doi:10.1155/2010/478291
Research Article
On Regularly Varying and History-Dependent Convergence Rates of Solutions of
a Volterra Equation with Infinite Memory
John A. D. Appleby
Edgeworth Centre for Financial Mathematics, School of Mathematical Sciences, Dublin City University, Dublin 9, Ireland
Correspondence should be addressed to John A. D. Appleby,john.appleby@dcu.ie Received 30 December 2009; Accepted 28 February 2010
Academic Editor: A ˘gacik Zafer
Copyrightq2010 John A. D. Appleby. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We consider the rate of convergence to equilibrium of Volterra integrodifferential equations with infinite memory. We show that if the kernel of Volterra operator is regularly varying at infinity, and the initial history is regularly varying at minus infinity, then the rate of convergence to the equilibrium is regularly varying at infinity, and the exact pointwise rate of convergence can be determined in terms of the rate of decay of the kernel and the rate of growth of the initial history.
The result is considered both for a linear Volterra integrodifferential equation as well as for the delay logistic equation from population biology.
1. Introduction
In this paper we consider the asymptotic behaviour of linear and nonlinear Volterra integrodifferential equations with infinite memory, paying particular attention to the connection between the asymptotic behaviour of the initial history ast → −∞and the rate of convergence of the solution to a limit. In fact we focus our attention on the cases where the initial historyφobeys limt→ ∞|φt|∞. We do not aim to be comprehensive in our analysis and focus only on scalar equations whose initial histories and kernels are regularly varying functions. However, we note that such history-dependent asymptotic behaviour does not seem to be generic behaviour for equations with a finite memory.
We consider both linear and nonlinear equations. In particular we consider the linear Volterra integrodifferential equation given by
xt −axt− t
−∞bt−sxsds, t >0; xt φt, t∈−∞,0, 1.1
as well as the nonlinear logistic equation with infinite delay given by
Nt Nt
r−aNt− t
−∞bt−sNsds
, t >0, Nt φt, t∈−∞,0.
1.2
In both cases, we presume that b is continuous, positive, and integrable, and that φ is continuous on −∞,0. When t → |φt|is bounded and a > ∞
0bsds, the solution of x of1.1obeysxt → 0 ast → ∞; moreover, Miller1has shown when in additionr >0, the solutionNof1.2obeysNt → r/a∞
0 bsds :Kast → ∞.
For definiteness, we concentrate in this introduction on solutions of1.2. In Appleby et al.2, an extension of Miller’s global asymptotic stability result was given to a class of initial functionsΦ, which can include initial historiesφwhich are unbounded in the sense that limt→ −∞φt ∞. Furthermore, the rate at whichNtconvergesKast → ∞is also of interest.
It was shown in2that for certain classes of kernelsbthat the rate of convergence of NttoK ast → ∞depends on the asymptotic behaviour ofφtast → −∞. Whenbis a type of slowly decaying function, called subexponential, it has been shown that when
0
−∞
φt−Kdt <∞ 1.3
thenNt−Ktends to zero likebtast → ∞. In the case whenφt−Ktends to a nonzero limit ast → −∞,Nt−Ktends to zero like∞
t bsdsast → ∞. Moreover, this rate of decay to zero is slower thanb. It is therefore of interest to consider how this history-dependent decay rate develops in the case thatφ∈Φis unbounded.
In our main results, we show that ifbdecays polynomiallyin the sense thatbis in the class of integrable and regularly varying functionsand the historyφtgrows polynomially ast → −∞in the sense thatφis a regularly varying function at−∞, and the rate of growth ofφis not too rapid relative to the rate of decay ofb, then problems1.1and1.2are well posed andxt → 0 andNt−K → 0 ast → ∞at the ratetbtφ−tast → ∞.
The question of history-dependent asymptotic behaviour is of interest not only in demography and population dynamics, but also in financial mathematics and time series, and this also motivates our study here. It is well known that certain discrete- and continuous- time stochastic processes have autocovariance functions which can be represented as linear difference or delay-differential equations see, e.g., K ¨uchler and Mensch 3. In the case of the so-called ARCH ∞ processes which are stationary, the resulting equation for the autocovariance function of the process can be represented as a Volterra summation equation with infinite memory. For details on the stationarity and autocovariance function of such ARCH processes; see, for example, Zaffaroni4, Giraitis et al.5, Robinson6, and Giraitis et al.7. In the nonstationary case, the process may be autocorrelated on−Nis a manner which is inconsistent with the autocovariance function in the stationary casewhich must be an even function, while the mean and variance of the process still converge. Therefore, the process can have a limiting autocovariance function which may differ from that of the stationary process. This phenomenon is impossible for processes with bounded memory, and the different convergence rates which depend on the asymptotic behaviour of the initial
history in this case is an exact analogue to the history-dependent decay rates recorded here.
Therefore, this paper also lays the groundwork for analysis of this phenomenon in finance from the perspective of infinite memory Volterra equations. The interest in such so-called long memory stems in part from the presence of inefficiency in financial markets and the applicability of ARCH-type processes in modelling the evolution of market volatility. Some of the fundamental papers in this direction are Comte and Renault8, Baillie et al.9, and Bollerslev and Mikkelsen10. An up-to-date survey of work on long memory processes is given by Cont11.
