Forward and Pullback Dynamics of Nonautonomous Integrodifference Equations: Basic Constructions.

In theoretical ecology, models describing the spatial dispersal and the temporal evolution of species having non-overlapping generations are often based on integrodifference equations. For various such applications the environment has an aperiodic influence on the models leading to nonautonomous integrodifference equations. In order to capture their long-term behaviour comprehensively, both pullback and forward attractors, as well as forward limit sets are constructed for general infinite-dimensional nonautonomous dynamical systems in discrete time. While the theory of pullback attractors, but not their application to integrodifference equations, is meanwhile well-established, the present novel approach is needed in order to understand their future behaviour.

Introduction

Integrodifference equations not only occur as temporal discretisations of integrodifferential equations or as time-1-maps of evolutionary differential equations, but are of interest in themselves. First and foremost, they are a popular tool in theoretical ecology to describe the dispersal of species having non-overlapping generations (see, for instance, [18] or [9, 17, 25]). While the theory of Urysohn or Hammerstein integral equations is now rather classical [19], both numerically and analytically, our goal is here to study their iterates from a dynamical systems perspective. This means one is interested in the long term behaviour of recursions based on a fixed nonlinear integral operator. In applications, the iterates for instance represent the spatial distribution of interacting species over a habitat. One of the central questions in this context is the existence and structure of an attractor. These invariant and compact sets attract bounded subsets of an ambient state space X and fully capture the asymptotics of an autonomous dynamical system [8, 23]. The dynamics inside the attractor can be very complicated and even chaotic [7].

Extending this situation, the main part of this paper is devoted to general nonautonomous difference equations in complete metric spaces. Their right-hand side can depend on time allowing to model the dispersal of species in temporally fluctuating environments [3, 9] being not necessarily periodic. Thus, the behaviour depends on both the initial and the actual time. This is why many dynamically relevant objects are contained in the extended state space (one speaks of nonautonomous sets) [10], rather than being merely subsets of the state space X as in the autonomous case. Furthermore, a complete description of the dynamics in a time-variant setting necessitates a strict distinction between forward and pullback convergence [10, 15]. For this reason only a combination of several attractor notions yields the full picture:The situation for forward attractors and limit sets is not as well-established as their pullback counterparts and deserves to be developed for the above reasons. Their initial construction in [12, 16] requires a locally compact state space, but recent continuous-time results in [6], which extend these to infinite-dimensional dynamical systems, will be transferred here. We indeed address nonautonomous difference equations in (not necessarily locally compact) metric and Banach spaces, introduce the mentioned attractor types and study their properties.

The pullback attractor [4, 11, 15, 20] is a compact, invariant nonautonomous set which attracts all bounded sets from the past. As fixed target problem, it is based on previous information, at a fixed time from increasingly earlier initial times. Since it consists of bounded entire solutions to a nonautonomous system (see [20, p. 17, Cor. 1.3.4]), a pullback attractor can be seen as an extension of the global attractor to nonautonomous problems and apparently captures the essential dynamics to a certain point. Meanwhile the corresponding theory is widely developed in discrete and continuous time. However, pullback attractors reflect the past rather than the future of systems (see [14]) and easy examples demonstrate that differential or difference equations with identical pullback attractors might have rather different asymptotics as and possibly feature limit sets, which are not captured by the pullback dynamics.

This led to the development of forward attractors, which are also compact and invariant nonautonomous sets [15]. This dual concept depends on information from the future and given a fixed initial time, the actual time increases beyond all bounds — they are a moving target problem. Forward attractors are not unique, independent of pullback attractors, but often do not exist. Nevertheless, we will describe forward attractors using a pullback construction, even though this has the disadvantage that information on the system over the entire time axis is required.

Therefore, it was suggested in [12] to work with forward limit sets, a concept related to the uniform attractor due to [24]. They correctly describe the asymptotic behaviour of all forward solutions to a nonautonomous difference equation. These limit sets have forward attraction properties, but different from pullback and forward attractors, they are not (even positively) invariant and constitute a single compact set, rather than a nonautonomous set. Nonetheless, asymptotic forms of positive (and negative) invariance do hold.

