- Research Article
- Open Access
On Regularly Varying and History-Dependent Convergence Rates of Solutions of a Volterra Equation with Infinite Memory
© John A. D. Appleby. 2010
- Received: 30 December 2009
- Accepted: 28 February 2010
- Published: 15 March 2010
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.
- Logistic Equation
- Volterra Equation
- Positive Continuous Function
- Volterra Operator
- Autocovariance Function
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 as and the rate of convergence of the solution to a limit. In fact we focus our attention on the cases where the initial history obeys . 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
as well as the nonlinear logistic equation with infinite delay given by
In both cases, we presume that is continuous, positive, and integrable, and that is continuous on . When is bounded and , the solution of of (1.1) obeys as ; moreover, Miller  has shown when in addition , the solution of (1.2) obeys as .
For definiteness, we concentrate in this introduction on solutions of (1.2). In Appleby et al. , an extension of Miller's global asymptotic stability result was given to a class of initial functions , which can include initial histories which are unbounded in the sense that . Furthermore, the rate at which converges as is also of interest.
It was shown in  that for certain classes of kernels that the rate of convergence of to as depends on the asymptotic behaviour of as . When is a type of slowly decaying function, called subexponential, it has been shown that when
then tends to zero like as . In the case when tends to a nonzero limit as , tends to zero like as . Moreover, this rate of decay to zero is slower than . It is therefore of interest to consider how this history-dependent decay rate develops in the case that is unbounded.
In our main results, we show that if decays polynomially (in the sense that is in the class of integrable and regularly varying functions) and the history grows polynomially as (in the sense that is a regularly varying function at ), and the rate of growth of is not too rapid relative to the rate of decay of , then problems (1.1) and (1.2) are well posed and and as at the rate as .
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., Küchler and Mensch ). 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, Zaffaroni , Giraitis et al. , Robinson , and Giraitis et al. . In the nonstationary case, the process may be autocorrelated on is a manner which is inconsistent with the autocovariance function in the stationary case (which 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 Renault , Baillie et al. , and Bollerslev and Mikkelsen . An up-to-date survey of work on long memory processes is given by Cont .
We introduce some standard notation. We denote by the set of real numbers. If is an interval in and a finite dimensional normed space, we denote by the family of continuous functions . The space of Lebesgue integrable functions will be denoted by . Where is clear from the context we omit it from the notation.
The convolution of and is denoted by and defined to be the function given by
If the domain of contains an interval of the form and , then denotes , if it exists.
2.1. Subexponential and Regularly Varying Functions
We make a definition, based on the hypotheses of Theorem of .
In  the terminology positive subexponential function was used instead of just subexponential 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 that (2.4) implies that, for every as . This is proved, for example, in . It is also true that
In Definition 2.1 above, condition (2.3) can be replaced by
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 for , for and . 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  that the class of subexponential functions includes all positive, continuous, integrable functions which are regularly varying at infinity. We recall that a function is said to be regularly varying at infinity with index if
and we write . When is subexponential. A useful property of a continuous function for is
In this paper, we also find it convenient to consider functions in for . 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 as . Indeed, if , then
Hence as if and as if .
If for (and as ), then is asymptotic to a continuous function , which is also in , such that is increasing on . Similarly, if for , and is ultimately positive, then is asymptotic to a continuous function , which is also in , such that is decreasing on .
A function is said to be regularly varying at minus infinity with index if the function defined for by is in . We denote the class of regularly varying functions at minus infinity with index by .
For further details on regularly varying functions, consult .
3.1. Hypotheses on and
We make the following standing hypotheses concerning the kernel and initial history of (1.1) and (1.2)
We introduce a function which depends on the continuous function . Suppose that
Following , we define by the space of initial functions for which such exists and has the properties ((3.3b)) and (3.3c):
The importance of and in this paper is the following. Suppose that we have an infinite memory integrodifferential equation with solution and initial history , that is, for . If the equation involves a convolution term of the form
on the right-hand side, the infinite memory equation is equivalent to an initial value integrodifferential equation with unbounded memory, provided that 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 by (3.3c). Therefore the class of initial histories helps 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 , to a certain extent, plays the role of a forcing term or perturbation.