2. Mathematical Preliminaries
We introduce some standard notation. We denote byR the set of real numbers. If J is an interval inRandV a finite dimensional normed space, we denote byCJ, Vthe family of continuous functionsφ : J → V. The space of Lebesgue integrable functionsφ : 0,∞ → V will be denoted byL10,∞, V. Where V is clear from the context we omit it from the notation.
The convolution off:0,∞ → Randg:0,∞ → Ris denoted byf∗gand defined to be the function given by
f∗g t
t
0
fsgt−sds, t≥0. 2.1
If the domain offcontains an interval of the formT,∞andγ :0,∞ → 0,∞, thenLγf denotes limt→ ∞ft/γt, if it exists.
2.1. Subexponential and Regularly Varying Functions We make a definition, based on the hypotheses of Theorem 3 of12.
Definition 2.1. Let β : 0,∞ → 0,∞ be a continuous function. Then we say that β is subexponential if
∞
0
βtdt <∞, 2.2
t→ ∞lim 1 βt
t
0
βt−sβsds2 ∞
0
βsds, 2.3
tlim→ ∞
βt−s
βt 1 uniformly for 0≤s≤S, ∀S >0. 2.4
In13the terminology positive subexponential function was used instead of just subex- ponential function. Because subexponential functions play the role here of weight functions, it is natural that they have strictly positive values. The nomenclature subexponential is
suggested by the fact that2.4implies that, for every >0, βtet → ∞ast → ∞. This is proved, for example, in14. It is also true that
t→ ∞limβt 0. 2.5
InDefinition 2.1above, condition2.3can be replaced by
Tlim→ ∞lim sup
t→ ∞
1 βt
t−T
T
βt−sβsds0, 2.6
and this latter condition often proves to be useful in proofs.
The properties of subexponential functions have been extensively studied, for example, in 12–15. Simple examples of subexponential functions are βt 1t−α for α > 1, βt e−1tα for 0 < α < 1 and βt e−t/logt2. The class of subexponential functions therefore includes a wide variety of functions exhibiting polynomial and slower- than-exponential decay: nor is the slower-than-exponential decay limited to a class of polynomially decaying functions.
In this paper, however, we restrict our attention to an important subclass of subexponential functions. It is noted in 13 that the class of subexponential functions includes all positive, continuous, integrable functions which are regularly varying at infinity.
We recall that a functionγ:0,∞ → Ris said to be regularly varying at infinity with index α∈Rif
t→ ∞lim γλt
γt λα, ∀λ >0, 2.7
and we write γ ∈ RV∞α. When α < −1, γ is subexponential. A useful property of a continuous functionγ∈RV∞αforα <−1 is
t→ ∞lim ∞
t γsds
tγt − 1
1α. 2.8
In this paper, we also find it convenient to consider functions in RV∞αforα≥ −1. We list some of the important properties used here. A characteristic of regularly varying functions of nonzero index is that they exhibit a type of power-law growth or decay ast → ∞. Indeed, ifγ∈RV∞α, then
t→ ∞lim
logγt
logt α. 2.9
Hence|γt| → ∞ast → ∞ifα >0 andγt → 0 ast → ∞ifα <0.
Ifγ∈RV∞αforα >0andγt → ∞ast → ∞, thenγis asymptotic to a continuous function δ, which is also in RV∞α, such that δ is increasing on 0,∞. Similarly, if γ ∈ RV∞αforα <0, andγis ultimately positive, thenγis asymptotic to a continuous function δ, which is also in RV∞α, such thatδis decreasing on0,∞.
A functionγ :−∞,0 → Ris said to be regularly varying at minus infinity with index αif the functionγ− :0,∞ → Rdefined fort ≥0 byγ−t γ−tis in RV∞α. We denote the class of regularly varying functions at minus infinity with indexαby RV−∞α.
For further details on regularly varying functions, consult16.
3. Existence and Asymptotic Behaviour of Functionals of the Initial History
3.1. Hypotheses onbandφ
We make the following standing hypotheses concerning the kernelband initial historyφof 1.1and1.2
b:0,∞−→0,∞is continuous and inL10,∞, 3.1
φ:−∞,0−→Ris continuous. 3.2
We introduce a function f : 0,∞ → R which depends on the continuous function φ.
Suppose that
For every t≥0, f t;φ
defined byf t;φ
: 0
−∞bt−sφsdsexists 3.3a f
·;φ
∈C0,∞,R 3.3b
t→ ∞limf t;φ
0. 3.3c
Following2, we define byΦthe space of initial functionsφfor which suchf·;φexists and has the properties3.3band3.3c:
Φ φ∈C−∞,0;R:f
·;φ
obeys 3.3
. 3.4
The importance off andΦin this paper is the following. Suppose that we have an infinite memory integrodifferential equation with solutionxand initial historyφ, that is,xt φt fort∈−∞,0. If the equation involves a convolution term of the form
t
−∞bt−sxsds, t≥0 3.5
on the right-hand side, the infinite memory equation is equivalent to an initial value integrodifferential equation with unbounded memory, provided that φ is such that 3.3a holds. The existence and uniqueness of a solution of the integrodifferential equation is essentially guaranteed by 3.3b, and asymptotic analysis and in particular stability is aided by3.3c. Therefore the class of initial historiesΦhelps us to recast questions about the existence, uniqueness, and asymptotic stability of solutions of an infinite memory
convolution equation in terms of a perturbed initial value convolution equation, where f, to a certain extent, plays the role of a forcing term or perturbation.