This brings us to our second purpose. The above abstract setting allows concrete applications to a particularly interesting class of infinite-dimensional dynamical systems in discrete time, namely integrodifference equations (IDEs for short). We provide sufficient criteria for the existence of pullback attractors tailor-made for a quite general class of IDEs. Their right-hand sides go beyond pure integral operators and might also include superposition operators, which are used to describe populations having a sedentary fraction. Such results follow from a corresponding theory of set contractions contained in [20, pp. 15ff], [19, pp. 79ff]. For completely continuous right-hand sides (i.e., Urysohn operators) we construct forward limit sets and provide an application to asymptotically autonomous IDEs. We restrict to rather simple IDEs in the space of continuous functions over a compact domain as state space. More complicated equations and the behaviour of attractors under spatial discretisation will be tackled in future papers.

The contents of this paper are as follows: In Sect. 2 we establish the necessary terminology and provide a useful dissipativity condition for nonautonomous difference equations. The key notions related to pullback convergence, i.e., limit sets and attractors are reviewed and established in Sect. 3. The subsequent Sect. 4 addresses the corresponding notions in forward time. In detail, it establishes forward limit sets and their (weakened) invariance properties. For a class of asymptotically autonomous equations it is shown that their forward limit sets coincide with the global attractor of the limit equation. Moreover, a construction of forward attractors is suggested. Finally, in Sect. 5 we provide some applications to various IDEs. In particular, we illustrate the above theoretical results by studying pullback attractors and forward limit sets.

Notation Let . A discrete interval is defined as the intersection of a real interval with the integers , and .

On a metric space (X, d), is the identity map, the open ball with center and radius , and denotes its closure. We write for the distance of x from a set and for its r-neighbourhood. The Hausdorff semidistance of bounded and closed subsets is defined asThe Kuratowski measure of noncompactness on X (cf. [19, pp. 16ff, I.5]) is denoted by , where stands for the family of bounded subsets of X.

A mapping is said to be bounded, if it maps bounded subsets of X into bounded sets and globally bounded, if is bounded. We say a bounded satisfies a Darbo condition, if there exists a real constant such thatThe smallest such k is the Darbo constant of . A completely continuous mapping is bounded, continuous and satisfies .

A subset with t-fibres , , is called nonautonomous set. If all fibres , , are compact, then is denoted as compact nonautonomous set and we proceed accordingly with other topological notions. Furthermore, one speaks of a bounded nonautonomous set , if there exists real and a point such that holds for all .

Finally, on a Banach space X, L(X) denotes the space of bounded linear operators and is the spectral radius of a .

Nonautonomous Difference Equations

Unless otherwise noted, let (X, d) be a complete metric space. We consider nonautonomous difference equations in the abstract form with continuous right-hand sides and defined on closed sets , . For an initial time , a forward solution to is a sequence with satisfyingfor all , , while an entire solution satisfies (2.1) on . The unique forward solution starting at in is denoted by ; it is denoted as general solution to and reads asas long as the compositions stay in . Under the inclusion , , the general solution exists for all and the process propertyholds; we introduce the nonautonomous set .

One denotes as -periodic with some , if , and tacitly hold for all . In this case the general solution satisfiesyielding a rather tame time-dependence. An autonomous equation is 1-periodic.

A nonautonomous set is called positively or negatively invariant (w.r.t. the difference equation ), if the respective inclusionholds; an invariant set is both positively and negatively invariant, that is, for all . One denotes as -periodic, if holds for all with .

The next two subsections provide some preparations on nonautonomous difference equations in Banach spaces :

Semilinear Difference Equations

Let , , be a sequence of bounded linear operators. For a linear difference equationwe define the transition operator byThen is understood as semilinear, if its right-hand side can be represented aswith continuous mappings , . The variation of constants formula [20, p. 100, Thm. 3.1.16] yields the general solution of in the formThe following result will be helpful in the construction of absorbing sets:

Lemma 2.1

Let be of semilinear form (2.5) and suppose there exist reals , withIf there exist reals , such that the nonlinearity fulfillsthen the general solution of satisfies the estimatefor all and .

Remark 2.2

(Linear growth) In case on one can choose , in (2.7) and the estimate (2.9) simplifies to

Proof

Let . It is convenient to abbreviate and we first assume that , . Given , from (2.6) and (2.7) we obtainand therefore the sequence satisfiesThus, the Grönwall inequality from [20, p. 348, Prop. A.2.1(a)] impliesand consequentlywhich is the claimed inequality (2.9).

Additive Difference Equations

We now address right-hand sideswhere is bounded and continuous, while , , is assumed to be completely continuous.