We now impose some additional conditions on and which enable us to demonstrate that and which are also central to the asymptotic analysis of solutions of (1.1) and (1.2). To this end suppose that obeys
In addition to (3.2), also obeys
Suppose further that
It is often convenient to work with the function defined by for , rather than with itself. An important property of is that it is in .
By virtue of the fact that is regularly varying at infinity with index , it follows that there exists a function such that
and is also forced to satisfy
Also, because is regularly varying at infinity with index , there exists a function which is increasing and which obeys as .
3.2. Existence and Asymptotic Behaviour of
Our results in this section demonstrate that, under the hypotheses (3.6) and (3.7), has the properties given in (3.3a), (3.3b), and (3.3c). The proofs of the main results are postponed to later in the paper.
It can be seen that (3.11) is necessary for the existence of for (i.e., for the validity of (3.3a)). This is because the integral in (3.11) is .
Condition (3.11) is close to being sufficient for the existence of and indeed is sufficient if and are nonnegative. In fact, because is integrable, we have that as .
and exists for all and therefore obeys (3.3a). Moreover obeys (3.3c).
It is notable that condition (3.12) does not require that be bounded, but merely that it cannot grow too quickly as , relative to the rate of decay of as . This is the significance of the parameter restriction (3.8). Scrutiny of the proof, which is in Section 7, reveals that the regular variation of and is used sparingly. Indeed, if one assumes (3.12), the properties (3.9) and (3.10) suffice to prove the result.
Conditions (3.9) and (3.10) will be used to establish the continuity of , as well as for later asymptotic analysis of as .
We notice that by virtue of (3.9) that as . Since is also continuous, it follows that it is uniformly continuous. This fact is used at important points in the proof of the following result.
Suppose that obeys (3.1) and (3.6) and that obeys (3.2) and (3.7). Since and also obey (3.12), then exists for all , is continuous and as (i.e., ).
A careful reading of the proof (again deferred to Section 7) reveals that it is the properties (3.9) and (3.10), together with (3.12), that are employed, and that the full strength of (3.6) and (3.7) is unnecessary.
Having shown that obeys all the properties in (3.3a), (3.3b), and (3.3c), including the fact that as , our first main result determines the exact rate of decay to zero of as . In contrast to the other results in this section, the proof of this result employs extensively the regular variation of and of .
The proof of Theorem 3.4 is postponed to Section 5. We note that the integral on the right-hand side of (3.13) exists because and . We also notice that as and , by (3.13), the function .
4.1. Linear Equations with Unbounded Initial History
Once Theorem 3.4 has been proven, we are able to determine the rate of decay of the solution of the following linear infinite memory convolution equation
If we suppose that and obey merely (3.1) and (3.2), and that (where is the space defined by (3.4)), the function defined in (3.3a), (3.3b), and (3.3c) is well defined and continuous on . Therefore, we see that (4.1) can be written in the equivalent form
where for . Since this initial value problem has a unique continuous solution, it follows that there is a unique continuous solution of (4.1). However, as we assume that and obey the hypotheses (3.6), (3.7), (3.1), (3.2), and (3.8) throughout, it follows that , and therefore (4.1) has a unique continuous solution .
We now investigate conditions under which as , and the rate of convergence to zero. To study this asymptotic behaviour, it is conventional to introduce the linear differential resolvent which is defined to be the unique continuous solution of the integrodifferential equation
The significance of is that it enables us to represent the unique continuous solution of (4.1) in terms of (defined in (3.3a)). Using (4.2), the formula for is given by
Moreover, as is in for , is subexponential, and so it is known by results of, for example, , that
due to (3.9). Therefore, as we already have good information about the rate of convergence of as from Theorem 3.4, the representation (4.4) together with (4.5) opens the prospect that the rate of convergence of as can be obtained. Our main result in this direction is as follows.
It is worth re-emphasising that the condition is not a merely a technical convenience; in the case when for and , problem (4.1) is not well posed, because , for example, is not well defined.
The proof of Theorem 4.1 is given in Section 6.1, and uses results from the admissibility theory of linear Volterra operators. These results are stated in Section 6, in advance of the proof of Theorem 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
where is continuous and integrable, is continuous, and and are real numbers. This equation, and related equations, have been used to study the population dynamics of a single species, where stands for the population at time .
