − η λ ρ dλ. Letting η and (cid:15) → + in the above then yieldslim sup t →∞ I ( t ) t a ( t ) b ( t ) ≤ (cid:90) λ ρ (1 − λ ) σ dλ. (5.8)Combining (5.8) with (5.4) gives the desired conclusion. (cid:3) The proof of Theorem 4 now begins in earnest by proving a “rough” lower bound on the solutionwhich we will later refine. Lemmas 2, 3 and 4 are all proven under the same set of hypotheses and arepresented separately purely for readability and clarity.
EMORY DEPENDENT GROWTH IN SUBLINEAR VOLTERRA DIFFERENTIAL EQUATIONS 15
Lemma 2.
Suppose the measure µ obeys (2.1) with M ∈ RV ∞ ( θ ) , θ ≥ and that f ∈ RV ∞ ( β ) , β ∈ [0 , .If β = 0 , let f be asymptotically increasing and obey lim x →∞ f ( x ) = ∞ . Then the unique continuoussolution, x , of (1.1) obeys lim inf t →∞ x ( t ) F − ( t M ( t )) > . Proof.
Let (cid:15) ∈ (0 ,
1) be arbitrary. By hypothesis there exists φ such that (5.1) holds and hence thereexists x ( (cid:15) ) > f ( x ) > (1 − (cid:15) ) φ ( x ) for all x > x ( (cid:15) ). Furthermore, there exists T ( (cid:15) ) > t ≥ T implies x ( t ) > x ( (cid:15) ). Similarly, there exists T ( (cid:15) ) > M ( t ) > t ≥ T . Since M ∈ RV ∞ ( θ ), there exists a C function M such that for all (cid:15) ∈ (0 ,
1) there exists T ( (cid:15) ) > t ≥ T , M ( t ) > (1 − (cid:15) ) M ( t ). Let T := T + T + T . Hence, for t ≥ T , estimate as follows x (cid:48) ( t ) = (cid:90) [0 ,t − T ] µ ( ds ) f ( x ( t − s )) + (cid:90) ( t − T ,t ] µ ( ds ) f ( x ( t − s )) ≥ (1 − (cid:15) ) (cid:90) [0 ,t − T ] µ ( ds ) φ ( x ( t − s ))= (1 − (cid:15) ) (cid:90) [0 , ( t − T ) / µ ( ds ) φ ( x ( t − s )) + (1 − (cid:15) ) (cid:90) (( t − T ) / ,t − T ] µ ( ds ) φ ( x ( t − s )) ≥ (1 − (cid:15) ) (cid:90) [0 , ( t − T ) / µ ( ds ) φ ( x ( t − s )) ≥ (1 − (cid:15) ) M ( ( t − T )) φ (cid:0) x ( ( t + T )) (cid:1) . Since M ∈ RV ∞ ( θ ), lim t →∞ M (( t − T ) / /M ( t − T ) = 2 − θ . Thus there exists a positive constant C and a time ˜ T ≥ T such that x (cid:48) ( t ) ≥ C M ( t − T ) φ (cid:0) x ( ( t + T )) (cid:1) , for all t ≥ ˜ T . (5.9)Furthermore, since t ≥ ˜ T implies t − T > T , there exists C > x (cid:48) ( t ) ≥ C M ( t − T ) φ ( x (( t + T ) / , for all t ≥ ˜ T . (5.10)Now define the C , positive, increasing function ¯ M ( t ) := (cid:82) t M ( s ) ds for t ≥
0. Let α ( t ) := ¯ M − ( t ) + T , t ≥ ¯ M ( ˜ T ) . (5.11)For t ≥ ¯ M ( ˜ T ), α ( t ) ≥ α ( ¯ M ( ˜ T )) = ˜ T + T > ˜ T since α is increasing. Define ˜ x ( t ) := x ( α ( t )) for t ≥ ¯ M ( ˜ T ). Note that ˜ x ∈ C ([ ¯ M ( ˜ T ) , ∞ ); (0 , ∞ )) and α (cid:48) ( t ) = 1 /M ( ¯ M − ( t )). For t ≥ ¯ M ( ˜ T ), use(5.10) to compute˜ x (cid:48) ( t ) = α (cid:48) ( t ) x (cid:48) ( α ( t )) ≥ C M ( α ( t ) − T ) M ( ¯ M − ( t )) φ (cid:0) x ( ( α ( t ) + T )) (cid:1) = C φ (cid:0) x ( ( α ( t ) + T )) (cid:1) . (5.12)Define τ ( t ) = t − ¯ M (cid:0) ¯ M − ( t ) / (cid:1) > , for t ≥ ¯ M ( ˜ T ). It follows that ( α ( t ) + T ) / α ( t − τ ( t )). Hence,for t ≥ ¯ M ( ˜ T ) ˜ x (cid:48) ( t ) ≥ C φ (cid:0) x ( ( α ( t ) + T ) (cid:1) = C φ ( x ( α ( t − τ ( t ))) = C φ (˜ x ( t − τ ( t ))) . (5.13)For t ≥ ¯ M ( ˜ T ) it is straightforward to show, using the monotonicity of ¯ M , that τ ( t ) >
0. Using that¯ M ∈ RV ∞ ( θ + 1) we havelim t →∞ t − τ ( t ) t = lim t →∞ ¯ M (cid:0) ¯ M − ( t ) (cid:1) ¯ M (cid:0) ¯ M − ( t ) (cid:1) = lim t →∞ ¯ M (cid:0) ¯ M − ( t ) (cid:1) ¯ M (cid:0) ¯ M − ( t ) (cid:1) = (cid:18) (cid:19) θ +1 . It follows that there exists T > t ≥ T , t − τ ( t ) > − ( θ +2) t for all (cid:15) > T := max( T , ¯ M ( ˜ T )) we have, for t ≥ T ˜ x (cid:48) ( t ) ≥ C φ (˜ x ( qt )) , q = 2 − ( θ +2) ∈ (0 , . (5.14)The following estimates will be needed to define a lower comparison solution. Since φ ◦ Φ − is inRV ∞ ( β/ (1 − β )) we have lim x →∞ ( φ ◦ Φ − )( xq )( φ ◦ Φ − )( x ) = (cid:18) q (cid:19) β − β . Thus there exists x > x ≥ x ( φ ◦ Φ − )( xq )( φ ◦ Φ − )( x ) < (cid:18) q (cid:19) β − β . Next let T (cid:48) > x ( qT (cid:48) )) − x > T := max( T , T (cid:48) ) + 1. Then Φ(˜ x ( qT )) > Φ(˜ x ( qT (cid:48) )) > x . Define c := min C q β − β , Φ(˜ x ( qT )) − x T (1 − q ) , Φ(˜ x ( qT ))2 T (5.15)and d := cT − Φ(˜ x ( qT )) . (5.16)Then define x := cqT − d = Φ(˜ x ( qT )) − cT (1 − q ) > x . Note that d < /q − > x ≥ x , x/q + (1 /q − d < x/q . Hence for x ≥ x ( φ ◦ Φ − ) (cid:16) xq + (cid:16) q − (cid:17) d (cid:17) ( φ ◦ Φ − )( x ) ≤ ( φ ◦ Φ − ) (cid:16) xq (cid:17) ( φ ◦ Φ − )( x ) < (cid:18) q (cid:19) β − β . (5.17)Letting t = ( x + d ) /cq in (5.17) and noting that (5.15) implies C /c ≥ /q ) β/ (1 − β ) we have( φ ◦ Φ − )( ct − d )( φ ◦ Φ − )( cqt − d ) < (cid:18) q (cid:19) β − β < C c , for all t ≥ T . (5.18)Define the lower comparison solution, x − , by x − ( t ) = Φ − ( ct − d ) , t ≥ qT . (5.19)Then for t ∈ [ qT , T ], by the monotonicity of Φ − and (5.16), x − ( t ) ≤ x − ( T ) = Φ − ( cT − d ) = ˜ x ( qT ) ≤ ˜ x ( t ) . Also x − ( T ) = ˜ x ( qT ) < ˜ x ( T ), because ˜ x is increasing. Hence x − ( t ) < ˜ x ( t ) , t ∈ [ qT , T ] . (5.20)Next, since Φ( x − ( t )) = ct − d , for t ≥ T x (cid:48)− ( t ) = c (cid:0) φ ◦ Φ − (cid:1) ( ct − d ) = c φ ( x − ( t )) = cC φ ( x − ( t )) φ ( x − ( qt )) C φ ( x − ( qt )) . Now for t ≥ T , by (5.18) cC φ ( x − ( t )) φ ( x − ( qt )) = cC ( φ ◦ Φ − )( ct − d )( φ ◦ Φ − )( cqt − d ) < cC C c = 1 . Thus x (cid:48)− ( t ) < C φ ( x − ( qt )) , t ≥ T . (5.21)Recalling (5.14), ˜ x (cid:48) ( t ) ≥ C φ (˜ x ( qt )) for all t ≥ T > T . Then by (5.20) and (5.21), because φ isincreasing, ˜ x ( t ) > x − ( t ) for all t ≥ qT . To see this suppose there is a minimal t > T such that x − ( t ) = ˜ x ( t ). Thus x (cid:48)− ( t ) ≥ ˜ x (cid:48) ( t ) and x − ( t ) < ˜ x ( t ) for all t ∈ [ qT , t ). Then, since t > T and qt > qT , φ increasing yields˜ x (cid:48) ( t ) ≥ C φ (˜ x ( qt )) > C φ ( x − ( qt )) > x (cid:48)− ( t ) ≥ ˜ x (cid:48) ( t ) , a contradiction. Now, for t ≥ qT , ˜ x ( t ) > x − ( t ) = Φ − ( ct − d ). Hence for t ≥ qT x ( α ( t )) = ˜ x ( t ) > Φ − ( ct − d ) . From the definition of α , in (5.11), α − ( t ) = ¯ M ( t − T ) and therefore x ( t ) = ˜ x ( α − ( t )) > Φ − ( c α − ( t ) − d ) = Φ − ( c ¯ M ( t − T ) − d ) , ¯ M ( t − T ) > qT . EMORY DEPENDENT GROWTH IN SUBLINEAR VOLTERRA DIFFERENTIAL EQUATIONS 17