Lemma 2.3

If is of additive form (2.10), then the general solution of satisfies

Since for every is continuous and bounded, their composition (2.2) is also continuous and bounded. The estimate for the Darbo constant of will be established by mathematical induction. For the assertion is clear, since and the Lipschitz constant of the identity mapping is 1; it provides an upper bound for the Darbo constant (see [19, p. 81, Prop. 5.3]). For times , from , which holds because is completely continuous (cf. [19, p. 82, Prop. 5.4]), it follows thatfrom [19, pp. 79–80, Prop. 5.1]. This establishes the claim.

Pullback Convergence

In this section, suppose that is unbounded below and that , , i.e., generates a process on .

A difference equation is said to be pullback asymptotically compact, if for every , every sequence in with and every bounded sequence with , the sequence possesses a convergent subsequence.

Pullback Limit Sets

The pullback limit set of a bounded subset is given by the fibresFor pullback asymptotically compact nonautonomous difference equations it is shown in [20, p. 14, Thm. 1.2.25] that is nonempty, compact, invariant and pullback attracts , i.e., the limit relationholds. For positively invariant sets the defining relation (3.1) simplifies toTherefore, as a fundamental tool for the construction of pullback limit sets and attractors, as well as for forward attractors in Sect. 4.3, we state

Proposition 3.1

Suppose that has a nonempty, positively invariant, closed and bounded subset . If is pullback asymptotically compact, then the fibresdefine a maximal invariant, nonempty and compact nonautonomous set , which pullback attracts .

If the nonautonomous set is even compact, then Proposition 3.1 applies without the asymptotic compactness assumption.

Since generates a continuous process in discrete time, the assertion results via an adaption of [13, Prop. 5], where pullback asymptotic compactness yields that the intersection of the nested sets in (3.4) is nonempty.

Pullback Attractors

A pullback attractor of is a nonempty, compact, invariant nonautonomous set which pullback attracts all bounded nonautonomous sets . Bounded pullback attractors are unique and allow the dynamical characterisation(cf. [20, p. 17, Cor. 1.3.4]). Despite being pullback attracting nonautonomous sets within , the set constructed in Proposition 3.1 needs not to be a pullback attractor, since nothing was assumed outside of . Remedy provides the notion of a pullback dissipative difference equation . This means there exists a bounded set such that for every and every bounded nonautonomous set there is an absorption time such thatFor a uniformly pullback dissipative equation the absorption time S is independent of . One denotes as a pullback absorbing set.

If is pullback absorbing, then the set obtained from Proposition 3.1 becomes a pullback attractor, i.e., , and one has the characterisationA possibility to construct pullback absorbing sets provides

Proposition 3.2

(Pullback absorbing set) On a Banach space X, let and be of semilinear form (2.5) satisfying (2.7), (2.8). If the limit relationshold for all , then the difference equation is pullback dissipative with absorbing set . In case holds, the difference equation is uniformly pullback dissipative.

The assertion follows from Lemma 2.1 by passing over to the pullback limit in the estimate (2.9).

A construction of pullback attractors based on set contractions, rather than asymptotic compactness, is suitable for later applications to integrodifference equations (see Sect. 5):

Theorem 3.3

If a difference equation of additive form (2.10) is uniformly pullback dissipative andholds, then there exists a unique bounded pullback attractor of .

Remark 3.4

(Periodic equations) For -periodic difference equations and sets , it results from (2.4) that also the pullback limit sets from (3.1), the set from Proposition 3.1 and the pullback attractor are -periodic (cf. [20, pp. 21ff, Sect. 1.4]). Furthermore, Theorem 3.3 applies when .

The terminology of [20] and results therein will be used. Let denote the family of all bounded sets in . Then Lemma 2.3 ensures that the general solution is -contracting in the sense of [20, p. 15, Def. 1.2.26(i)].

Since has a bounded absorbing set , for every bounded nonautonomous set , there exists an such that holds for all . This implies that the S-truncated orbit , fibrewise given byis bounded. Hence, [20, p. 16, Prop. 1.2.30] implies that is -asymptotically compact, so has a pullback attractor by [20, p. 19, Thm. 1.3.9].

Finally, the pullback attractor is contained in the closure of the absorbing set , so is bounded and thus uniquely determined.

Forward Convergence

In the previous section, we constructed pullback attractors of pullback asymptotically compact nonautonomous difference equations as pullback limit sets of such absorbing sets. Our next aim is to provide related notions in forward time. Due to the conceptional difference between pullback and forward convergence some modifications are necessary, yet.

Above all, this requires a discrete interval to be unbounded above. Now the right-hand sides , , are defined on a common closed subset , i.e., the extended state space has constant fibres. Therefore, the general solution is well-defined.