If is the function given in (3.3a), it is seen that the existence of a solution of (4.7) is equivalent to the existence of a solution of
Therefore, it is necessary that the function be well defined in order for solutions of (4.7) to exist. In the case that , then the function given in (3.3a) is well defined and is moreover continuous. Therefore standard results on existence, uniqueness, and continuation of solutions of Volterra integral equations (cf., e.g., Burton , Gripenberg et al. , Miller ) ensure that there is a unique solution of (4.7) (up to a possible explosion time). For the proof of positivity of the solution see, for example, Miller . We state [2, Theorem ] which concerns on the asymptotic behaviour of solutions of (4.7).
This theorem extends a result of  which deals with the case when is a bounded continuous function. We remark once more that the condition does not require to be bounded. Some other recent papers which employ Volterra equations with unbounded delay to model stable population dynamics include [22, 23].
With Theorem 4.2 in hand, we can determine the convergence rate of the solution of (4.7) to defined in (4.9).
where is defined by (4.9).
Once again, the proof appeals to results from the admissibility theory of linear Volterra operators. The proof of Theorem 4.3 is deferred to Section 6.2.
It is interesting to compare this result with those obtained for (4.7) under different conditions on and subexponential in . Suppose, as in Theorem 4.3 above, that , is continuous and positive, and is positive, continuous, and integrable. In the case when a fortiori obeys (3.6), and there exists such that
then by [2, Theorem ] there exists such that
On the other hand, by [2, Theorem ] we have that
In both these cases, there is a history-dependent (i.e., a -dependent) rate of convergence to the equilibrium; moreover, it appears that the larger the "size" of the history (as measured by the discrepancy of from ), the slower the rate of convergence to . In Theorem 4.3 we show that the rate of convergence is slower ( as ) than in both (4.12) ( as ) and (4.14) ( as ). This is consistent with the picture that a "larger" history leads to a slower rate of convergence, as the history in Theorem 4.3 obeys , in contrast to the "bounded" histories in (4.11) and (4.13). Unbounded histories are studied in , but only for equations in which decays exponentially fast to zero, in the sense that is subexponential for some , in which case, can grow as according to
and results similar to (4.14) or (4.12) can be established.
An interesting question, which we do not address here, is to determine the rate of convergence to the equilibrium for solutions of (4.7) in the case when for . In this case, is not integrable, but tends to as . Therefore, these cases cover histories whose discrepancy from is intermediate between those covered by conditions (4.13) and (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 when for . Obviously, the key ingredient to proving such results is an analysis of the rate of convergence of as .
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 and obey (3.1) and (3.2) throughout. In the first lemma supporting Theorem 3.4, we show that the requirement that and be nonmonotone can essentially be lifted. The key result is the following.
The next result shows that, subject to a technical condition, the conclusion of Theorem 3.4 holds for monotone and .
To prove Lemma 5.2, we need the following auxiliary result.
Finally, we need to prove the suppositions (5.3) and (5.4).
If is a decreasing and continuous function in for and for and , then (5.3) and (5.4) hold.
The proofs of these lemmas are given in the following subsections. It is readily seen that by taking the results of Lemmas 5.2 and 5.4 together with the result of Lemma 5.1 with , Theorem 3.4 is true.
5.2. Proof of Lemma 5.1
Since as and as for every there exists such that for all and for all . Therefore
Therefore as is in and , we have as , and so
Hence by (5.1) we have
so by (5.7) and (5.9)
Similarly by (5.8) and (5.9) we have
Letting in (5.15) and (5.14) gives (5.2).
5.3. Proof of Lemma 5.2
Fix . Since is decreasing and is increasing, we have
Suppose we can show that (5.4) holds, then
Also by (5.3)
By (5.20), using the facts that and , and by employing (5.6), we have (5.5) as required.
5.4. Proof of Lemma 5.3
The required results are
We pause to remark that the integrals on the right-hand side of both (5.21) and (5.22) are finite. To start, notice that
Let . Let be such that . Then for we have and so
Also, as , for every we have so because , we have
By (5.23), we have
Hence by (5.25) and (5.27) and the fact that and , we get
so we have
By (5.23) and (5.35), we have
Similarly, by (5.24) we have
Combining these inequalities gives (5.21) as required. By (5.34) and (5.21), we get
On the other hand, by (5.23) we have
so by combining these inequalities, we get (5.22) as required.