We now impose some additional conditions on b and φ which enable us to demonstrate thatφ ∈ Φand which are also central to the asymptotic analysis of solutions of1.1and1.2. To this end suppose thatbobeys
b∈RV∞δ for someδ∈−∞,−1, 3.6
In addition to3.2,φalso obeys φ∈RV−∞
η
for someη >0 with lim
t→ −∞φt ∞. 3.7
Suppose further that
δη1<0. 3.8
It is often convenient to work with the functionφ:0,∞ → Rdefined byφt φ−tfor t≥0, rather than withφitself. An important property ofφis that it is in RV∞η.
By virtue of the fact that b is regularly varying at infinity with indexδ <−1, it follows that there exists a functionβsuch that
β∈C0,∞;0,∞is decreasing, lim
t→ ∞βt 0, lim
t→ ∞
bt
βt/0 exists, 3.9 andβis also forced to satisfy
βt−s
βt −→1 ast−→ ∞uniformly for 0≤s≤T, ∀T ≥0. 3.10 Also, becauseφ is regularly varying at infinity with indexη > 0, there exists a functionϕ which is increasing and which obeysφt/ϕt → 1 ast → ∞.
3.2. Existence and Asymptotic Behaviour off·;φ
Our results in this section demonstrate that, under the hypotheses3.6and3.7,f·;φhas the properties given in3.3a,3.3b, and3.3c. The proofs of the main results are postponed to later in the paper.
Remark 3.1. Condition3.8implies that ∞
0
bsφ−sdsis finite. 3.11
It can be seen that3.11is necessary for the existence offt;φfort≥0i.e., for the validity of3.3a. This is because the integral in3.11isf0;φ.
Condition3.11is close to being sufficient for the existence off·;φand indeed is sufficient ifbandφare nonnegative. In fact, becausebis integrable, we have thatft;φ → 0 ast → ∞.
Proposition 3.2. Suppose thatbobeys3.1and3.6and thatφobeys3.2and3.7. Ifδandη obey3.8, thenbandφobey
∞
0
|bs|φ−sds <∞, 3.12
andf·;φexists for allt≥0 and therefore obeys3.3a. Moreoverf·;φobeys3.3c.
It is notable that condition 3.12does not require that |φ|be bounded, but merely that it cannot grow too quickly ast → −∞, relative to the rate of decay of bt → 0 as t → ∞. This is the significance of the parameter restriction3.8. Scrutiny of the proof, which is inSection 7, reveals that the regular variation ofbandφis used sparingly. Indeed, if one assumes3.12, the properties3.9and3.10suffice to prove the result.
Conditions3.9and3.10will be used to establish the continuity oft → ft;φ, as well as for later asymptotic analysis offt;φast → ∞.
We notice that by virtue of3.9thatbt → 0 ast → ∞. Sincebis also continuous, it follows that it is uniformly continuous. This fact is used at important points in the proof of the following result.
Proposition 3.3. Suppose thatbobeys3.1and3.6and thatφobeys3.2and3.7. Sinceband φalso obey3.12, thenft;φexists for allt ≥ 0,t → ft;φis continuous andft;φ → 0 as t → ∞(i.e.,φ∈Φ).
A careful reading of the proof again deferred to Section 7 reveals that it is the properties3.9and3.10, together with3.12, that are employed, and that the full strength of3.6and3.7is unnecessary.
Having shown thatf obeys all the properties in3.3a,3.3b, and3.3c, including the fact thatft;φ → 0 ast → ∞, our first main result determines the exact rate of decay to zero offt;φast → ∞. In contrast to the other results in this section, the proof of this result employs extensively the regular variation ofband ofφ.
Theorem 3.4. Suppose thatbis a positive continuous function which obeys3.6for someδ <−1.
Letφbe a function which obeys3.7for someη >0, and suppose thatδη1<0. Thenφ∈Φand fdefined in3.3a,3.3b, and3.3cobeys
t→ ∞lim f
t;φ
tbtφ−t
∞
0
xη1xδdx. 3.13
The proof ofTheorem 3.4is postponed toSection 5. We note that the integral on the right-hand side of3.13exists becauseη > 0 andδη1 < 0. We also notice that asb ∈ RV∞δandφ∈RV∞η, by3.13, the functionf·;φ∈RV∞δη1.
4. Statement and Discussion of Main Results
4.1. Linear Equations with Unbounded Initial HistoryOnceTheorem 3.4has been proven, we are able to determine the rate of decay of the solution of the following linear infinite memory convolution equation
xt axt t
−∞bt−sxsds, t >0, vφt, t∈−∞,0.