Given a nonautonomous set , a difference equation is said to be

-asymptotically compact, if there exists a compact set such that K forward attracts , i.e.,

strongly -asymptotically compact, if there exists a compact set so that every sequence in with , as yields

Remark 4.1

If is positively invariant, then strong -asymptotic compactness (needed in Theorem 4.10 below) is a tightening of -asymptotic compactness (required in Theorem 4.9). Indeed, suppose that the sequence does not converge to 0. Hence, the strong -asymptotic compactness of and positive invariance of yields the contradiction

Forward Limit Sets

Let us investigate the forward dynamics of inside a nonautonomous set . We first capture the forward limit points from a single fibre :

Lemma 4.2

Suppose that is a bounded nonautonomous set. If is -asymptotically compact with a compact subset , then the fibresare nonempty, compact, and forward attract , i.e.,

An analogous result for pullback limit sets is given in [20, p. 9, Lemma 1.2.12].

Remark 4.3

(Characterisation of ) The fibres , , consist of points v such that there is a sequence with , andThis readily yields the monotonicity for all .

Let . Given a sequence with and , by the -asymptotic compactness of , we obtainSince is compact, . This implies that there exist a sequence in K satisfyingand a subsequence converging to . Thus,which implies that the subsequence converges to . Hence, by the characterisation (4.3), , i.e., is nonempty.

Now choose a sequence in . By Remark 4.3, for each fixed , there is a sequence satisfying and for all such thati.e., for every , there is a such thatSince , there is a satisfying for all . Pick , as a subsequence of and as a subsequence of in such a way thatClearly, for . Hence, we constructed a sequence with and such thatTherefore,Similarly as above, because K is compact, there is a subsequence converging to . Moreover, since is closed by definition, , which implies that is compact. Also note that , so , i.e., .

Suppose that does not forward attract for some , i.e., there exist a real and a sequence in with andAlthough the supremum in the Hausdorff semidistance in the left-hand side of (4.4) may not be attained due to no condition ensuring that the image is compact, there still exists a point for each with such thatThe above inequalities in fact give for all . On the other hand, since , we obtainand thus . Moreover, since K is compact, there is a convergent subsequence with limit . This shows by definition, and thusa contradiction to (4.4). Hence, every must forward attract .

Corollary 4.4

If in addition is positively invariant w.r.t. , then the inclusions hold for all .

Owing to the positive invariance of , every can also be written asComparing the respective relations (4.1) and (3.1), (4.2) and (3.2), (4.5) and (3.3) shows that the fibres are counterparts to the pullback limit set . However, their invariance property is missing. This is easily demonstrated by

Example 4.5

Let and . The difference equation in possesses the positively invariant and bounded set . This yields the apparently not even positively invariant sets for all .

Let . Given a point , thanks to and the process property (2.3), we obtainThis implies .

While the fibres from Lemma 4.2 yield the long term behaviour starting from a single fibre of , the following result addresses all forward limit sets of originating from within an entire nonautonomous set .

Theorem 4.6

(Forward -limit sets) Suppose that is positively invariant and bounded. If is -asymptotically compact with a compact subset , thenare nonempty and compact. In particular, forward attracts , i.e.,and called forward -limit set of .

Due to Corollary 4.4, is a union over nondecreasing sets and actually a limit.

Remark 4.7

(Characterisation of ) The forward -limit set consists of all points v such that there is a sequence with , and satisfying

Remark 4.8

(Periodic equations) For -periodic difference equations and sets the fibres are -periodic due to (2.4) and (4.1). If is moreover positively invariant, then are even constant and thus for all .

Proof of Theorem 4.6

Since is nonempty, there exists a point for all . This implies that v is also contained in , i.e., the forward -limit set is nonempty. From Lemma 4.2, we know that for each . This yields . Moreover, since K is compact and is closed, is also compact. The claimed limit relation is a consequence of (4.2) and .

The properties of are an immediate consequence of the fact that is an intersection of nested compact sets (cf. [23, p. 23, Lemma 22.2(5)]).