Hence, recalling that d < x ( t )) > c ¯ M ( t − T ) − d > c ¯ M ( t − T ) , ¯ M ( t − T ) > qT . (5.22)Note that for t > T , t/ < t − T . Since ¯ M is increasing this implies that ¯ M ( t/ ≤ ¯ M ( t − T ). Thus(5.22) implies lim inf t →∞ Φ( x ( t ))¯ M ( t ) ≥ lim inf t →∞ c ¯ M (cid:0) t (cid:1) ¯ M ( t ) = c − ( θ +1) > . By Karamata’s Theorem lim t →∞ ¯ M ( t ) /tM ( t ) = 1 / (1 + θ ) and thereforelim inf t →∞ Φ( x ( t )) t M ( t ) ≥ c (1 + θ ) 2 − ( θ +1) > . Finally, since Φ − ∈ RV ∞ (1 / (1 − β )) and M is asymptotic to M , we conclude thatlim inf t →∞ x ( t )Φ − ( t M ( t )) > , as required. (cid:3) Lemma 3.
Suppose the hypotheses of Lemma 2 hold. Then the unique continuous solution, x , of (1.1) obeys lim sup t →∞ F ( x ( t )) t M ( t ) ≤ − β B (cid:18) θ + 1 , θβ + 11 − β (cid:19) . Proof.
Once again let φ satisfying (5.1) obey f ( x ) /φ ( x ) < (1 + (cid:15) ) for all x > x ( (cid:15) ), for any (cid:15) > x ( (cid:15) ) >
0. Owing to the fact that lim t →∞ x ( t ) = ∞ there exists T ( (cid:15) ) such that t ≥ T ( (cid:15) ) implies x ( t ) > x ( (cid:15) ). Since lim t →∞ M ( t ) = ∞ there exists T ( (cid:15) ) such that M ( t ) > t ≥ T . Hence, forall t ≥ T , T ), (1.3) becomes x ( t ) φ ( x ( t )) ≤ x (0) φ ( x ( t )) + (cid:82) T M ( t − s ) f ( x ( s )) dsφ ( x ( t )) + (1 + (cid:15) ) t M ( t ) , (5.23)where the upper bound on the term (cid:82) tT M ( t − s ) φ ( x ( s )) ds was obtained by exploiting the fact that t (cid:55)→ x ( t ) and t (cid:55)→ M ( t ) are non-decreasing. By Karamata’s Theorem and the regular variation of φ , itis true that lim x →∞ (1 − β ) φ ( x )Φ( x ) /x = 1. Thus for all (cid:15) > x ( (cid:15) ) such thatΦ( x ) < (1 + (cid:15) ) x (1 − β ) φ ( x ) , for all x > x ( (cid:15) ) . Once more the divergence of x ( t ) yields the existence of a T ( (cid:15) ) such that x ( t ) > x ( (cid:15) ) for all t ≥ T ( (cid:15) ).Letting T = 2 max( T , T , T ) we obtainΦ( x ( t )) t M ( t ) < (1 + (cid:15) ) x ( t )(1 − β ) φ ( x ( t )) t M ( t ) , for all t ≥ T . Combining the above estimate with (5.23) yieldsΦ( x ( t )) t M ( t ) < (1 + (cid:15) ) x (0)(1 − β ) φ ( x ( t )) t M ( t ) + (1 + (cid:15) ) (cid:82) T M ( t − s ) f ( x ( s )) ds (1 − β ) φ ( x ( t )) t M ( t ) + (1 + (cid:15) ) (1 − β ) , t ≥ T ( (cid:15) ) . Hence, letting t → ∞ and then sending (cid:15) → + , we getlim sup t →∞ Φ( x ( t )) t M ( t ) ≤ − β . Since Φ − ∈ RV ∞ (1 / (1 − β )) the above estimate can be restated aslim sup t →∞ x ( t )Φ − ( t M ( t )) ≤ (1 − β ) β − < ∞ . We now seek to refine the “crude” upper bound on the growth of the solution obtained above. From theabove construction and Lemma 2 we may suppose thatlim sup t →∞ x ( t )Φ − ( t M ( t )) =: η ∈ (0 , ∞ ) . (5.24)From (5.24) it follows that for all (cid:15) > T ( (cid:15) ) > t ≥ T ( (cid:15) ), x ( t ) < ( η + (cid:15) )Φ − ( t M ( t )). By monotonicity of φ it follows that φ ( x ( t )) φ (Φ − ( t M ( t ))) < φ (cid:0) ( η + (cid:15) )Φ − ( t M ( t )) (cid:1) φ (Φ − ( t M ( t ))) , t ≥ T ( (cid:15) ) . Since φ ∈ RV ∞ ( β ) lim sup t →∞ φ ( x ( t )) φ (Φ − ( t M ( t ))) ≤ ( η + (cid:15) ) β . Thus for all (cid:15) > T ( (cid:15) ) > t ≥ T , φ ( x ( t )) < (1 + (cid:15) )( η + (cid:15) ) β φ (cid:0) Φ − ( t M ( t )) (cid:1) . Integrating this estimate yields (cid:90) tT M ( t − s ) φ ( x ( s )) ds ≤ (1 + (cid:15) )( η + (cid:15) ) β (cid:90) tT M ( t − s ) φ (cid:0) Φ − ( s M ( s )) (cid:1) ds, t ≥ T ( (cid:15) ) . (5.25)Since ( φ ◦ Φ − )( t M ( t )) ∈ RV ∞ ( β (1 + θ ) / (1 − β )) and M ∈ RV ∞ ( θ ), Lemma 1 can be applied to obtainlim t →∞ (cid:82) t M ( t − s ) φ (cid:0) Φ − ( s M ( s )) (cid:1) dst M ( t ) φ (Φ − ( t M ( t ))) = B (cid:18) θ + 1 , θβ + 11 − β (cid:19) . (5.26)Hence combining (5.25) and (5.26) yieldslim sup t →∞ (cid:82) tT M ( t − s ) φ ( x ( s )) dst M ( t ) φ (Φ − ( t M ( t ))) ≤ (1 + (cid:15) )( η + (cid:15) ) β B (cid:18) θ + 1 , θβ + 11 − β (cid:19) . Apply the above estimate to (1.3) as follows η = lim sup t →∞ x ( t )Φ − ( t M ( t )) ≤ lim sup t →∞ (cid:82) T M ( t − s ) f ( x ( s )) ds Φ − ( t M ( t )) + lim sup t →∞ (1 + (cid:15) ) (cid:82) tT M ( t − s ) φ ( x ( s )) ds Φ − ( t M ( t )) ≤ (1 + (cid:15) ) ( η + (cid:15) ) β B (cid:18) θ + 1 , θβ + 11 − β (cid:19) lim sup t →∞ t M ( t ) φ (cid:0) Φ − ( t M ( t )) (cid:1) Φ − ( t M ( t ))= (1 + (cid:15) ) ( η + (cid:15) ) β B (cid:18) θ + 1 , θβ + 11 − β (cid:19) lim sup x →∞ x φ (cid:0) Φ − ( x ) (cid:1) Φ − ( x ) . Letting (cid:15) → + and using Karamata’s Theorem to the remaining limit on the right-hand side η − β = lim sup y →∞ Φ( y ) φ ( y ) y B (cid:18) θ + 1 , θβ + 11 − β (cid:19) = 11 − β B (cid:18) θ + 1 , θβ + 11 − β (cid:19) , with y = Φ − ( x ) so that y → ∞ as x → ∞ . Thus η = lim sup t →∞ x ( t )Φ − ( t M ( t )) ≤ (cid:26) − β B (cid:18) θ + 1 , θβ + 11 − β (cid:19)(cid:27) − β . Using Φ ∈ RV ∞ (1 − β ) and Φ( x ) ∼ F ( x ) as x → ∞ the above upper bound can be reformulated aslim sup t →∞ F ( x ( t )) t M ( t ) ≤ − β B (cid:18) θ + 1 , θβ + 11 − β (cid:19) , which is the required estimate. (cid:3) EMORY DEPENDENT GROWTH IN SUBLINEAR VOLTERRA DIFFERENTIAL EQUATIONS 19
Lemma 4.