4.1
If we suppose that b andφ obey merely 3.1 and 3.2, and that φ ∈ Φ where Φ is the space defined by3.4, the functionfdefined in3.3a,3.3b, and3.3cis well defined and continuous on0,∞. Therefore, we see that4.1can be written in the equivalent form
xt axt t
0
bt−sxsdsf t;φ
, t >0, x0 φ0, 4.2
wherext φtfort ∈ −∞,0. Since this initial value problem has a unique continuous solution, it follows that there is a unique continuous solution of4.1. However, as we assume that b andφobey the hypotheses3.6,3.7,3.1,3.2, and3.8throughout, it follows that φ∈Φ, and therefore4.1has a unique continuous solutionx.
We now investigate conditions under whichxt → 0 as t → ∞, and the rate of convergence to zero. To study this asymptotic behaviour, it is conventional to introduce the linear differential resolventr which is defined to be the unique continuous solution of the integrodifferential equation
rt art t
0
bt−srsds, t >0, r0 1. 4.3
The significance ofris that it enables us to represent the unique continuous solutionxof4.1 in terms offt;φ defined in3.3a. Using4.2, the formula forxis given by
xt rtφ0
t
0
rt−sf s;φ
ds, t≥0. 4.4
In the case whena∞
0 |bs|ds <0, it is known thatr ∈L10,∞andrt → 0 ast → ∞.
Therefore, in this casext → 0 ast → ∞. Some recent results on the asymptotic stability of Volterra equations with unbounded delay include17,18.
Moreover, asbis in RV∞δforδ < −1,bis subexponential, and so it is known by results of, for example,19, that
t→ ∞lim rt
βt exists and is finite, 4.5
due to3.9. Therefore, as we already have good information about the rate of convergence offt;φ → 0 ast → ∞fromTheorem 3.4, the representation4.4together with4.5opens the prospect that the rate of convergence ofxt → 0 ast → ∞can be obtained. Our main result in this direction is as follows.
Theorem 4.1. Suppose thatbis a continuous and integrable function which obeys 3.6for some δ <−1. Letφbe a continuous function which obeys3.7for someη >0 and suppose thatδη1<0.
Ifa∞
0|bs|ds <0, thenx, the unique continuous solution of 4.1, obeys
t→ ∞lim xt
tbtφ−t−
∞
0
uη1uδ du· 1 a∞
0bsds. 4.6
It is worth re-emphasising that the conditionδη1 <0 is not a merely a technical convenience; in the case whenφt>0 fort∈−∞,0andδη1>0, problem4.1is not well posed, becausef0;φ, for example, is not well defined.
The proof ofTheorem 4.1is given inSection 6.1, and uses results from the admissibility theory of linear Volterra operators. These results are stated inSection 6, in advance of the proof ofTheorem 4.1.
4.2. Delay Logistic Equation with Unbounded Initial History
In this section, we state and discuss a result similar to Theorem 4.1 for a nonlinear integrodifferential equation with infinite memory. We consider the logistic equation with infinite delay
Nt Nt
r−aNt− t
−∞bt−sNsds
, t >0, Nt φt, t∈−∞,0,
4.7
wherebis continuous and integrable,φ is continuous, andaand r are real numbers. This equation, and related equations, have been used to study the population dynamics of a single species, whereNtstands for the population at timet.
Iff·;φis the function given in3.3a, it is seen that the existence of a solution of4.7 is equivalent to the existence of a solution of
Nt Nt
r−aNt− t
0
bt−sNsds−f t;φ
, t >0, N0 φ0. 4.8
Therefore, it is necessary that the functionf be well defined in order for solutions of4.7to exist. In the case thatφ ∈ Φ, then the functionf·;φgiven in3.3ais well defined and is moreover continuous. Therefore standard results on existence, uniqueness, and continuation of solutions of Volterra integral equationscf., e.g., Burton20, Gripenberg et al.21, Miller 1ensure that there is a unique solution of4.7 up to a possible explosion time. For the proof of positivity of the solution see, for example, Miller1. We state2, Theorem 1which concerns on the asymptotic behaviour of solutions of4.7.
Theorem 4.2. Letb∈C0,∞,0,∞∩L10,∞,0,∞,a >∞
0 bsds,r >0. LetΦbe defined by3.4, and suppose thatφ ∈ C−∞,0,0,∞is also inΦ. Then there is a unique continuous positive solutionNof 4.7which obeys
tlim→ ∞Nt r a∞
0 bsds :K. 4.9
This theorem extends a result of1which deals with the case whenφ is a bounded continuous function. We remark once more that the conditionφ∈Φdoes not requireφto be bounded. Some other recent papers which employ Volterra equations with unbounded delay to model stable population dynamics include22,23.
WithTheorem 4.2in hand, we can determine the convergence rate of the solutionN of4.7toKdefined in4.9.
Theorem 4.3. Suppose thatbis a positive continuous function which obeys3.6for someδ <−1.
Leta >∞
0 bsdsandr >0. Letφ∈C−∞,0,0,∞obey3.7for someη >0, and suppose that δη1<0. ThenN, the unique continuous positive solution of4.7, obeys
tlim→ ∞
K−Nt
tbtφ−t
∞
0
uη1uδ du· 1 a∞
0 bsds, 4.10
whereKis defined by4.9.
Once again, the proof appeals to results from the admissibility theory of linear Volterra operators. The proof ofTheorem 4.3is deferred toSection 6.2.