Note that Example 4.5 demonstrates that both the set , as well as the forward limit sets constructed in Theorem 4.6 are not invariant or even positively invariant. Yet, under additional assumptions weaker forms of invariance hold:

Theorem 4.9

(Asymptotic positive invariance) Suppose that is -asymptotically compact with a compact subset for a bounded, positively invariant . If for every sequence in with , one hasthen the forward -limit set is asymptotically positively invariant, that is, for every strictly decreasing sequence , there exists a strictly increasing sequence in as such that

Recall the definition of the neighborhoods and thus (4.6) reads as

Suppose by contradiction that there exists a fixed so that there is a sequence with and as satisfyingSince is continuous and is compact due to Theorem 4.6, is also a compact set. This implies that there exists awith such thatOn the other hand, with , by the assumption, we obtainimplying that . Additionally, since the set K is compact, there is a subsequence converging to . Therefore, by definition, the inclusion leads to , i.e., for all , a contradiction to (4.7). Thus, for this , there exists an integer large enough such thatRepeating inductively with and for all , we then obtain that is asymptotically positively invariant.

Theorem 4.10

(Asymptotic negative invariance) Suppose that is strongly -asymptotically compact with a compact subset for a bounded, positively invariant . If for every and , there exists a real such that for all , one has the implicationthen the forward -limit set is asymptotically negatively invariant, that is, for all , and , there are integers satisfying and such that

Consider reals and and take a point . Thanks to Remark 4.7, there is a sequence with , as , , and , and an integer withGiven a sequence with , , as , and , by the strong -asymptotic compactness of , we obtainSince K is compact, . This implies that there exist a sequence in K such thatand a subsequence converging to . Thus,implying with . Hence, by Remark 4.7, one has . Moreover, with and , by the assumption, we obtain for an integer large enough,Now the triangle inequality and the process property (2.3) yieldSetting , we then obtain that is asymptotically negatively invariant.

Asymptotically Autonomous Difference Equations

In general it is difficult to obtain the forward limit set given as limit of the fibres explicitly. This situation simplifies, if behaves asymptotically as an autonomous difference equation with right-hand side in a sense to be specified below. Here, it is common to denote the iterates of by , . A maximal, invariant and nonempty compact set attracting all bounded subsets of U is called global attractor of (cf. [8, p. 17]).

For a class of nonautonomous equations introduced next, the sets , , turn out to be constant and determined by the global attractor of .

Theorem 4.11

(Asymptotically autonomous difference equations) Suppose that has a bounded absorbing set and a global attractor . If is a forward absorbing set of and the conditionholds, then for all and in particular .

Remark 4.12

Asymptotically autonomous difference equations were also studied in [5] in order to show that the fibres of a pullback attractor to converge to the global attractor of the limit equation as . In these results, however, asymptotic autonomy is based on e.g. the limit relation(see [5, Thm. 1]) with sequences converging to some . This condition is clearly different from (4.9).

Given any , we have to show two inclusions:

Let . Due to Remark 4.3 there exist sequences and as withand it follows from (4.9) thatsince the global attractor of attracts the absorbing set A. This implies that , and since v was arbitrary, the inclusion holds for .

Conversely, since is compact, there exists an withfor all and we separately estimate the two terms on the right-hand side of this inequality. First, due to the invariance of there exists with and thereforeSecond, from one haswhich guarantees the remaining inclusion .

Hence, all are constant, thus and .

The following simple example illustrates the condition (4.9):

Example 4.13

(Beverton–Holt equation) If , then it is well known that all solutions to the autonomous Beverton–Holt equation starting with a positive initial value converge to (see [18, pp. 13ff]). We establish that an asymptotically autonomous, but nonautonomous Beverton–Holt equationshares this behaviour, whenever the sequence in satisfiesand grows at most polynomially. For instance, the relation (4.11) holds for the sequences with , . Indeed, the explicit representationof the general solution to (4.10) yields that holds uniformly in for any . Consequently, one hasand therefore (4.9) is valid with arbitrary subsets .

We continue with two sufficient criteria for the condition (4.9) to hold. Thereto we assume in the remaining subsection that is a Banach space.

Theorem 4.14

(Asymptotically autonomous linear difference equations) Suppose that , , , satisfyIf , then with right-hand side and with right-hand side fulfill the limit relation (4.9) on every bounded subset .

We note that (4.12) implies that is uniformly exponentially stable and that the sequences , are bounded. Thus, [2, Cor. 5 with ] implies that is bounded and the representation for all , shows that is bounded uniformly in from bounded subsets of X. Now it is easy to see that the difference solves the initial value problemwhose inhomogeneity satisfies uniformly in a from bounded subsets of X. Now using [2, Cor. 5 with ] guarantees that the sequence converges to 0 as uniformly in a from bounded subsets of X, that is, in particular (4.9) holds.

