In this paper, we consider the dynamics of multi-valued processes generated by nonautonomous lattice systems with delays. In particular, the effects of small delays on the asymptotic behavior of multi-valued nonautonomous lattice systems and finite lattice approximation of infinite delay lattice systems are presented. We do not assume any Lipschitiz condition on the nonlinear term, just a continuity assumption together with growth condition, so that uniqueness of solutions of the problem fails to be true.
The asymptotical behavior of multi-valued dynamical systems has been extensively studied by a lot of authors due to its importance in theory and applications. The theory of global attractors for multi-valued dynamical systems was initiated by Babin and Vishik [1] to study evolution differential equations without the assumption that the solution of the initial value problem is unique. Later, the existence of global attractors for multi-valued semidynamical systems was explored by Ball [2], Kapustyan and Melnik [10], and trajectory attractors of equations of mathematical physics were developed by Chepyzhov and Vishik [7] (see also [16] and [15]). For the nonautonomous case, the general theory and applications of pullback attractors or kernel sections (which are very close to the pullback attractors) for deterministic nonautonomous differential equations without the uniqueness of solutions were studied in [5,11–13,19] and other references, and global random attractors for stochastic differential equations with non-Lipschitz nonlinearities were investigated in [4,8,9] and the references cited therein.
The existence of attractors for single-valued lattice systems has been investigated by [3,14,21,22,24] (see [20] for the first-order retarded lattice systems as well). Recently, a rather general theorem about existence of solutions for an abstract differential equation in a Banach with application to a lattice dynamical system with delays was presented in [6]. Very recently, a new method to check the asymptotical upper-semicompactness was given in [18] for the multi-valued process generated by reaction-diffusion delay equations on an unbounded domain for which the uniqueness of solutions need not hold.
In the present paper, we first extend the results given in [20] by proving the existence of a pullback attractor for the first-order retarded lattice systems without the uniqueness of solutions. It is worth mentioning that here we consider the operator , p times, where p is any positive integer and △ denotes the discrete one-dimensional Laplace operator, and we assume that the external force satisfies
where , and L are given in Section 3. Therefore, in order to obtain the pullback attractor we use the general theory of attractors for multi-valued nonautonomous dynamical systems developed in [4] and some techniques to deal with the non-Lipschitz nonlinearities in [6]. In addition, we introduce a concept of norm-to-weak upper-semicontinuity for multi-valued processes with delays (see Theorem 8 below) which is an extension of the definition of upper-semicontinuity for such systems.
Then, we show the effects of small delays on the asymptotic behavior of multi-valued nonautonomous lattice systems, and prove the upper semicontinuity of pullback attractors when the infinite lattice system is approached by the finite lattice system. The stability result of pullback attractors for multi-valued processes was established in [17]. However, there is little reference on the upper semicontinuity of pullback attractors for the first-order nonautonomous retarded lattice systems without uniqueness of solutions. The main new difficulty comes from lack of the uniqueness of solutions and time dependence of the absorbing set.
The paper is organized as follows. In Section 2 we recall basic concepts and some necessary results concerning multi-valued processes. In Sections 3–6, we prove the existence of pullback attractors for retarded lattice reaction-diffusion equations in the context of multi-valued nonautonomous dynamical systems. The upper semicontinuity of pullback attractors are consider in Sections 7 and 8.
Preliminaries
Let X be a Banach space with norm , and denote by the class of nonempty closed subsets of X. Denote by the Hausdorff semidistance between two nonempty subsets of a Banach space , which are defined by
where . Finally, denote by the open neighborhood of radius of a subset A of a Banach space X.
A family of mappings , , , is called to be a multi-valued process (MVP in short) if it satisfies:
, , ;
, , , .
Let be a nonempty class of parameterized sets .
A collection of some families of nonempty closed subsets of X is said to be inclusion-closed if for each ,
also belongs to , see, e.g., [9].
Let be a multi-valued process on X.
is called a -pullback absorbing set for if for any and each , there exists a such that
is said to be -pullback asymptotically upper-semicompact in X with respect to if for any fixed , any sequence has a convergent subsequence in X whenever (), with .
A family of nonempty compact subsets is called to be a -pullback attractor for the multi-valued process , if it satisfies
is invariant, i.e.,
attracts every member of , that is, for every and any fixed ,
A mapping is called a complete orbit of the multi-valued process if for every and , the following holds:
If, in addition, there exists such that belongs to for every , then ψ is called a -complete orbit of .