It is interesting to compare this result with those obtained for4.7 under different conditions on φ and subexponential bin 2. Suppose, as inTheorem 4.3 above, thata >
∞
0bsds,φ∈Φis continuous and positive, andbis positive, continuous, and integrable. In the case whenba fortiori obeys3.6, and there existsL /0 such that
t→ ∞lim φ−t−K
L, 4.11
then by2, Theorem 2there existsc /0 such that
tlim→ ∞
K−Nt
tbt c. 4.12
On the other hand, by2, Theorem 2we have that 0
−∞
φt−Kdt <∞ 4.13
implies
t→ ∞lim
K−Nt
bt exists. 4.14
In both these cases, there is a history-dependenti.e., aφ-dependentrate of convergence to the equilibrium; moreover, it appears that the larger the “size” of the historyas measured by the discrepancy ofφfromK, the slower the rate of convergence toK. InTheorem 4.3we show that the rate of convergence is slowertbtφ−tast → ∞than in both4.12 tbtas t → ∞and4.14 btast → ∞. This is consistent with the picture that a “larger” history leads to a slower rate of convergence, as the history inTheorem 4.3obeys limt→ −∞φt ∞, in contrast to the “bounded” histories in4.11and4.13. Unbounded histories are studied in2, but only for equations in whichbdecays exponentially fast to zero, in the sense that t → bteλt is subexponential for some λ > 0, in which case, φt can grow as t → −∞
according to
t→ −∞lim eλt
φt−K
L /0 or 0
−∞eλφt−Kdt <∞, 4.15 and results similar to4.14or4.12can be established.
An interesting question, which we do not address here, is to determine the rate of convergence to the equilibrium for solutions of4.7in the case whenφ−K ∈RV−∞ηfor η∈−1,0. In this case,φ−Kis not integrable, butφttends toKast → −∞. Therefore, these cases cover historiesφwhose discrepancy fromKis intermediate between thoseφcovered by conditions 4.13and 4.11. It might be expected that a similar rate of convergence to zero would be found for solutions of 4.1 in the case whenφ ∈ RV−∞ηforη ∈ −1,0.
Obviously, the key ingredient to proving such results is an analysis of the rate of convergence offt;φ → 0 ast → ∞.
5. Proof of Theorem 3.4
Theorem 3.4 follows by a number of lemmas. The first part of this section discusses and presents these results; the rest of the section is devoted to their proofs.
5.1. Discussion of Supporting Lemmas
We suppose that b and φ obey3.1 and 3.2 throughout. In the first lemma supporting Theorem 3.4, we show that the requirement thatbandφbe nonmonotone can essentially be lifted. The key result is the following.
Lemma 5.1. Suppose thatbobeys3.6andφobeys3.7, then there exist a decreasing continuous functionβsuch thatbt/βt → 1 ast → ∞and an increasing functionϕsuch thatφ−t/ϕt → 1 ast → ∞. If there existsL >0 such that
t→ ∞lim ∞
0ϕsβstds
tβtϕt L, 5.1
andf·;φis the function defined by3.3a, then
tlim→ ∞
f t;φ
tβtϕt L. 5.2
The next result shows that, subject to a technical condition, the conclusion of Theorem 3.4holds for monotoneβandϕ.
Lemma 5.2. Suppose thatβis a decreasing and continuous function in RV∞δforδ <−1, and that ϕis an increasing and continuous function in RV∞ηforη >0. Letδη1<0. If
tlim→ ∞
1 βtϕt
∞ j0
ϕ j1
ht β
jhtt
Λh:∞
j0
j1 hη
jh1δ
, for each fixedh >0,
5.3
tlim→ ∞
1 βtϕt
∞ j1
ϕ jht
β j1
htt
Λh:∞
j1
jhη j1
h1δ
, for each fixedh >0,
5.4
then
t→ ∞lim ∞
0ϕsβstds
tβtϕt
∞
0
xη1xδdx. 5.5
To proveLemma 5.2, we need the following auxiliary result.
Lemma 5.3. IfΛis defined by5.3andΛby5.4, then
h→lim0hΛh ∞
0
xη1xδdx,
h→lim0hΛh ∞
0
xη1xδdx.
5.6
Finally, we need to prove the suppositions5.3and5.4.
Lemma 5.4. Ifβis a decreasing and continuous function in RV∞δforδ <−1 andϕ∈RV∞ηfor η >0 andδη1<0, then5.3and5.4hold.
The proofs of these lemmas are given in the following subsections. It is readily seen that by taking the results of Lemmas5.2and5.4together with the result ofLemma 5.1with L∞
0 xη1xδdx,Theorem 3.4is true.
5.2. Proof ofLemma 5.1
Sincebt/βt → 1 ast → ∞andφt/ϕt → 1 ast → ∞for everyε∈0,1there exists Tε>0 such that 1−ε < bt/βt<1εfor allt≥Tεand 1−ε < φt/ϕt<1εfor all t≥Tε. Therefore
∞
T
φsbstds≤1ε2 ∞
T
ϕsβstds, 5.7
∞
T
φsbstds≥1−ε2 ∞
T
ϕsβstds. 5.8
Now
tlim→ ∞
1 βt
T
0
φsbstds lim
t→ ∞
T
0
φsbst βst
βst
βt ds
T
0
φsds.