Theorem 4.15

(Asymptotically autonomous semilinear difference equations) Let be of semilinear form (2.5) such thatandwith , and . If and satisfy then with right-hand side and fulfill the limit relation (4.9) even exponentially on every bounded subset .

there exists a such that for all ,

for every there exists a such that

The assumption (4.13) holds in case with (see [22, Thm. 5]).

We proceed in two steps:

(I) Claim: All solutions to the autonomous equation are bounded, i.e.,This is a consequence of [20, p. 155, Thm. 3.5.8(a)].

(II) Let and we abbreviate , . Due to step (I) the sequence is bounded and we choose so large that for all . It is easy to see that the difference satisfies the equationand fulfills the initial condition . Using the variation of constants formula [20, p. 100, Thm. 3.1.16] resultsconsequentlyand the Grönwall inequality [20, p. 348, Prop. A.2.1(a)] yieldsIf we replace t by , then it resultsand therefore even exponential convergence holds in (4.9).

Forward Attractors

In the previous Sect. 3.2 we constructed pullback attractors of nonautonomous difference equations by means of Proposition 3.1 applied to a pullback absorbing, positively invariant nonautonomous set. Now it is our goal is to obtain a corresponding concept in forward time.

Mimicking the approach for pullback attractors, we define a forward attractor of as a nonempty, compact and invariant nonautonomous set forward attracting every bounded subset , i.e.,As demonstrated in e.g. [16, Sect. 4], forward attractors need not to be unique. They are Lyapunov asymptotically stable, that is, Lyapunov stable and attractive in the sense of (4.17) (see [12, Prop. 3.1]). While it is often claimed in the literature that there is no counterpart to the characterisation (3.5) of pullback attractors for forward attractors of nonautonomous equations, a suitable construction will be given now.

A nonautonomous difference equation is denoted as forward dissipative, if there exists a bounded set such that for every and bounded there is an absorption time such thatone says that is as a forward absorbing set.

Proposition 4.16

(Forward absorbing set) On a Banach space X, let be of the semilinear form (2.5) satisfying (2.7), (2.8) and let . If the limit relationshold for all and , then the difference equation is forward dissipative with absorbing set .

For constant positive sequences , , in (2.8) satisfying , both the pullback absorbing set from Proposition 3.2 and the forward absorbing set from Proposition 4.16 have constant fibres and simplify to .

The claim follows readily from relation (2.9) in Lemma 2.1.

Using [12, Prop. 3.2 with compact replaced by open and bounded] one shows

Proposition 4.17

Every bounded forward attractor has a nonempty, positively invariant, closed and bounded forward absorbing set.

First, this Proposition 4.17 allows us to choose a closed and bounded, positively invariant set . We then deduce a nonempty, invariant and compact nonautonomous set from Proposition 3.1.

Second, the construction of forward attractors requires . Different from the pullback situation (with being pullback absorbing), having an forward absorbing set does not ensure the forward convergence within , i.e.,and in particular not forward convergence of a general bounded nonautonomous set to . This is because may have forward limit points starting in which are not forward limit points from within . Corresponding examples illustrating this are given in [14].

Now on the one hand, the set of forward -limit points for the dynamics starting in is given byand is nonempty and compact as intersection of nested compact sets. It consists of all points such that there is a sequence with and with satisfyingOn the other hand, the set of forward limit points from within was constructed in Theorem 4.6. With being positively invariant, the chain of inclusions holds, while is not necessarily contained in (see Example 4.20 for an illustration), as well as

Theorem 4.18

Suppose that has a positively invariant, closed and bounded set . If the assumptions in Proposition 3.1 and Theorems 4.9–4.10 hold, then the following statements are equivalent:

is forward attracting , that is,

.

() Suppose that is forward attracting from within and that . Since , there exists a point , i.e., there are and such that andSince , there exists a sequence with and points satisfying . Moreover, by the forward attraction of , there exists an such thatCombining all of them, we obtainSince by definition, it then followsa contradiction to (4.20). Hence, holds.

() Suppose that , i.e., for all , and that is not forward attracting from within , i.e., there exist a real and a sequence in with andAlthough there is no condition ensuring the set is compact, which means the supremum in the Hausdorff semidistance may not be attained, there still exists a point for all and such thatThe above inequalities in fact give for all . Moreover, take a point , thenOn the other hand, by assumptions and definitions, is -asymptotically compact and both and are in for all , soimplying that both and are in K as well. Additionally, since K is compact, there are convergent subsequences with limit and with limit . This implies and by definitions. Combining this with for all , we arrive at the contradictionto the assumption. Thus, is forward attracting from within .