Suppose the hypotheses of Lemma 2 hold. Then the unique continuous solution, x , of (1.1) obeys lim inf t →∞ F ( x ( t )) t M ( t ) ≥ − β B (cid:18) θ + 1 , θβ + 11 − β (cid:19) . Proof.
By Lemma 2 and Lemma 3lim inf t →∞ x ( t )Φ − ( t M ( t )) =: η ∈ (0 , ∞ ) . Then for all (cid:15) ∈ (0 , η ) ∩ (0 ,
1) there exists T ( (cid:15) ) > t ≥ T η − (cid:15) < x ( t ) / Φ − ( t M ( t )).Since lim t →∞ M ( t ) = ∞ there exists T such that M ( t ) > t ≥ T . Hence x ( t ) > ( η − (cid:15) )Φ − ( t M ( t )) , t ≥ T := max( T , T ) . (5.27)Using monotonicity and regular variation of φ it follows from (5.27) thatlim inf t →∞ φ ( x ( t ))( φ ◦ Φ − )( t M ( t )) ≥ ( η − (cid:15) ) β . Now, because φ ( x ) ∼ f ( x ) as x → ∞ , for all (cid:15) ∈ (0 , η ) ∩ (0 ,
1) there exists T ( (cid:15) ) > f ( x ( t )) > (1 − (cid:15) ) φ ( x ( t )) > (1 − (cid:15) ) ( η − (cid:15) ) β ( φ ◦ Φ − )( t M ( t )) , t ≥ T ( (cid:15) ) . Integration then yields (cid:90) t M ( t − s ) f ( x ( s )) ds > (1 − (cid:15) ) ( η − (cid:15) ) β (cid:90) tT M ( t − s )( φ ◦ Φ − )( s M ( s )) ds. Hence, as in the proof of Lemma 3, applying Lemma 1 giveslim inf t →∞ (cid:82) t M ( t − s ) f ( x ( s )) dst M ( t ) ( φ ◦ Φ − )( t M ( t )) ≥ (1 − (cid:15) ) ( η − (cid:15) ) β B (cid:18) θ + 1 , θβ + 11 − β (cid:19) . (5.28)Now apply the estimate from (5.28) to (1.3) as follows η = lim inf t →∞ x ( t )Φ − ( t M ( t )) ≥ lim inf t →∞ (cid:82) t M ( t − s ) f ( x ( s )) ds Φ − ( t M ( t ))= (1 − (cid:15) ) ( η − (cid:15) ) β B (cid:18) θ + 1 , θβ + 11 − β (cid:19) lim inf t →∞ t M ( t ) ( φ ◦ Φ − )( t M ( t ))Φ − ( t M ( t ))= (1 − (cid:15) ) ( η − (cid:15) ) β B (cid:18) θ + 1 , θβ + 11 − β (cid:19) lim inf x →∞ x φ (cid:0) Φ − ( x ) (cid:1) Φ − ( x ) . The limit of the final term on the right-hand side is 1 / (1 − β ) by Karamata’s Theorem and sending (cid:15) → + yields η = η β − β B (cid:18) θ + 1 , θβ + 11 − β (cid:19) . Hence lim inf t →∞ x ( t ) F − ( t M ( t )) ≥ (cid:26) − β B (cid:18) θ + 1 , θβ + 11 − β (cid:19)(cid:27) − β . Since F ∈ RV ∞ (1 − β ) this can be rewritten in the formlim inf t →∞ F ( x ( t )) t M ( t ) ≥ − β B (cid:18) θ + 1 , θβ + 11 − β (cid:19) , which is the desired bound. (cid:3) As with Theorem 4, the proof of Theorem 5 is split into a series of lemmata. A final consolidatingargument then establishes the result as stated in Section 2.
Lemma 5.
Suppose the measure µ obeys (2.1) with M ∈ RV ∞ ( θ ) , θ ≥ and that f ∈ RV ∞ ( β ) , β ∈ [0 , .If β = 0 , let f be asymptotically increasing and obey lim x →∞ f ( x ) = ∞ . Let x ( t ) denote the uniquecontinuous solution of (1.4) and suppose H ∈ C ((0 , ∞ ); (0 , ∞ )) . Then lim inf t →∞ x ( t ) F − ( t M ( t )) ≥ L := (cid:26) − β B (cid:18) θ, θβ − β (cid:19)(cid:27) − β > . (5.29) Proof.