With regard to the existence of pullback attractors, we have
(Caraballo et al. [4], Wang and Zhou [19]).
Letbe a inclusion-closed collection of some families of nonempty closed subsets of X andbe a multi-valued process on X. Also letis norm-to-weak upper-semicontinuous in x for fixed,(i.e., ifin X, then for any, there exists asuch that(weak convergence)). Suppose thatis-pullback asymptotically upper-semicompact in X andhas a-pullback absorbing setin. Then, the-pullback attractoris unique and is given by, for each,
Multi-valued lattice systems with delays
Let
and equip it with the inner product and norm as
The norm of is usually written as . Let be a given positive number, which will denote the delay time. We use to denote the set of continuous functions from into . Clearly, is a Banach space with norm for . For any , we use to denote the set of continuous functions from into . Given , and a function , for each we denote by the function defined on by the relation , . In the sequel C denotes an arbitrary positive constant, which may be different from line to line and even in the same line. In order to state the existence result of pullback attractors for multi-valued lattice systems with delays, we need to use the following remark.
Let E be a real Banach space and let , with norms and , respectively, where . Let be the space E endowed with the weak topology. We consider the space . From the definition in [6], we know that in if
We shall show that in if and only if
Indeed, if in , in view of
then it is easy to see that (2.4) holds true. Conversely, if (2.4) holds true, we need to prove in . Suppose not. Then there exist and sequences with () and with , such that
Without loss of generality, we assume that . Then (2.4) and imply that for all k sufficiently large,
which contradicts with (2.5).
Finally, from the above and the definition of the convergence in , it is clear that if in , then in .
By slightly modifying the proof of Theorem 3.4 in [19], in view of Remark 7, we have the following existence result of pullback attractors.
Letbe a inclusion-closed collection of some families of nonempty closed subsets ofandbe a multi-valued process on. Also letis norm-to-weak upper-semicontinuous in x for fixed,(i.e., ifin, then for any, there exists asuch thatin). Suppose thatis-pullback asymptotically upper-semicompact inandhas a-pullback absorbing setin. Then, the-pullback attractoris unique and is given by, for each,
The following theorem presents a method to check the pullback asymptotically upper-semicompactness in .
Letbe a multi-valued process on, and letbe a inclusion-closed collection of some families of nonempty closed subsets of. Suppose that for any fixed, everyand any, there exist, aand asuch that
for each fixed,
for all,,,with,
for all,,
Thenis-pullback asymptotically upper-semicompact in.
It suffices to prove that for each fixed , and any , any sequence with (), and with , this last sequence is relatively compact in . Now let , , sequences (), and be given arbitrarily. Without generality, we assume that for all . Then we will show that is relatively compact in .
Noticing that assumptions (1) and (3) imply that for each fixed , is relatively compact in . On the other hand, by assumptions (2) and (3), we deduce that for all , , , with ,
This implies that for all , with , we have
Then by the Arzelà–Ascoli theorem, we can conclude that is precompact in . □
Multi-valued lattice systems with delays: Setting of the problem
Let us consider the following first order lattice dynamical system with finite delays
with the initial condition
where . In (3.1), p is any positive integer and , p times, where △ denotes the discrete one-dimensional Laplace operator, which is defined by . We also write , and
It is easy to see by induction that
for all in .
We consider the following conditions:
There exist two positive constants and such that
are continuous and satisfy that , for all , where , and .
, for all , where , and .
are continuous and verify that , for all , where , .
The delay function satisfies that , for all , where .
, for all and a.a. , where are Caratheodory, that is, measurable in r and continuous in x.
Also, , , , and defining
We also assume that , .
The external force is such that
which implies that
where , and L is given in Lemma 12.
If is defined by follows: , ,
then we can extend the following results to the case of lattices. For more details please see [23] and the references cited therein.
Let (H1)–(H7) hold. Then for each, there existssuch that ifand, then Eqs (
3.1
) and (
3.2
) admit at least a solutiondefined on, and v belongs to the space.
First, we prove that is well defined and bounded. By (H3) we have
Then,
By the similar arguments in [6], in view of (H4) and (H6), we find that
and
Note that
Since is fixed, then using (3.4)–(3.7) we obtain that maps the bounded sets of into the bounded sets of .