Corollary 4.19

Suppose in addition that is forward absorbing. If holds, then is a forward attractor of .

Due to Proposition 3.1 the set is already nonempty, compact, invariant and thus it suffices to show that is forward attracting. Thereto, suppose that is bounded and choose arbitrarily. With the forward absorption time we obtain from Theorem 4.18 thatand this yields the assertion.

We close this section with a simple, yet illustrative example:

Example 4.20

(Beverton–Holt equation) Given reals we consider the asymptotically autonomous Beverton–Holt equationin having the general solutionIt possesses the absorbing set and the forward -limit set . Depending on the constellation of the parameters one obtains the following capturing the forward dynamics:

For all fibres are constant. For two cases arise:Except for , where the pullback and forward dynamics of (4.21) differ, Corollary 4.19 applies and yields that the pullback attractor is the forward attractor .

: Solutions starting in at time first decay until time and then increase again, which yields

: Solutions starting in decay to and thus the fibres are constant on .

As a conclusion, in case is a positively invariant, forward absorbing nonautonomous set this section provided two concepts to capture the forward dynamics of , namely the limit set from Theorem 4.6 and the forward attractor constructed in Corollary 4.19. On the one hand, the limit set is asymptotically positively invariant, forward attracts and is contained in all other sets with these properties. It depends only on information in forward time. On the other hand, the forward attractor shares these properties, but is actually invariant. Its construction is based on information on the entire axis and more restrictively, relies on the condition from Theorem 4.18(b). The latter might be hard to verify in concrete examples, unless rather strict assumptions like asymptotic autonomy hold [5].

Integrodifference Equations

The above abstract results will now be applied to nonautonomous IDEs. For this purpose let be a measure space satisfying . Suppose additionally that is equipped with a metric such that it becomes a compact metric space.

We consider the Banach space of continuous -valued functions over equipped with the norm If is a nonempty, closed set, then is a complete metric space. Furthermore, we have , where is an unbounded discrete interval.

Given functions and , the Nemytskii operator is defined byand the Urysohn integral operators byWith these operators, a nonautonomous difference equation of the additive form (2.10) is called an integrodifference equation and explicitly reads as Such problems are well-motivated from applications:Hypothesis: For every we suppose: Then the Nemytskii operator satisfieswhile the Urysohn operators are globally bounded byand completely continuous due to [19, p. 166, Prop. 3.2].

For an integrodifferential equation with, e.g., a continuous kernel function , the forward Euler discretisation with step-size gives the IDE matching () with a compact and the Lebesgue measure .

Population genetics or ecological models of the form are investigated in [25], where is a parameter and e.g. continuous functions , . These problems are of the from () with a compact and the Lebesgue measure .

Let the compact set be countable, and be reals. Then defines a measure on the family of all countable subsets . The assumption ensures that . W.r.t. the resulting -integral the IDE () becomes Such difference equations occur as Nyström methods with nodes and weights as used in numerical discretizations and simulations [1] of IDEs ().

The function is such that is continuous and there exist reals with

The kernel function is such that is measurable for all , , and the following holds for almost all : is continuous for all and the limit holds uniformly in .

There exists a function , measurable in the second argument with and for almost all one has

In the following we tacitly suppose for all .

Proposition 5.1

(Dissipativity for ()) If – withhold, then the bounded and closed setis positively invariant, forward absorbing (if is unbounded above), pullback absorbing (if is unbounded below) w.r.t. () with absorption time 1.

Proof

Clearly the set is closed and due to (5.3) also bounded. Let with . Thus, and our assumptions readily imply thatand consequently holds for all . Thanks to for any bounded this inclusion guarantees that is positively invariant, but also forward and uniformly absorbing with absorption time .

Theorem 5.2

(Pullback attractor for ()) Let be unbounded below. If – are satisfied with (5.3) and there exists a such that hold, then the IDE () has a unique and bounded pullback attractor .

We aim to apply Theorem 3.3 to (). Thereto, Proposition 5.1 guarantees that () is uniformly pullback absorbing. Moreover, since the Lipschitz constant of the Nemytskii operator is an upper bound for its Darbo constant and because is completely continuous, the assertion follows.