5.5. Proof of Lemma 5.4
Let . Since is decreasing for we have
Since and are continuous and are in and , respectively, there exists such that and for all . Hence with and we have
We see that and is decreasing, while and is increasing. Therefore, we have
Since , we may choose so small that . By (5.42), for every sufficiently small, there exists such that
Let be an integer. Then for we have
For every integer there exists a unique integer such that . Suppose that . Then as is decreasing, and is increasing, we have
Hence by (5.41) for , we have
Then for and we have
Since , the sequence is summable. Next, as and , we have
Since for all , by the summability of and (5.49), the Dominated Convergence Theorem gives
This is equivalent to
which implies (5.3).
To prove that (5.4) holds, note that as is decreasing and is increasing, we have
Thus, for and all . Also as and , we have
Now by the summability of , the last limit and the fact that , by the Dominated Convergence Theorem we have
and therefore (5.4) holds, as required.
The proofs of Theorems 4.1 and 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 from  are restated.
Let be a continuous function on . Associated with is the linear operator defined by
Firstly we restate a theorem, which is a variant of part of a result in Corduneanu [24, page 74].
Theorem 6.1 (see [2, Theorem ]).
The next result is [2, Theorem ]. It extends Appleby et al. [19, Theorem ] to nonconvolution integral equations (cf. [19, Theorem ]); it is also the counterpart of Appleby et al. [25, Theorems and ], and Győri and Horváth [26, Theorem ] which concerns linear nonconvolution difference equations.
Theorem 6.2 (see [2, Theorem ]).
6.1. Proof of Theorem 4.1
The method of  is now used to prove Theorems 4.1 and 4.3.
Let be the positive function defined in (3.9), which is decreasing on for some . As remarked the solution of (4.3) is in ; it also obeys
holds, then by (4.4), (4.5), and the fact that as , we have
which is nothing other than (4.6).
It therefore remains to establish (6.10). Since for some there exists such that is increasing on for some , positive, differentiable and obeys as . Define . Then by Theorem 3.4, we have
Note also that as as , we have
Our strategy here is to use Theorem 6.1 to show that exists and to determine it. To this end write
where we identify
In the notation of Theorem 6.1, here plays the role of and the role of . Evidently and are continuous. By (6.12), as , it follows that . By (6.13) and (4.5), we have . Using this and the fact that as uniformly on compact intervals (by (2.4)), we obtain
where the convergence is uniform for , for any .
Next let . For , we have the identity
Since obeys (2.4), the second term on the right-hand side has zero limit as . As for the first term, by the continuity of and (4.5), the facts that and that is increasing, we deduce the estimate
Since is subexponential, and therefore obeys (2.3), we have
so it follows that
Therefore by (6.17) and (6.20), we arrive at the estimate
Since , we have
Returning to (6.17) we get
Hence by using (6.20), we obtain
Finally, letting yields
Since all of the hypotheses of Theorem 6.1 are satisfied with we have, by (6.12) and (6.9),
which is nothing but (6.10), and so the theorem is proven.
6.2. Proof of Theorem 4.3
First, define for , where is given by (4.9). Then by Theorem 4.2, as . Our strategy, as in , is to show that satisfies a linear integral equation (where nonlinearities are subsumed into the kernel and forcing function). Once this is done, we scale the resulting integral equation appropriately and apply Theorem 6.2 to determine the asymptotic behaviour of .
Although the same derivation of the integral equation for is given in , we give it afresh here, partly to make the exposition self contained, and partly because it enables us to define and analyse a number of auxiliary functions that will be important in the proof.
Substitution of into (4.7) with defined by
leads to the initial-value problem
where . We note by (3.3a) that the function is well defined, continuous and obeys as . Define
Then is continuous and as , and obeys
Define the differential resolvent by
Therefore by the variation of constants formula, we have
and we have suppressed the -dependence in and . Since , it follows that as and with
Moreover we have that exists and is finite. Define as in the proof of Theorem 4.1. Dividing (6.32) by , and defining and for we arrive at
where is given by
Clearly , , and are continuous. Our strategy now is to apply Theorem 6.2 to determine the integral equation (6.35), showing that tends to a finite limit as . From this fact, we will be able to deduce the speed of convergence of to as .