5.9
Therefore ast→tϕtis in RV∞η1andη1>0, we havetϕt → ∞ast → ∞, and so
t→ ∞lim 1 tβtϕt
T
0
φsbstds0. 5.10
Similarly
t→ ∞lim 1 tβtϕt
T
0
ϕsβstds0. 5.11
Hence by5.1we have
t→ ∞lim 1 tβtϕt
∞
T
ϕsβstdsL. 5.12
Now
f t;φ
tβtϕt 1
tβtϕt T
0
φsbstds 1
tβtϕt ∞
T
φsbstds. 5.13
so by5.7and5.9
lim sup
t→ ∞
f t;φ
tβtϕt ≤L1ε2. 5.14
Similarly by5.8and5.9we have
lim inf
t→ ∞
f t;φ
tβtϕt ≥L1−ε2. 5.15
Lettingε → 0in5.15and5.14gives5.2.
5.3. Proof ofLemma 5.2
Fixh∈0,1. Sinceβis decreasing andϕis increasing, we have ∞
0
ϕsβstds∞
j0
j1ht
jht
ϕsβstds
≤∞
j0
j1ht
jht
ϕ j1
ht β
jhtt ds
≤∞
j0
htϕ j1
ht β
jhtt .
5.16
Similarly
∞
0
ϕsβstds≥∞
j0
htϕ jht
β j1
htt
. 5.17
Suppose we can show that5.4holds, then
lim inf
t→ ∞
1 tβtϕt
∞
0
ϕsβstds
≥lim inf
t→ ∞
1
tβtϕt·∞
j0
htϕ jht
β j1
htt
hΛh.
5.18
Also by5.3
lim sup
t→ ∞
1 tβtϕt
∞
0
ϕsβstds
≤lim sup
t→ ∞
1 tβtϕt
∞ j0
htϕ j1
ht β
jhtt
hΛh.
5.19
Therefore
lim sup
t→ ∞
1 tβtϕt
∞
0
ϕsβstds≤hΛh,
lim inf
t→ ∞
1 tβtϕt
∞
0
ϕsβstds≥hΛh.
5.20
By5.20, using the facts thatδη1<0 andη >0, and by employing5.6, we have5.5 as required.
5.4. Proof ofLemma 5.3 The required results are
hlim→0
∞ j0
h
j1hη
jh1δ
∞
0
xη1xδdx, 5.21
hlim→0
∞ j0
h jhη
j1h1δ
∞
0
xη1xδdx. 5.22
We pause to remark that the integrals on the right-hand side of both5.21and 5.22are finite. To start, notice that
∞
0
xη1xδdx∞
j0
j1h
jh
xη1xδdx≥∞
j0
h jhη
1 j1hδ
, 5.23
∞
0
xη1xδdx≤∞
j0
h
j1hη
1jhδ
. 5.24
Letε∈0,1. LetJε∈Nbe such thatJε>1−ε/ε. Then forj > Jεwe havej >1−ε/ε and so
j
j1 >1−ε, j > Jε. 5.25
Also, asε∈0,1, for everyh >0 we haveεh1−ε< ε 1−εhso becausej > Jε>
1−ε/ε, we have
1jhh
1jh 1 h
1jh <1 h
1 1−ε/ε·h < 1
1−ε. 5.26
This implies
1jh
1jhh>1−ε, j > Jε. 5.27
By5.23, we have ∞
0
xη1xδdx≥∞
j0
h j1
hη
1jhδ j j1
η 1jh 1 j1h
−δ
. 5.28
Hence by5.25and5.27and the fact thatη >0 and−δ >0, we get ∞
jJε1
h jhη
1 j1hδ
>1−εη1−ε−δ ∞
jJε1
h
j1hη
1jhδ
. 5.29
Therefore ∞ j0
h jhη
1 j1hδ
>
Jε
j0
h jhη
1 j1hδ
1−εη−δ ∞
jJε1
h
j1hη
1jhδ .
5.30
Hence
lim inf
h→0
∞ j0
h jhη
1 j1hδ
≥1−εη−δlim inf
h→0
∞ jJε1
h
j1hη
1jhδ
,
lim sup
h→0
∞ j0
h jhη
1 j1hδ
≥1−εη−δlim sup
h→0
∞ jJε1
h
j1hη
1jhδ .
5.31
Now
hlim→0 Jε
j0
h
j1hη
1jhδ
0, 5.32
so we have lim inf
h→0
∞ j0
h jhη
1 j1hδ
≥1−εη−δlim inf
h→0
∞ j0
h
j1hη
1jhδ ,
lim sup
h→0
∞ j0
h jhη
1 j1hδ≥1−εη−δlim sup
h→0
∞ j0
h
j1hη
1jhδ
.
5.33
Lettingε → 0 yields
lim inf
h→0
∞ j0
h jhη
1 j1hδ
≥lim inf
h→0
∞ j0
h
j1hη
1jhδ
, 5.34
lim sup
h→0
∞ j0
h jhη
1 j1hδ
≥lim sup
h→0
∞ j0
h
j1hη
1jhδ
. 5.35
By5.23and5.35, we have ∞
0
xη1xδdx≥lim sup
h→0
∞ j0
h jhη
1 j1hδ
≥lim sup
h→0
∞ j0
h
j1hη
1jhδ .