With (cid:15) ∈ (0 ,
1) arbitrary and T ( (cid:15) ) and T ( (cid:15) ) defined as in Lemma 2, (1.4) admits the initial lowerestimate x ( t ) > x (0) + H ( t ) + (1 − (cid:15) ) (cid:90) tT M ( t − s ) φ ( x ( s )) ds, t ≥ T ( (cid:15) ) := T ( (cid:15) ) + T ( (cid:15) ) . Letting y ( t ) = x ( t + T ) and noting that H ( t ) > t > y ( t ) > x (0) + (1 − (cid:15) ) (cid:90) t + TT M ( t + T − s ) φ ( x ( s )) ds = x (0) + (1 − (cid:15) ) (cid:90) t M ( t − u ) φ ( x ( u + T )) du = x (0) + (1 − (cid:15) ) (cid:90) t M ( t − u ) φ ( y ( u )) du, t ≥ T ( (cid:15) ) . Now consider the comparison equation defined by x (cid:48) (cid:15) ( t ) = (1 − (cid:15) ) (cid:90) [0 ,t ] µ ( ds ) φ ( x (cid:15) ( t − s )) , t > , x (cid:15) (0) = x (0) / . (5.30)In contrast to (1.4), the solution to (5.30) will be non-decreasing. Integrating (5.30) using Fubini’sTheorem yields x (cid:15) ( t ) = x (0) / − (cid:15) ) (cid:90) t M ( t − u ) φ ( x (cid:15) ( u )) du, t ≥ . By construction x (cid:15) ( t ) < y ( t ) = x ( t + T ) for all t ≥
0, or x ( t ) > x (cid:15) ( t − T ) for all t ≥ T . Applying Theorem4 to x (cid:15) then yields lim t →∞ F ( x (cid:15) ( t )) t M (cid:15) ( t ) = 11 − β B (cid:18) θ, θβ − β (cid:19) , where M (cid:15) ( t ) = (1 − (cid:15) ) M ( t ). Hencelim t →∞ F ( x (cid:15) ( t )) t M ( t ) = 1 − (cid:15) − β B (cid:18) θ, θβ − β (cid:19) . Therefore lim inf t →∞ F ( x ( t )) t M ( t ) ≥ lim inf t →∞ F ( x (cid:15) ( t − T )) t M ( t ) = lim inf t →∞ F ( x (cid:15) ( t − T ))( t − T ) M ( t − T ) ( t − T ) M ( t − T ) t M ( t )= 1 − (cid:15) − β B (cid:18) θ, θβ − β (cid:19) , where the final equality follows from the trivial fact that t − T ∼ t as t → ∞ and noting that M preservesasymptotic equivalence because M ∈ RV ∞ ( θ ). Finally, letting (cid:15) → + and using the regular variation of F − yields lim inf t →∞ x ( t ) F − ( t M ( t )) ≥ (cid:26) − β B (cid:18) θ, θβ − β (cid:19)(cid:27) − β = L, which finishes the proof. (cid:3) Lemma 6.
Suppose the hypotheses of Lemma 5 hold and lim t →∞ H ( t ) /F − ( t M ( t )) = λ ∈ [0 , ∞ ) . Then,with x denoting the unique continuous solution of (1.4) , lim sup t →∞ x ( t ) F − ( t M ( t )) ≤ U := (cid:18) λL β + 11 − β (cid:19) − β , (5.31) where L is defined by (5.29) . EMORY DEPENDENT GROWTH IN SUBLINEAR VOLTERRA DIFFERENTIAL EQUATIONS 21
Proof.
We begin by constructing a monotone comparison solution which will majorise the solution of(1.4) and to which Lemma 5 can be applied. Let (cid:15) ∈ (0 ,
1) be arbitrary and define T ( (cid:15) ) and T ( (cid:15) ) as inthe proof of Lemma 3.By hypothesis lim t →∞ H ( t ) /F − ( t M ( t )) = λ ∈ [0 , ∞ ) and so there exists a T ( (cid:15) ) > t ≥ T ( (cid:15) ) implies H ( t ) < ( λ + (cid:15) )Φ − ( t M ( t )). M ∈ RV ∞ ( θ ) implies there exists M ∈ C asymptotic to M and T ( (cid:15) ) > T such that M ( t ) < (1 + (cid:15) ) M ( t ) for all t ≥ T . For t ≥ T , because Φ − is increasing,Φ − ( t M ( t )) < Φ − ( t (1 + (cid:15) ) M ( t )) and since Φ − ∈ RV ∞ (1 / (1 − β )) there exists T ∗ > T such thatΦ − ( t M ( t )) < (1 + (cid:15) ) (2 − β ) / (1 − β ) Φ − ( t M ( t )) for all t ≥ T ∗ .For notational convenience define the quantity (cid:15) ∗ by letting (1 + (cid:15) ∗ ) := (1 + (cid:15) ) (2 − β ) / (1 − β ) ; note that(1 + (cid:15) ∗ ) → (cid:15) → + . Defining T (cid:48) := T ∗ + T + T , we have the estimate x ( t ) < x (0) + H ( t ) + (cid:90) T (cid:48) M ( t − s ) f ( x ( s )) ds + (1 + (cid:15) ) (cid:90) tT (cid:48) M ( t − s ) φ ( x ( s )) ds ≤ x (0) + H ( t ) + M ( t ) T (cid:48) F ∗ + (1 + (cid:15) ) (cid:90) tT (cid:48) M ( t − s ) φ ( x ( s )) ds< x (0) + ( λ + (cid:15) )(1 + (cid:15) ∗ )Φ − ( t M ( t )) + (1 + (cid:15) ) M ( t ) T (cid:48) F ∗ + (1 + (cid:15) ) (cid:90) tT (cid:48) M ( t − s ) φ ( x ( s )) ds, (5.32)for all t ≥ T (cid:48) and where F ∗ := max ≤ s ≤ T (cid:48) f ( x ( s )). Now define the constant x ∗ := max ≤ s ≤ T (cid:48) x ( s ) andthe function¯ H ( t ) := ( λ + (cid:15) )(1 + (cid:15) ∗ )Φ − ( t M ( t )) + (1 + (cid:15) ) M ( t ) T (cid:48) F ∗ − ( λ + (cid:15) )(1 + (cid:15) ∗ ) , t ≥ . Since Φ − (0) = 1 and M (0) = 0, H (0) = 0 and by construction H ∈ C ((0 , ∞ ); (0 , ∞ )). The initialupper estimate (5.32) motivates the definition of the following upper comparison equation: y (cid:48) (cid:15) ( t ) := ¯ H (cid:48) ( t ) + (1 + (cid:15) ) (cid:90) [0 ,t ] µ ( ds ) φ ( y (cid:15) ( t − s )) ds, t ≥ , y (cid:15) (0) = x (0) + x ∗ + ( λ + (cid:15) )(1 + (cid:15) ∗ ) . Integration using Fubini’s theorem quickly shows that y (cid:15) ( t ) = x (0) + x ∗ + ( λ + (cid:15) )(1 + (cid:15) ∗ ) + ¯ H ( t ) + (1 + (cid:15) ) (cid:90) t M ( t − s ) φ ( y (cid:15) ( s )) ds, t ≥ . Since y (cid:15) ( t ) is non-decreasing it is immediately clear that x ( t ) ≤ y (cid:15) ( t ) for all t ∈ [0 , T (cid:48) ]. A simple time ofthe first breakdown argument using the estimate (5.32) then yields that x ( t ) ≤ y (cid:15) ( t ) for all t ≥
0. Wenow compute an explicit upper bound on lim sup t →∞ y (cid:15) ( t ) /F − ( t M ( t )). Monotonicity readily yields y (cid:15) ( t ) ≤ x (0) + x ∗ + ( λ + (cid:15) )(1 + (cid:15) ∗ )Φ − ( t M ( t )) + (1 + (cid:15) ) M ( t ) T (cid:48) F ∗ + (1 + (cid:15) ) M ( t ) t φ ( y (cid:15) ( t )) , t ≥ . Hence, with C ( t ) suitably defined, y (cid:15) ( t ) t M ( t ) φ ( y (cid:15) ( t )) ≤ C ( t ) + ( λ + (cid:15) )(1 + (cid:15) ∗ )Φ − ( t M ( t )) t M ( t ) φ ( y (cid:15) ( t )) + (1 + (cid:15) ) , t ≥ . A short calculation reveals that lim t →∞ C ( t ) = 0. By Karamata’s Theorem there exists a T ( (cid:15) ) suchthatΦ( y (cid:15) ( t )) t M ( t ) < (1 + (cid:15) ) C ( t )1 − β + (1 + (cid:15) )( λ + (cid:15) )(1 + (cid:15) ∗ )Φ − ( t M ( t ))(1 − β ) t M ( t ) φ ( y (cid:15) ( t )) + (1 + (cid:15) ) − β , t ≥ T := T + T (cid:48) . (5.33)By applying Lemma 5 to y (cid:15) we conclude thatlim inf t →∞ y (cid:15) ( t )Φ − ( t M ( t )) =: L ∈ (0 , ∞ ] . If L ∈ (0 , ∞ ) then there exists a T ( (cid:15) ) such that for all t ≥ T := T + T Φ( y (cid:15) ( t )) t M ( t ) < (1 + (cid:15) ) C ( t )1 − β + (1 + (cid:15) )( λ + (cid:15) )(1 + (cid:15) ∗ )Φ − ( t M ( t ))(1 − β ) t M ( t ) φ ((1 − (cid:15) ) L Φ − ( t M ( t ))) + (1 + (cid:15) ) − β< (1 + (cid:15) ) C ( t )1 − β + (1 + (cid:15) )( λ + (cid:15) )(1 + (cid:15) ∗ )Φ − ( t M ( t ))(1 − β ) t M ( t ) (1 − (cid:15) ) β L β φ (Φ − ( t M ( t ))) + (1 + (cid:15) ) − β . (5.34)By Karamata’s Theorem the following asymptotic equivalence holds(1 − β ) t M ( t ) φ (cid:0) Φ − ( t M ( t )) (cid:1) ∼ Φ − ( t M ( t )) as t → ∞ . Therefore taking the limit superior across (5.34) yieldslim sup t →∞ Φ( y (cid:15) ( t )) t M ( t ) ≤ (1 + (cid:15) )( λ + (cid:15) )(1 + (cid:15) ∗ )(1 − (cid:15) ) β L β + (1 + (cid:15) ) − β . By letting (cid:15) → + and using the regular variation of Φ − lim sup t →∞ x ( t )Φ ( t M ( t )) ≤ (cid:18) λL β + 11 − β (cid:19) − β =: U. If L := lim inf t →∞ y (cid:15) ( t ) / Φ − ( t M ( t )) = ∞ the above construction will yieldlim sup t →∞ y (cid:15) ( t ) / Φ − ( t M ( t )) < ∞ , a contradiction. Hence L ∈ (0 , ∞ ) and the claim is proven. (cid:3) Lemma 7.