First we show that exists and is finite. Since is finite, and , we have . Also with defined by (3.3a), for . Since for , it obeys
Therefore , and so we get
by Theorem 3.4. Therefore, by also using the fact that , we see that
by using the argument used to prove (6.26). Hence by (6.34), obeys
With defined by (6.36), we use the facts that and the fact that as uniformly on compact intervals to establish that
for all and any . Next, as is uniformly bounded on , for any , we have
Therefore by (6.20), we deduce that
Since is bounded and , it follows that obeys
Hence we also have
and therefore all the conditions of Theorem 6.2 hold, with . Hence the solution of (6.35) obeys . Therefore by (6.40) we have
which implies (4.10).
7.1. Proof of Proposition 3.2
By (3.9), (3.12), is finite, where
Hence as is nonincreasing, we have
so , where we define
Hence is uniformly bounded.
Since , is nonincreasing, and we have that as . Returning to (7.3) with given by (7.4), we get
Let . Then as is nonincreasing, we have
The second integral on the right-hand side is finite, and bounded above by . Since is continuous, is finite. Hence, as as , we have
Since is arbitrary, and we have . Hence , as claimed.
7.2. Proof of Proposition 3.3
Fix and let . We will show for every that there exists such that implies
Assume temporarily that this holds. Then for , we have
where we used (7.8) and (7.1) at the last step. This establishes the continuity of . It remains to prove (7.8). The identity
applied with and gives
For every there exists such that
where is given by (7.1). Suppose also that so that . Since obeys (3.10), we have
Hence for every there exists such that
where is given by (7.4). Hence as we have
Let . Then as and are nonnegative, using (7.4), (7.12), and (7.15) in (7.11), for we have
Set . Since is uniformly continuous on , for every there is a such that implies
The -dependence here is permissible as is fixed. Thus, for ,
Let . By the last inequality and (7.16), for we get
which is nothing other than (7.8), proving the result.
The author is grateful to the referees of this paper for their thorough and thoughtful reviews, as well as to the Editor-in-Chief for his suggestions. In addition to the support of Science Foundation Ireland, the author is delighted to thank David Reynolds for many fruitful discussions concerning this work. His criticism and insight of earlier drafts of the manuscript, and particularly his suggestion to streamline the proofs of Theorems 4.1 and 4.3, have greatly improved the final paper. The author is also very grateful to John Daniels for his collaboration and discussions on long-memory time series processes. This has informed the discussion and choice of problems presented in the paper. Finally, the author wishes to thank the editors of this Special Issue of Advances in Difference Equations on "Recent Trends in Differential and Difference Equations" for the opportunity to present his research in this volume. The author gratefully acknowledges Science Foundation Ireland for the support of this research under the Mathematics Initiative 2007 Grant 07/MI/008 "Edgeworth Centre for Financial Mathematics."
- Miller RK: On Volterra's population equation. SIAM Journal on Applied Mathematics 1966, 14: 446-452. 10.1137/0114039MathSciNetView ArticleMATHGoogle Scholar
- Appleby JAD, Győri I, Reynolds DW: History-dependent decay rates for a logistic equation with infinite delay. submittedGoogle Scholar
- Küchler U, Mensch B: Langevin's stochastic differential equation extended by a time-delayed term. Stochastics and Stochastics Reports 1992,40(1-2):23-42.MathSciNetView ArticleMATHGoogle Scholar
- Zaffaroni P:Stationarity and memory of models. Econometric Theory 2004,20(1):147-160.MathSciNetView ArticleMATHGoogle Scholar
- Giraitis L, Kokoszka P, Leipus R: Stationary ARCH models: dependence structure and central limit theorem. Econometric Theory 2000,16(1):3-22.MathSciNetView ArticleMATHGoogle Scholar
- Robinson PM: Testing for strong serial correlation and dynamic conditional heteroskedasticity in multiple regression. Journal of Econometrics 1991,47(1):67-84. 10.1016/0304-4076(91)90078-RMathSciNetView ArticleMATHGoogle Scholar