5.36
Similarly, by5.24we have ∞
0
xη1xδdx≤lim inf
h→0
∞ j0
h
j1hη
1jhδ
. 5.37
Combining these inequalities gives5.21as required. By5.34and5.21, we get
lim inf
h→0
∞ j0
h jhη
1 j1hδ≥ lim
h→0
∞ j0
h
j1hη
1jhδ ∞
0
xη1xδdx. 5.38
On the other hand, by5.23we have ∞
0
xη1xδdx≥lim sup
h→0
∞ j0
h jhη
1 j1hδ
, 5.39
so by combining these inequalities, we get5.22as required.
5.5. Proof ofLemma 5.4
Leth∈0,1. Sinceβis decreasing forj ≥0 we have β
1jh t
βt ·ϕ
1j ht
ϕt ≤ β
1j ht
βht ·βht
βt ·ϕ 1j
ht
ϕht ·ϕht
ϕt . 5.40
Sinceβandϕare continuous and are in RV∞δand RV∞η, respectively, there existsKh>
0 such thatβht/βt≤ Khandϕht/ϕt≤ Khfor allt ≥0. Hence withβht βht andϕht ϕhtwe have
β 1jh
t
βt ·ϕ
1j ht
ϕt ≤Kh2βh
1j t βht ·ϕh
1j t
ϕht , t≥0. 5.41
We see thatβh ∈ RV∞δ and βh is decreasing, whileϕh ∈ RV∞ηand ϕh is increasing.
Therefore, we have
t→ ∞lim βh2t
βht 2δ, lim
t→ ∞
ϕh2t
ϕht 2η. 5.42
Sinceδη1<0, we may chooseε >0 so small thatAε : 1ε22δη1 <1. By5.42, for everyε >0 sufficiently small, there existsTε, h>1 such that
βh2t
βht <1ε2δ, ϕh2t
ϕht <1ε2η, t > Tε, h. 5.43 Letn≥0 be an integer. Then fort > Tε, hwe have
βh2nt βht n
l1
βh 2lt βh
2l−1t <
1ε2δn
, ϕh
2n1t ϕht n1
l1
ϕh 2lt ϕh
2l−1t <1ε2ηn1.
5.44
For every integerj≥0 there exists a unique integern≥0 such that 2n≤j1<2n1. Suppose thatt > Tε, h. Then asβhis decreasing, andϕhis increasing, we have
βh
j1 t
βht ≤ βh2nt βht <
1ε2δn , ϕh
j1 t ϕht ≤ ϕh
2n1t
ϕht <1ε2ηn1.
5.45
Hence by5.41for 2n≤j1<2n1, we have β
1jh t
βt ·ϕ
1j ht
ϕt ≤Kh2
1ε2δn
1ε2ηn1, t > Tε, h. 5.46
Define forn≥0
Cnt:2
n1−2 j2n−1
β 1jh
t
βt ·ϕ
1j ht
ϕt . 5.47
Then forn≥0 andt > Tε, hwe have
Cnt≤2
n1−2 j2n−1
Kh2
1ε22ηδn
1ε2ηKh21ε2ηAnε. 5.48
SinceAε < 1, the sequenceMn :Kh21ε2ηAnε is summable. Next, asβ ∈RV∞δand ϕ∈RV∞η, we have
t→ ∞limCnt 2
n1−2 j2n−1
tlim→ ∞
β 1jh
t
βt ·ϕ
1j ht
ϕt 2
n1−2 j2n−1
1jhδ
1jhη
. 5.49
Since 0 ≤ Cnt ≤ Mn for all t > Tε, h, by the summability of Mnn≥1 and 5.49, the Dominated Convergence Theorem gives
tlim→ ∞
∞ n0
Cnt ∞
n0
tlim→ ∞Cnt. 5.50
This is equivalent to
t→ ∞lim ∞ n0
2n1−2 j2n−1
β 1jh
t
βt ·ϕ
1j ht
ϕt ∞
n0 2n1−2 j2n−1
1jhδ
1jhη
, 5.51
which implies5.3.
To prove that5.4holds, note that asβis decreasing andϕis increasing, we have β
1 j1
h t
βt ·ϕ
jht
ϕt ≤ β
1jh t
βt ·ϕ
1j ht
ϕt . 5.52
Define
Dnt: 2
n1−2 j2n−1
β 1
j1 h
t
βt ·ϕ
jht
ϕt . 5.53
Thus, 0 ≤ Dnt ≤ Cnt ≤ Mn for t > Tε, h and all n ≥ 0. Also as β ∈ RV∞δ and ϕ∈RV∞η, we have
t→ ∞limDnt 2
n1−2
j2n−1 t→ ∞lim
β 1
j1 h
t
βt ·ϕ
jht
ϕt 2
n1−2
j2n−1
1 j1hδ jhη
. 5.54
Now by the summability ofMnn≥1, the last limit and the fact that 0 ≤Dnt ≤Mn, by the Dominated Convergence Theorem we have
t→ ∞lim ∞ n0
2n1−2 j2n−1
β 1
j1 h
t
βt ·ϕ
jht
ϕt ∞
n0 2n1−2 j2n−1
1 j1hδ jhη
, 5.55
and therefore5.4holds, as required.