Suppose β ∈ [0 , , λ ∈ [0 , ∞ ) and consider the iterative scheme defined by x n +1 = g ( x n ) := x βn − β B (cid:18) θ, θβ − β (cid:19) + λ, n ≥ x ∈ [ L, C ∗ ] , (5.35) with L defined by (5.29) , U defined by (5.31) and C ∗ := max (cid:18) U, L + λ − β (cid:19) . (5.36) Then there exists a unique x ∞ ∈ [ L, C ∗ ] such that lim n →∞ x n = x ∞ .Proof. By inspection, g ∈ C ([ L, ∞ ); (0 , ∞ )). We calculate as follows g (cid:48) ( x ) = β − β x β − B (cid:18) θ, θβ − β (cid:19) > , x > , and similarly g (cid:48)(cid:48) ( x ) = − βx β − B (cid:18) θ, θβ − β (cid:19) < , x > . Therefore g (cid:48) ( L ) = β > g (cid:48) ( x ) > x > L and | g (cid:48) ( x ) | ≤ β < x ∈ [ L, ∞ ). Since g is monotoneincreasing it is sufficient check that g maps [ L, C ∗ ] to [ L, C ∗ ] as follows. Firstly, g ( L ) = L β − β B (cid:18) θ, θβ − β (cid:19) + λ = L + λ ∈ [ L, C ∗ ] . (5.37)By the Mean Value Theorem there exists ξ ∈ [ L, C ∗ ] such that g ( C ∗ ) − g ( L ) C ∗ − L = g (cid:48) ( ξ ) ≤ β. Therefore g ( C ∗ ) ≤ β ( C ∗ − L ) + g ( L ) and thus a sufficient condition for g ( C ∗ ) ≤ C ∗ is β ( C ∗ − L ) + g ( L ) ≤ C ∗ or C ∗ ≥ ( g ( L ) − Lβ ) / (1 − β ) = L + λ/ (1 − β ), using (5.37). Thus with C ∗ as defined in (5.36), g : [ L, C ∗ ] → [ L, C ∗ ]. Hence (5.35) has a unique fixed point in [ L, C ∗ ] and the claim follows. (cid:3) With the preceding auxiliary results proven we are now in a position to supply the proof of Theorem5, as promised.
EMORY DEPENDENT GROWTH IN SUBLINEAR VOLTERRA DIFFERENTIAL EQUATIONS 23
Proof of Theorem 5.
Suppose that ( ii. ) holds, or that lim t →∞ H ( t ) /F − ( t M ( t )) = λ ∈ [0 , ∞ ). The ideahere is to combine the crude bounds on the solution from Lemmas 5 and 6 with a fixed point argumentbased on Lemma 7 to complete the proof that ( ii. ) implies ( i. ). We compute lim sup t →∞ x ( t ) /F − ( t M ( t ))in detail only as the calculation of the corresponding limit inferior proceeds in an analogous manner. Tobegin make the following induction hypothesis( H n ) lim sup t →∞ x ( t )Φ − ( M t ) ≤ ζ n , ζ n +1 := ζ βn − β B (cid:18) θ, θβ − β (cid:19) + λ, n ≥ , and choose ζ := U . ( H ) is true by Lemma 6. Suppose that ( H n ) holds. Thus there exists T ( (cid:15) ) > x ( t ) < ( ζ n + (cid:15) )Φ − ( t M ( t )) for all t ≥ T . Hence φ ( x ( t )) φ (Φ − ( t M ( t ))) < φ (( ζ n + (cid:15) )Φ − ( M t )) φ (Φ − ( t M ( t ))) , t ≥ T. The regular variation of φ thus yields lim sup t →∞ φ ( x ( t )) /φ (Φ − ( t M ( t ))) ≤ ( ζ n + (cid:15) ) β . Therefore thereexists a T ( (cid:15) ) > t ≥ T implies f ( x ( t )) < (1 + (cid:15) )[( ζ n + (cid:15) ) β + (cid:15) ] φ (Φ − ( t M ( t ))). From (1.6)lim sup t →∞ x ( t )Φ − ( t M ( t )) = lim sup t →∞ (cid:82) t M ( t − s ) f ( x ( s )) ds Φ − ( t M ( t )) + lim t →∞ H ( t )Φ − ( t M ( t )) . Using the upper bound derived from our induction hypothesis this becomeslim sup t →∞ x ( t ) F − ( t M ( t )) ≤ (1 + (cid:15) )[( ζ n + (cid:15) ) β + (cid:15) ] lim sup t →∞ (cid:82) tT M ( t − s ) φ (Φ − ( s M ( s )))Φ − ( t M ( t )) + λ. Applying Karamata’s Theorem and Lemma 1lim sup t →∞ x ( t ) F − ( t M ( t )) ≤ (1 + (cid:15) )[( ζ n + (cid:15) ) β + (cid:15) ] lim sup t →∞ (cid:82) tT M ( t − s ) φ (Φ − ( s M ( s )))(1 − β ) t M ( t ) φ (Φ − ( t M ( t ))) + λ = (1 + (cid:15) )[( ζ n + (cid:15) ) β + (cid:15) ]1 − β B (cid:18) θ, θβ − β (cid:19) + λ. Letting (cid:15) → + yieldslim sup t →∞ x ( t ) F − ( t M ( t )) ≤ ζ β − β B (cid:18) θ, θβ − β (cid:19) + λ = ζ n +1 , proving the induction hypothesis ( H n +1 ). Hence ( H n ) holds for all n , or Hencelim sup t →∞ x ( t ) F − ( t M ( t )) ≤ ζ n , for all n ≥ . By Lemma 7, lim n →∞ ζ n = ζ , where ζ is the unique solution in [ L, U ] of the “characteristic” equation(3.2). Thus lim sup t →∞ x ( t ) F − ( M t ) ≤ ζ. In the case of the corresponding limit inferior the only modification is to the induction hypothesis, take ζ := L , and the argument then proceeds as above to yield lim inf t →∞ x ( t ) /F − ( M t ) ≥ ζ, completingthe proof.Now suppose that ( i. ) holds, or that lim t →∞ x ( t ) /F − ( t M ( t )) = ζ ∈ [ L, ∞ ). It follows that thereexists T ( (cid:15) ) > t ≥ T , φ (( ζ − (cid:15) )Φ − ( t M ( t ))) < φ ( x ( t )) < φ (( ζ + (cid:15) )Φ − ( t M ( t ))) . Hence for t ≥ T (cid:90) tT M ( t − s ) φ (( ζ − (cid:15) )Φ − ( s M ( s ))) ds ≤ (cid:90) tT M ( t − s ) φ ( x ( s )) ds ≤ (cid:90) tT M ( t − s ) φ (( ζ + (cid:15) )Φ − ( s M ( s ))) ds. Using the regular variation of φ the above estimate can be reformulated as( ζ − (cid:15) ) β (cid:82) tT M ( t − s ) φ (Φ − ( s M ( s ))) ds Φ − ( t M ( t )) ≤ (cid:82) tT M ( t − s ) φ ( x ( s )) ds Φ − ( t M ( t )) ≤ ( ζ + (cid:15) ) β (cid:82) tT M ( t − s ) φ (( ζ + (cid:15) )Φ − ( s M ( s ))) ds Φ − ( t M ( t )) , t ≥ T . Using Lemma 1 and letting (cid:15) → + thus yieldslim t →∞ (cid:82) t M ( t − s ) φ ( x ( s )) ds Φ − ( t M ( t )) = ζ β − β B (cid:18) θ, θβ − β (cid:19) . Therefore assuming ( i. ) and taking the limit across (1.6) we obtain ζ = ζ β − β B (cid:18) θ, θβ − β (cid:19) + lim t →∞ H ( t )Φ − ( t M ( t )) , as claimed. (cid:3) We now give the proof of Theorem 6 in which the perturbation is large. The reader will note that thisproof makes much less use of properties of regular varying functions: in fact, we establish the asymptoticresult by observing that a key functional of the solution is well approximated by a linear non–autonomousdifferential inequality.