Without further assumptions not much can be said about the detailed structure of the pullback attractor . Nevertheless, in case the functions satisfy monotonicity assumptions in the second resp. third argument, it is possible to construct “extremal” solutions in the attractor [21]. We illustrate this in

Example 5.3

(spatial Beverton–Holt equation) Let and , , be continuous functions describing the space- and time-dependent growth rates and a compact habitat . The spatial Beverton–Holt equationfor all fits into the framework of () with , ,and a continuous kernel function . Then – hold withand . If holds for some , then Theorem 5.2 yields the existence of a pullback attractor for (5.4). Since the functions are strictly increasing more can be said on the structure of . As in [21, Prop. 8], there exists an “extremal” entire solution being pullback attracting from above such thatWe illustrate both the pullback convergence to the solution , as well as the sets containing solutions in the pullback attractor in Fig. 1, where is equipped with the 1-dimensional Lebesgue measure, (artificial) and the Laplace kernel for the dispersal rate .

Fig. 1

Pullback convergence to the fibre (, initial function , left) and sequence of sets containing the pullback attractor (, right) for

The remaining section addresses forward attraction. For simplicity we restrict to the class of Urysohn IDEs

Hypothesis: For every we suppose:

For all there exists a function , measurable in the second argument with and for almost all one has

Proposition 5.4

(Dissipativity for ()) If – with hold, then the bounded and compact nonautonomous setis positively invariant, forward absorbing w.r.t. () with absorption time 2.

Let with and thus . Since the Urysohn operators are completely continuous, the fibres are compact. Thanks toit follows that is bounded. Moreover, holds for all and is positively invariant. Furthermore, from the inclusionwe deduce that is absorbing.

Theorem 5.5

(Forward limit set for ()) Suppose that – hold with additionally . If is relatively compact and is the forward absorbing set from Proposition 5.4, then the following are true:

is asymptotically positively invariant,

is asymptotically negatively invariant, provided is satisfied with

The relative compactness of the union holds for instance, if the kernel functions stem from a finite set or the images form a nonincreasing/nondecreasing sequence of sets.

By assumption the set is compact and this implies that () is strongly -asymptotically compact.

(a) By construction of the assertion results from Theorem 4.9.

(b) Let and choose so large that . We concludefrom assumption . After passing to the least upper bound over it follows that is a Lipschitz constant for on . Hence, the assumption (5.5) implies (4.8) and therefore Theorem 4.10 yields the claim.

The above results do apply to the following

Example 5.6

(spatial Ricker equation) Suppose that the compact is equipped with the -dimensional Lebesgue measure and that holds. Let denote a bounded sequence of positive reals, be continuous and be a bounded sequence in , . The spatial Ricker equationfits in the framework of () with and the kernel functionhence, (5.6) is defined on the cone . If is unbounded below, then (5.6) possesses a pullback attractor ; see Fig. 2 for an illustration.

Fig. 2

Functions contained in the fibres of the pullback attractor over the times (blue) and the forward limit set (red) for the spatial Ricker equation (5.6) with Laplace kernel (, ), , and the constant inhomogeneity . More detailed, depicted are the 4-periodic orbits (blue) of the spatial Ricker equation, which is autonomous for . In addition, the fibres also contain 0, a nontrivial fixed point and a 2-periodic orbit (Color figure online)

For our subsequent analysis it is convenient to set . We begin with some preparatory estimates. Above all, (5.6) satisfies the assumption with , which guarantees the global Lipschitz conditionIf we represent the right-hand side of (5.6) in semilinear form (2.5) withthen holds, as well as the global Lipschitz conditionIn order to obtain information on the forward attractor, we suppose that and that (5.6) is asymptotically autonomous in forward time, i.e., there exist and such thatIf , then it follows from the contraction mapping principle and (5.7) that the autonomous limit equationhas a unique, globally attractive fixed-point .

We choose and an absorbing set of the limit equation (5.9) such that is forward absorbing w.r.t. (5.6). If we assume , then the growth estimate (4.13) holds with due toIt follows from (5.8) that every nonlinearity has the Lipschitz constant . Consequently, if furthermore converges exponentially to with rate , then Theorem 4.15 applies under the assumptionand thus (4.9) holds. Hence, we derive from Theorem 4.11 the relationsIf we concretely definethen this implies the elementary estimateHence, we set and (5.10) simplifies to . This assumption can be fulfilled for sufficiently close to 0, which requires the kernel data or the asymptotic growth rate to be small.

Even more concretely, on the habitat with some real we again consider the Laplace kernel with dispersal rate , which yields . In this framework, an illustration of the forward limit set and subfibres of the pullback attractor is given in Fig. 2. Here, both the pullback attractor and the forward limit set capture the long term behaviour of (5.6).