Now we show that is continuous. Let in . Then for any , there exists such that
and
Also, by (H2) and (H4), we can choose such that if , then
and
Therefore, using (H4) we obtain
and similarly, by (H3) we deduce that
From (H6) and Lebesgue’s theorem that for all n sufficiently large, we have
and consequently,
Then it follows from , (3.3) and (3.8)–(3.10) that is continuous. Thus, Theorem 10 and Corollary 13 in [6] imply that for any , there exists at least one solution in a maximal interval . □
Estimate of solutions
Now, we shall obtain some estimates of solutions which are needed for proving the global existence of solutions and defining a pullback absorbing set.
In addition to the assumptions (H1)–(H7), suppose thatandThen for any initial data, every solutionof Eqs (
3.1
) and (
3.2
) satisfieswhereandis the maximal time of existence.
Taking the inner product of Eq. (3.1) with in , we have
Let and be positive parameters to be fixed later on. Multiplying (4.4) by , and using Young’s inequality, (H2) and (H4), we find that
Integrating (4.5) over and using (H5)–(H6) and Young’s inequality, we obtain that
and
Therefore,
Let . Then we can neglect one term since by (4.1)
if is chosen small enough. Now set instead of t, where . Multiplying by and neglecting the negative terms, it yields
Then we can rewrite this expression as
where we have used the notation , . Since , we have . Using Gronwall’s Lemma, we have
Hence,
which completes the proof of this lemma. □
Assuming the conditions of Lemma 12, it follows from Lemma 11 that every local solution of Eqs (3.1) and (3.2) can be defined globally, and thus there is a multi-valued process on corresponding to the system (3.1)–(3.2) defined by
We denote by the set of all functions such that
and denote by the class of all families such that , for some , where denotes the family of all nonempty closed subsets of and denotes the closed ball in centered at zero with radius . It is evident that is inclusion-closed. Then we have the following result.
Suppose the same hypotheses of Lemma
12
. Then the multi-valued processcorresponding to Eqs (
3.1
) and (
3.2
) possesses a closed-pullback absorbing setin.
Denote by the nonnegative number given for each by
and consider the family of closed bounded balls in defined by
It is straightforward to check that , and moreover, by (4.12) and (4.13), the family of is -pullback absorbing for the multi-valued process on and thus the proof is completed. □
Estimate of the tails
In order to obtain an estimate of the tails of solutions, we first need the following lemma.
Letand χ be the smooth function given in the proof of Theorem
15
. Thenwhere, for some positive constantdepending on p.
We will argue by induction. Let us prove the lemma for p odd. The proof for p even is similar. For , we have
where and obviously we have .
Now, suppose that Lemma 14 holds for . If , then we obtain
Note that
where . Then, using the induction hypothesis we find that
where
By (5.3) and (5.4), we get
with
Estimating the three last terms in (5.6), in view of (3.3) and (5.5), we have
which completes the proof of Lemma 14. □
Let (H1)–(H7) and (
4.1
)–(
4.2
) hold. Then for any, anyand every, there existandsuch that for any solutionsatisfies
Choose a smooth function χ such that for , and
Then there exists a constant such that for . Let M be a positive integer which will be specified later, and set with
Taking the inner product of Eq. (3.1) with , in view of (5.1), we get
Then, arguing as in the proof of Lemma 12 we have
Integrating (5.10) from to , similar to the arguments in (4.6) and (4.7), we find that
Let . By assumption (4.1), we can neglect the forth term in (5.11) if is chosen small enough. Setting now instead of (where ), multiplying by , it yields
By Lemmas 12 and 14, we deduce that
Then
Since , and , taking into account assumption (H7), we can choose M and s sufficiently large such that for any and every , we get that for that
and
Thanks to (5.14)–(5.16), when M and s are sufficiently large, we obtain for that
where . Applying Gronwall’s inequality and using (4.2), when M and s are sufficiently large, we find that for any solution ,
The proof of this theorem is finished. □
Existence of the pullback attractor
Let E be a real Banach space with its dual , with norms , respectively, and let be the space E endowed with the weak topology.
The following result will be used in the proof of norm-to-weak upper-semicontinuity of the multi-valued process .
Assume that E is reflexive and separable. Letbe a sequence of continuous functions frominto E. If for any fixed,is relatively compact in, and equicontinuous, that is, for every, there exists a, independent of n, such thatthen there exist aand a subsequenceofsuch thatin, i.e.,
The proof is similar to the arguments of Theorem 4 in [6], so it is omitted here. □
For convenience, let be the space endowed with the weak topology. We consider the space , and we say that in if in for all .