Proof of Theorem 6.
As always (cid:15) ∈ (0 ,
1) is arbitrary. From (5.1) there exists a φ such thatlim x →∞ f ( x ) /φ ( x ) = 1 , lim x →∞ x φ (cid:48) ( x ) /φ ( x ) = β (see e.g., [8, Theorem 1.3.3]). Therefore there exists x ( (cid:15) ) > f ( x ) < (1 + (cid:15) ) φ ( x ) for all x ≥ x ( (cid:15) ) and x ( (cid:15) ) such that φ (cid:48) ( x ) < ( β + (cid:15) ) φ ( x ) /x for all x ≥ x ( (cid:15) ). Similarly, since lim t →∞ x ( t ) = ∞ ,there exists T ( (cid:15) ) > x ( t ) > max( x ( (cid:15) ) , x ( (cid:15) )) for all t ≥ T ( (cid:15) ). The regular variation of M means that there exists a non-decreasing function M ∈ C and T ( (cid:15) ) > − (cid:15) ) M ( t )
T > T ( (cid:15) ), max ≤ s ≤ T M ( s ) < (1 + (cid:15) ) M ( T ) < (1 + (cid:15) ) M ( t ). Hence x ( t ) < x (0) + H ( t ) + (cid:90) T M ( t − s ) f ( x ( s )) ds + (1 + (cid:15) ) M ( t ) (cid:90) tT φ ( x ( s )) ds, t ≥ T. For t ≥ T > T , max ≤ s ≤ T M ( t − s ) = max t − T ≤ u ≤ t M ( u ) ≤ max ≤ u ≤ t M ( u ) < (1 + (cid:15) ) M ( t ). Thus, for t ≥ T , x ( t ) < x (0) + H ( t ) + (1 + (cid:15) ) M ( t ) (cid:90) T f ( x ( s )) ds + (1 + (cid:15) ) M ( t ) (cid:90) tT φ ( x ( s )) ds. (5.38)For t ∈ [ T, T ], x ( t ) ≤ max s ∈ [0 , T ] x ( s ) := x ∗ ( (cid:15) ). Combining this with (5.38) x ( t ) < x ∗ ( (cid:15) ) + H ( t ) + (1 + (cid:15) ) M ( t ) x ∗ ( (cid:15) ) + (1 + (cid:15) ) M ( t ) (cid:90) tT φ ( x ( s )) ds, t ≥ T, (5.39)where x ∗ ( (cid:15) ) := (cid:82) T f ( x ( s )) ds . Define for t ≥ TH (cid:15) ( t ) := x ∗ ( (cid:15) ) + H ( t ) + (1 + (cid:15) ) M ( t ) x ∗ ( (cid:15) ) . (5.40)Note that by construction lim t →∞ H (cid:15) ( t ) /H ( t ) = 1. Consolidating (5.39) and (5.40) we have x ( t ) < H (cid:15) ( t ) + (1 + (cid:15) ) M ( t ) (cid:90) tT φ ( x ( s )) ds, t ≥ T. (5.41)By defining I (cid:15) ( t ) := (cid:90) tT φ ( x ( s )) ds, t ≥ T. we can formulate an advantageous auxiliary differential inequality as follows. Since x is continuous and φ ∈ C (0 , ∞ ), I (cid:48) (cid:15) ( t ) = φ ( x ( t )) , t ≥ T . Moreover, lim t →∞ I (cid:15) ( t ) = ∞ . By (5.41) I (cid:48) (cid:15) ( t ) = φ ( x ( t )) < φ (cid:0) H (cid:15) ( t ) + (1 + (cid:15) ) M ( t ) I (cid:15) ( t ) (cid:1) , t ≥ T. (5.42)By the Mean Value Theorem, for each t ≥ T , there exists ξ (cid:15) ( t ) ∈ [0 ,
1] such that φ (cid:0) H (cid:15) ( t ) + (1 + (cid:15) ) M ( t ) I (cid:15) ( t ) (cid:1) = φ ( H (cid:15) ) + φ (cid:48) (cid:0) H (cid:15) ( t ) + ξ (cid:15) ( t )(1 + (cid:15) ) M ( t ) I (cid:15) ( t ) (cid:1) (1 + (cid:15) ) M ( t ) I (cid:15) ( t ) . Let a (cid:15) ( t ) := H (cid:15) ( t ) + ξ (cid:15) ( t )(1 + (cid:15) ) M ( t ) I (cid:15) ( t ) , t ≥ T. For t ≥ T , a (cid:15) ( t ) ≥ H (cid:15) ( t ) > x ∗ ( (cid:15) ) := max s ∈ [0 , T ] x ( s ) > x ( (cid:15) ) . Therefore, with ψ ∈ RV ∞ ( β −
1) a decreasing function asymptotic to φ ( x ) /x , φ (cid:48) ( a (cid:15) ( t )) < ( β + (cid:15) ) φ ( a (cid:15) ( t )) a (cid:15) ( t < ( β + (cid:15) )(1 + (cid:15) ) ψ ( a (cid:15) ( t )) < ( β + (cid:15) )(1 + (cid:15) ) ψ ( H (cid:15) ( t )) , t ≥ T. But since ψ ( x ) ∼ φ ( x ) /x we also have ψ ( H (cid:15) ( t )) / (1 + (cid:15) ) < φ ( H (cid:15) ( t )) /H (cid:15) ( t ) and hence φ (cid:48) ( a (cid:15) ( t )) < ( β + (cid:15) )(1 + (cid:15) ) φ ( H (cid:15) ( t )) H (cid:15) ( t ) , t ≥ T. Combining this estimate with (5.42) yields I (cid:48) (cid:15) ( t ) < φ ( H (cid:15) ( t )) + ( β + (cid:15) )(1 + (cid:15) ) φ ( H (cid:15) ( t )) H (cid:15) ( t ) M ( t ) I (cid:15) ( t ) , t ≥ T. Letting α (cid:15) ( t ) = ( β + (cid:15) )(1+ (cid:15) ) M ( t ) φ ( H (cid:15) ( t )) /H (cid:15) ( t ), this becomes I (cid:48) (cid:15) ( t ) < φ ( H (cid:15) ( t ))+ α (cid:15) ( t ) I (cid:15) ( t ) for t ≥ T. Thus the variation of constants formula yields I (cid:15) ( t ) ≤ e (cid:82) tT α (cid:15) ( s ) ds (cid:90) tT e − (cid:82) sT α (cid:15) ( u ) du φ ( H (cid:15) ( s )) ds, t ≥ T. We reformulate this as I (cid:15) ( t ) (cid:82) tT φ ( H (cid:15) ( s )) ds ≤ (cid:82) tT e − (cid:82) sT α (cid:15) ( u ) du φ ( H (cid:15) ( s )) dse − (cid:82) tT α (cid:15) ( s ) ds (cid:82) tT φ ( H (cid:15) ( s )) ds =: C (cid:15) ( t ) B (cid:15) ( t ) , t ≥ T. (5.43)Since C (cid:48) (cid:15) ( t ) = φ ( H (cid:15) ( t )) e − (cid:82) tT α (cid:15) ( u ) du >
0, we have lim t →∞ C (cid:15) ( t ) = C ∗ ( (cid:15) ) ∈ (0 , ∞ ) or lim t →∞ C (cid:15) ( t ) = ∞ .Also, for t ≥ T , B (cid:48) (cid:15) ( t ) = φ ( H (cid:15) ( t )) e − (cid:82) tT α (cid:15) ( u ) du − α (cid:15) ( t ) e − (cid:82) tT α (cid:15) ( u ) du (cid:90) tT φ ( H (cid:15) ( s )) ds = C (cid:48) (cid:15) ( t ) − α (cid:15) ( t ) C (cid:48) (cid:15) ( t ) (cid:82) tT φ ( H (cid:15) ( s )) dsφ ( H (cid:15) ( t )) = C (cid:48) (cid:15) ( t ) (cid:40) − α (cid:15) ( t ) (cid:82) tT φ ( H (cid:15) ( s )) dsφ ( H (cid:15) ( t )) (cid:41) . Therefore, recalling the definition of α (cid:15) ( t ) and rearranging, B (cid:48) (cid:15) ( t ) C (cid:48) (cid:15) ( t ) = 1 − ( β + (cid:15) )(1 + (cid:15) ) (cid:32) M ( t ) (cid:82) tT φ ( H (cid:15) ( s )) dsH (cid:15) ( t ) (cid:33) , t ≥ T. Letting t → ∞ and using the hypothesis (3.3), and that H (cid:15) ( t ) ∼ H ( t ) and M ( t ) ∼ M ( t ) as t → ∞ , yieldslim t →∞ B (cid:48) (cid:15) ( t ) /C (cid:48) (cid:15) ( t ) = 1, or that lim t →∞ C (cid:48) (cid:15) ( t ) /B (cid:48) (cid:15) ( t ) = 1. Hence there exists T such that B (cid:48) (cid:15) ( t ) > t ≥ T and either lim t →∞ B (cid:15) ( t ) = B ∗ ( (cid:15) ) ∈ (0 , ∞ ) or lim t →∞ B (cid:15) ( t ) = ∞ . Furthermore, asymptoticintegration shows that lim t →∞ C (cid:15) ( t ) = ∞ implies lim t →∞ B (cid:15) ( t ) = ∞ and lim t →∞ C (cid:15) ( t ) = C ∗ ( (cid:15) ) implieslim t →∞ B (cid:15) ( t ) = B ∗ ( (cid:15) ). Hence,Λ( (cid:15) ) := lim t →∞ C (cid:15) ( t ) B (cid:15) ( t ) = (cid:40) , lim t →∞ C (cid:15) ( t ) = ∞ , C ∗ ( (cid:15) ) B ∗ ( (cid:15) ) , lim t →∞ C (cid:15) ( t ) = C ∗ , where the first limit is calculated using L’Hˆopital’s rule. Taking the limit superior across equation (5.43)then yields lim sup t →∞ (cid:82) tT φ ( x ( s )) ds (cid:82) tT φ ( H (cid:15) ( s )) ds = lim sup t →∞ I (cid:15) ( t ) (cid:82) tT φ ( H (cid:15) ( s )) ds ≤ Λ( (cid:15) ) ∈ (0 , ∞ ) . (5.44)Since H (cid:15) ( t ) ∼ H ( t ) as t → ∞ and φ is increasing we can apply L’Hˆopital’s rule once more to computelim t →∞ (cid:82) tT φ ( H (cid:15) ( s )) ds (cid:82) t φ ( H (cid:15) ( s )) ds = lim t →∞ φ ( H (cid:15) ( t )) φ ( H ( t )) = 1 β = 1 , using that φ ∈ RV ∞ ( β ). A similar argument relying on the divergence of φ ( x ( t )) and L’Hˆopital’s ruleyields (cid:82) tT φ ( x ( s )) ds ∼ (cid:82) t φ ( x ( s )) ds as t → ∞ . Therefore (5.44) is equivalent tolim sup t →∞ (cid:82) t φ ( x ( s )) ds (cid:82) t φ ( H ( s )) ds ≤ Λ( (cid:15) ) ∈ (0 , ∞ ) . (5.45)Therefore there exists a Λ ∗ ∈ (0 , ∞ ) such that lim sup t →∞ (cid:82) t φ ( x ( s )) ds/ (cid:82) t φ ( H ( s )) ds ≤ Λ ∗ , with Λ ∗ independent of (cid:15) . Thus there exists a T ( (cid:15) ) such that (cid:82) t φ ( x ( s )) ds < (Λ ∗ + (cid:15) ) (cid:82) t φ ( H ( s )) ds for all t ≥ T ( (cid:15) ). Letting ¯ T = 1 + max(2 T, T ) we apply this estimate to (5.41) as follows x ( t ) H ( t ) < H (cid:15) ( t ) H ( t ) + (1 + (cid:15) ) M ( t ) (cid:82) tT φ ( x ( s )) dsH ( t ) < H (cid:15) ( t ) H ( t ) + (1 + (cid:15) ) M ( t ) (Λ ∗ + (cid:15) ) (cid:82) t φ ( H ( s )) dsH ( t ) , t ≥ ¯ T .