6. Proofs of Theorems 4.1 and 4.3
The proofs of Theorems 4.1and 4.3, which concern the asymptotic behaviour of Volterra equations, are greatly facilitated by applying extant results on the admissibility of certain linear Volterra operators. For the convenience of the reader, two results from2are restated.
LetH : Δ → Rbe a continuous function onΔ {s, t ∈ R2 : 0 ≤ s ≤ t < ∞}.
Associated withHis the linear operatorH:C0,∞ → C0,∞defined by
Hξt t
0
Ht, sξsds, t≥0. 6.1
Firstly we restate a theorem, which is a variant of part of a result in Corduneanu24, page 74.
Theorem 6.1see2, Theorem 3. Suppose that for allT >0,
Ht, s−→H∞s ast−→ ∞uniformly with respect tos∈0, T. 6.2
Further assume that
W : lim
T→ ∞lim sup
t→ ∞
t
T
|Ht, s|ds <∞, 6.3
Tlim→ ∞lim sup
t→ ∞
t
T
Ht, sds−V
0 for someV ∈R, 6.4
then limt→ ∞Hξtexists for allξfor which limt→ ∞ξt :ξ∞exists, and
t→ ∞limHξt V ξ∞
∞
0
H∞sξsds. 6.5
The next result is 2, Theorem 4. It extends Appleby et al. 19, Theorem 5 to nonconvolution integral equations cf. 19, Theorem A.1; it is also the counterpart of Appleby et al.25, Theorems 3.1 and 5.1, and Gy˝ori and Horv´ath26, Theorem 3.1which concerns linear nonconvolution difference equations.
Theorem 6.2see2, Theorem 4. Suppose that6.2and6.4hold, and that6.3holds with
W <1. 6.6
Assume that ξ is in C0,∞ and that limt→ ∞ξt : ξ∞ exists. If η : 0,∞ → Rn is the continuous solution of
ηt ξt
t
0
Ht, sηsds, t≥0, 6.7
then limt→ ∞ηt :η∞exists and satisfies the limit formula η∞ I−V−1
ξ∞
∞
0
H∞sηsds
. 6.8
6.1. Proof ofTheorem 4.1
The method of2is now used to prove Theorems4.1and4.3.
Let β ∈ RV∞δ be the positive function defined in 3.9, which is decreasing on Θ1,∞for someΘ1>0. As remarked the solutionrof4.3is inr∈L10,∞; it also obeys
∞
0
rsds− 1
a∞
0bsds. 6.9
If
tlim→ ∞
r∗f
·;φ t
tbtφ−t −
∞
0
uη1uδ du· 1 a∞
0 bsds 6.10
holds, then by4.4,4.5, and the fact thattφ−t → ∞ast → ∞, we have
tlim→ ∞
xt
tbtφ−t φ0lim
t→ ∞
rt bt · 1
tφ−t lim
t→ ∞
r∗f
·;φ t tbtφ−t −
∞
0
uη1uδdu· 1 a∞
0 bsds,
6.11
which is nothing other than4.6.
It therefore remains to establish6.10. Sinceφ ∈RV∞ηfor someη >0 there exists ϕ such thatϕ is increasing onΘ2,∞for some Θ2 > 0, positive, differentiable and obeys φt/ϕt → 1 ast → ∞. Defineγt tβtϕt. Then byTheorem 3.4, we have
Lγf
·;φ : lim
t→ ∞
f t;φ
γt lim
t→ ∞
f t;φ
tbtφt·btφt
βtϕt
∞
0
uη1uδ du. 6.12
Note also that astϕt → ∞ast → ∞, we have
Lγβ lim
t→ ∞
βt γt lim
t→ ∞
1
tϕt 0. 6.13
Our strategy here is to useTheorem 6.1to show that limt→ ∞r∗f·;φt/γtexists and to determine it. To this end write
1 γt
t
0
rt−sf s;φ
ds t
0
rt−sγs
γt ·f
s;φ
γs ds
t
0
H1t, sξ1sds, 6.14
where we identify
H1t, s rt−sγs
γt , ξ1s f
s;φ
γs . 6.15
In the notation ofTheorem 6.1,H1here plays the role ofHandξ1the role ofξ. EvidentlyH1 andξ1are continuous. By6.12, ast → ∞, it follows thatξ1t → ∞
0uη1uδdu:ξ1∞.
By6.13and4.5, we haveLγrLβr·Lγβ0. Using this and the fact thatγt−s/γt → 1 ast → ∞uniformly on compact intervalsby2.4, we obtain
|H1t, s| |rt−s|
γt−s ·γt−s
γt ·γs−→0, ast−→ ∞, 6.16
where the convergence is uniform fors∈0, T, for anyT >0.
Next letT >maxΘ1,Θ2. Fort >2T, we have the identity
t
T
H1t, sds− ∞
0
rsds
t−T
T
rt−sγs
γt ds
T
0
rs
γt−s γt −1
ds−
∞
T
rsds.
6.17