- Giraitis L, Robinson PM, Surgailis D: A model for long memory conditional heteroscedasticity. The Annals of Applied Probability 2000,10(3):1002-1024.MathSciNetView ArticleMATHGoogle Scholar
- Comte F, Renault E: Long memory in continuous-time stochastic volatility models. Mathematical Finance 1998,8(4):291-323. 10.1111/1467-9965.00057MathSciNetView ArticleMATHGoogle Scholar
- Baillie RT, Bollerslev T, Mikkelsen HO: Fractionally integrated generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 1996,74(1):3-30. 10.1016/S0304-4076(95)01749-6MathSciNetView ArticleMATHGoogle Scholar
- Bollerslev T, Mikkelsen HO: Modeling and pricing long memory in stock market volatility. Journal of Econometrics 1996,73(1):151-184. 10.1016/0304-4076(95)01736-4MathSciNetView ArticleMATHGoogle Scholar
- Cont R: Long range dependence in financial markets. In Fractals in Engineering. Edited by: Lutton E, Vehel J. Springer, New York, NY, USA; 2005:159-180.View ArticleGoogle Scholar
- Chover J, Ney P, Wainger S: Functions of probability measures. Journal d'Analyse Mathématique 1973, 26: 255-302.MathSciNetView ArticleMATHGoogle Scholar
- Appleby JAD, Reynolds DW: Subexponential solutions of linear integro-differential equations and transient renewal equations. Proceedings of the Royal Society of Edinburgh. Section A 2002,132(3):521-543.MathSciNetView ArticleMATHGoogle Scholar
- Appleby JAD, Reynolds DW: Subexponential solutions of linear integro-differential equations. In Dynamic Systems and Applications. Volume 4. Dynamic, Atlanta, Ga, USA; 2004:488-494.Google Scholar
- Chistyakov VP: A theorem on sums of independent positive random variables and its applications to branching random processes. Theory of Probability and Its Applications 1964, 9: 640-648. 10.1137/1109088View ArticleGoogle Scholar
- Bingham NH, Goldie CM, Teugels JL: Regular Variation, Encyclopedia of Mathematics and Its Applications. Volume 27. Cambridge University Press, Cambridge, UK; 1989:xx+494.Google Scholar
- Murakami S, Nagabuchi Y: Uniform asymptotic stability and robust stability for positive linear Volterra difference equations in Banach lattices. Advances in Difference Equations 2008, 2008:-15.Google Scholar
- Philos ChG, Purnaras IK: On linear Volterra difference equations with infinite delay. Advances in Difference Equations 2006, 2006:-28.Google Scholar
- Appleby JAD, Győri I, Reynolds DW: On exact rates of decay of solutions of linear systems of Volterra equations with delay. Journal of Mathematical Analysis and Applications 2006,320(1):56-77. 10.1016/j.jmaa.2005.06.103MathSciNetView ArticleMATHGoogle Scholar
- Burton TA: Stability and Periodic Solutions of Ordinary and Functional-Differential Equations, Mathematics in Science and Engineering. Volume 178. Academic Press, Orlando, Fla, USA; 1985:x+337.Google Scholar
- Gripenberg G, Londen S-O, Staffans O: Volterra Integral and Functional Equations, Encyclopedia of Mathematics and Its Applications. Volume 34. Cambridge University Press, Cambridge, UK; 1990:xxii+701.View ArticleGoogle Scholar
- Choi SK, Koo N: Almost periodic solutions of nonlinear discrete Volterra equations with unbounded delay. Advances in Difference Equations 2008, 2008:-15.Google Scholar
- Itokazu T, Hamaya Y: Almost periodic solutions of prey-predator discrete models with delay. Advances in Difference Equations 2009, 2009:-19.Google Scholar
- Corduneanu C: Integral Equations and Stability of Feedback Systems, Mathematics in Science and Engineering. Volume 104. Academic Press, New York, NY, USA; 1973:ix+238.Google Scholar
- Applelby JAD, Győri I, Reynolds DW: On exact convergence rates for solutions of linear systems of Volterra difference equations. Journal of Difference Equations and Applications 2006,12(12):1257-1275. 10.1080/10236190600986594MathSciNetView ArticleMATHGoogle Scholar
- Győri I, Horváth L: Asymptotic representation of the solutions of linear Volterra difference equations. Advances in Difference Equations 2008, 2008:-22.Google Scholar
This article is published under license to BioMed Central Ltd. 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.