EMORY DEPENDENT GROWTH IN SUBLINEAR VOLTERRA DIFFERENTIAL EQUATIONS 27
Now, since H (cid:15) ( t ) ∼ H ( t ) as t → ∞ and M ∼ M , applying (3.3) to the above estimate yieldslim sup t →∞ x ( t ) /H ( t ) ≤ . By positivity (1.6) admits the trivial bound x ( t ) > H ( t ) for all t ≥ t →∞ x ( t ) /H ( t ) ≥
1, completing the proof. (cid:3) Proofs of Miscellaneous Propositions and Examples
Proof of Proposition 1. ( i. ) is clear from inspection. For ( ii. ) recall the following form of Sterling’sapproximation (see [13, eq. 5.11.7, p.141])Γ( az + b ) ∼ √ πe − az ( az ) az + b − , as z → ∞ , for a ∈ (0 , ∞ ) , b ∈ R . (6.1)Hence, as θ → ∞ ,Λ( β, θ ) ∼ (cid:18) √ π e − θ θ θ + 12 (cid:19) (cid:16) √ π e − βθ − β { βθ − β } βθ − β + − β − (cid:17)(cid:16) √ π e θβ − { θ − β } θ − β + − β − (cid:17) ∼ √ π θ (cid:16) (1 − β ) β β − β (cid:17) θ β − β − . Therefore, since (1 − β ) β β − β ∈ (0 ,
1) for β ∈ (0 , θ →∞ Λ( β, θ ) = 0, for each fixed β ∈ (0 , θ ∈ (0 , ∞ ), let z ( β ) = ( βθ + 1) / ( θ (1 − β )) and note that lim β ↑ z ( β ) = ∞ . Applying (6.1) thenyields Λ( β, θ ) = Γ(1 + θ )Γ ( θz ( β ))Γ( θz ( β ) + θ ) ∼ Γ(1 + θ )( θz ( β )) − θ ∼ Γ(1 + θ ) (cid:18) βθ − β (cid:19) − θ as β ↑ . Therefore lim β ↑ Λ( β, θ ) = 0 for each fixed θ ∈ (0 , ∞ ).To see ( iii. ) compute ∂∂β Λ( β, θ ) as follows ∂∂β Λ( β, θ ) = ( θ + 1)Γ( θ + 1)(1 − β ) Γ (cid:48) (cid:16) βθ +11 − β (cid:17) Γ (cid:16) θ − β (cid:17) − Γ (cid:48) (cid:16) θ +11 − β (cid:17) Γ (cid:16) βθ +11 − β (cid:17)(cid:16) Γ (cid:16) θ +11 − β (cid:17)(cid:17) = ( θ + 1)Γ( θ + 1)Γ (cid:16) βθ +11 − β (cid:17) (1 − β ) Γ (cid:16) θ +11 − β (cid:17) Γ (cid:48) (cid:16) βθ +11 − β (cid:17) Γ (cid:16) βθ − β (cid:17) − Γ (cid:48) (cid:16) θ +11 − β (cid:17) Γ (cid:16) θ +11 − β (cid:17) = − ( θ + 1)Γ( θ + 1)Γ (cid:16) βθ +11 − β (cid:17) (cid:110) ψ (cid:16) θ +11 − β (cid:17) − ψ (cid:16) βθ +11 − β (cid:17)(cid:111) ( β − Γ (cid:16) θ +11 − β (cid:17) , where ψ ( x ) := Γ (cid:48) ( x ) / Γ( x ) (as in [13, eq. 5.2.2, p.137]). Hence ∂∂β Λ( β, θ ) < β ∈ (0 ,
1) and for eachfixed θ ∈ (0 , ∞ ) if and only if ψ (cid:18) θ + 11 − β (cid:19) − ψ (cid:18) βθ + 11 − β (cid:19) > . This holds because ψ is monotone increasing on R + (see [13, eq. 5.7.6, p.139]). Similarly, to prove claim( iv. ), it can be shown that ∂∂θ Λ( β, θ ) = Γ( θ + 1)Γ (cid:16) βθ +11 − β (cid:17) (cid:110) ( β − ψ ( θ + 1) + ψ (cid:16) θ +11 − β (cid:17) − βψ (cid:16) βθ +11 − β (cid:17)(cid:111) ( β − (cid:16) θ +11 − β (cid:17) . Hence ∂∂β Λ( β, θ ) < θ ∈ (0 , ∞ ) and for each fixed β ∈ (0 ,
1) if and only if( β − ψ ( θ + 1) + ψ (cid:18) θ + 11 − β (cid:19) − βψ (cid:18) βθ + 11 − β (cid:19) > . (6.2) By monotonicity of ψ , (6.2) is implied by( β − ψ ( θ + 1) + ψ (cid:18) θ + 11 − β (cid:19) − βψ (cid:18) θ + 11 − β (cid:19) > . However, this inequality is equivalent to ψ (cid:18) θ + 11 − β (cid:19) > ψ ( θ + 1) . Since ψ is increasing and β ∈ (0 ,
1) this holds for all θ ∈ (0 , ∞ ). Thus (6.2) holds and the claim is proven.Finally, claim ( v. ) follows from the continuity of Λ and claims ( i. ) − ( iv. ). (cid:3) Proof of Proposition 2.
Applying Karamata’s Theorem to F yields the first part of the claim. We restatethis in terms of (cid:96) as follows F ( x ) ∼ − β x − β (cid:96) ( x ) − β , as x → ∞ . Note that L ( x ) := 1 / (1 − β ) − β (cid:96) ( x ) ∈ RV ∞ (0). Hence applying [8, Theorem 1.5.15] F − ( x ) ∼ x − β L (cid:16) x − β (cid:17) , as x → ∞ . (6.3)where L denotes the de Bruijn conjugate of L (see [8, eq. 1.5.12, p.29]). It can be shown that (4.1) isequivalent to lim x →∞ L ( x/L ( x )) /L ( x ) = 1 , and by [8, Corollary 2.3.4] this is a sufficient condition for L ( x ) ∼ /L ( x ) as x → ∞ . Combining this fact with (6.3) we conclude that F − ( x ) ∼ x − β (cid:110) L ( x − β ) (cid:111) − ∼ (1 − β ) − β (cid:96) (cid:16) x − β (cid:17) x − β , as x → ∞ , completing the proof. (cid:3) Proof of Proposition 3.
By hypothesis there exists a monotone decreasing, C function ˜ µ such that µ ( s ) ∼ ˜ µ ( s ) as s → ∞ . Define ˜ M ( t ) := (cid:82) t ˜ µ ( ds ) and note that ˜ M ∈ RV ∞ ( θ ). We claim that M ( t ) ∼ ˜ M ( t ) as t → ∞ . To see this first note that for an arbitrary (cid:15) ∈ (0 ,
1) 1 − (cid:15) < µ ( jτ ) / ˜ µ ( jτ ) < (cid:15), for all jτ ≥ J ( (cid:15) ) , for some J ( (cid:15) ) ∈ Z + . Hence(1 − (cid:15) ) (cid:98) t/τ (cid:99) (cid:88) j = J ( (cid:15) ) ˜ µ ( jτ ) ≤ (cid:98) t/τ (cid:99) (cid:88) j = J ( (cid:15) ) µ ( jτ ) ≤ (1 + (cid:15) ) (cid:98) t/τ (cid:99) (cid:88) j = J ( (cid:15) ) ˜ µ ( jτ ) , (6.4)for all t ≥ τ (1 + J ( (cid:15) )). Noting that M is given by (4.3), we can write (6.4) as M (( J ( (cid:15) ) − τ ) + (1 − (cid:15) ) (cid:98) t/τ (cid:99) (cid:88) j = J ( (cid:15) ) ˜ µ ( jτ ) ≤ M ( t ) ≤ (1 + (cid:15) ) (cid:98) t/τ (cid:99) (cid:88) j = J ( (cid:15) ) ˜ µ ( jτ ) + M (( J ( (cid:15) ) − τ ) , (6.5)for all t ≥ τ (1 + J ). It follows that M ( t ) ≤ (1 + (cid:15) ) (cid:98) t/τ (cid:99)− (cid:88) l = J − (cid:90) ( l +1) τlτ ˜ µ ( s ) ds + M (( J − τ ) ≤ (1 + (cid:15) ) (cid:90) (cid:98) t/τ (cid:99) τ + τ ( J − τ ˜ µ ( s ) ds + M (( J − τ ) ≤ (1 + (cid:15) ) (cid:90) t + τ ( J − τ ˜ µ ( s ) ds + M (( J − τ ) , t ≥ τ (1 + J ) . (6.6)Similarly, from (6.5), M ( t ) ≥ M (( J − τ ) + (1 − (cid:15) ) (cid:98) t/τ (cid:99) (cid:88) j = J ( (cid:15) ) ˜ µ ( jτ ) ≥ M (( J − τ ) + (1 − (cid:15) ) (cid:98) t/τ (cid:99) (cid:88) j = J ( (cid:15) ) (cid:90) ( j +1) τjτ ˜ µ ( s ) ds ≥ M (( J − τ ) + (1 − (cid:15) ) (cid:90) (cid:98) t/τ (cid:99) τ + τjτ ˜ µ ( s ) ds ≥ M (( J − τ ) + (1 − (cid:15) ) (cid:90) (cid:98) t/τ (cid:99) τjτ ˜ µ ( s ) ds, (6.7) EMORY DEPENDENT GROWTH IN SUBLINEAR VOLTERRA DIFFERENTIAL EQUATIONS 29 for t ≥ τ (1 + J ). Hence combining the upper estimate (6.6) and the lower estimate (6.7) yields M (( J − τ ) + (1 − (cid:15) ) (cid:90) (cid:98) t/τ (cid:99) τjτ ˜ µ ( s ) ds ≤ M ( t ) ≤ (1 + (cid:15) ) (cid:90) t + τ ( J − τ ˜ µ ( s ) ds + M (( J − τ ) , t ≥ τ (1 + J ) . Therefore lim t →∞ M ( t ) / ˜ M ( t ) = 1 , and M ∈ RV ∞ ( θ ). (cid:3) References [1] J. A. D. Appleby. On regularly varying and history-dependent convergence rates of solutions of a Volterra equationwith infinite memory.
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School of Mathematical Sciences, Dublin City University, Glasnevin, Dublin 9, Ireland
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