Let (H1)–(H7) and (
4.1
)–(
4.2
) hold. Then the multi-valued processassociated to Eqs (
3.1
) and (
3.2
) possesses a unique pullback attractorin.
It is worthy mentioning that in fact we can prove the map is upper-semicontinuous by using the tail estimates and the Arzelà–Ascoli theorem. Here we use the concept of the norm-to-weak upper-semicontinuity, it is also useful for partial functional differential equations.
We divide the proof into two steps.
Step 1. Since the fact that the map has closed graph follows easily from the norm-to-weak upper-semicontinuity of the map, now we first only need to prove that the map is norm-to-weak upper-semicontinuous. Suppose not. Then there exist , , ϕ, a neighborhood of in and sequences , such that . For any , let be such that . By Lemma 12 and in , there exists a such that
Fix now . Taking into account (6.1), passing to a subsequence, we can state that in . In order to apply Lemma 16, we need to obtain the equicontinuity property. To do this, without loss of generality, we assume that with , and then in a similar way as in Lemma 11, using (3.3), (6.1), and assumptions (H1), (H3), (H4) and (H6), we have for all ,
Then Lemma 16 implies the existence of a subsequence converging in to some function , and by the similar arguments of Theorem 4 in [6], we can see that v is a solution of Eq. (3.1). Recall that in , hence in where for any . Obviously, in , and thus in , where , which is a contradiction.
Step 2. Invoking Theorems 8 and 9, in view of Lemma 12 and Theorem 15, now it suffices to check condition (2) of Theorem 9.
Let be given arbitrarily, and let be a fixed positive constant which is given in Theorem 15. Without loss of generality, we assume that with . Then for any fixed , any and ,
Since for all v in , by assumption (H7) and (4.12) we have for all s sufficiently large,
Using the similar arguments as in Lemma 11, assumptions (H3)–(H4), (H6)–(H7) and (4.12) imply that
and
if s is sufficiently large. It follows from that
From (6.3)–(6.8), we conclude that for any fixed and every , when s is sufficiently large, we have
and thus this completes the proof of Theorem 17. □
Lattice dynamical systems with small delays
We consider a family of lattice systems with delays perturbed by ,
with the initial condition
where satisfy
In this section, and in assumption (H6) belong to , , and denote . We also assume that , .
Denote by and , respectively, the two classes of all families and such that and , for some , where is given by (4.13), and denote the nonempty closed subsets of and , and denote the closed balls in and centered at zero with radius , respectively.
Assuming the conditions (H1)–(H4), (H6)–(H7), (7.3) and
hold true, it follows from Theorem 17 that for any fixed , the multi-valued process associated to Eqs (7.1) and (7.2) possesses a unique pullback attractor in . This section is devoted to show the limiting behavior of the pullback attractor as .
When , we consider the following lattice system:
with the initial condition
Let (H1)–(H4), (H7) andhold.
Then there exists a unique pullback attractorfor the multi-valued processongenerated by Eqs (
7.6
) and (
7.7
), and for each,
Letbe a multi-valued process ongiven byand for each, let
Thenis a pullback attractor for the multi-valued process.
By slightly modifying the arguments in Sections 3–6, the conclusion (1) follows immediately. Then by the definition of , the invariance of for the multi-valued process and (7.8), we can deduce that , , .
Arguing as in the proofs of Theorems 15 and 17, we can conclude that for any , any and every , there exist , and a such that
for all , , , with ,
for all , , ,
Then analogous to the arguments of Theorem 3.8 in [17], we can see that for any fixed , is compact in , and for any , there exists a such that for all , , and all , we have
Hence is a pullback attractor for the multi-valued process . □
Assume (H1)–(H4), (H6)–(H7) and (
7.3
)–(
7.5
) hold. Letbe a sequence of positive numbers withwhen, and let,, then there exists a subsequenceofandsuch that
Let be a -complete orbit of with , and let , where and
Then by the similar arguments in Lemmas 12 and 13, we deduce that the family of closed bounded balls in belongs to , and for any , , . Hence for any , is uniformly (with respect to ) bounded in . Since is a Hilbert space, there exist a subsequence of (still denote ) and such that
Then we prove that the weak convergence in (7.12) is a strong one. In fact, for any and any fixed , by Theorem 15 and (7.10), there exists an such that for all ,
On the other hand, (7.12) implies that for any fixed ,
and consequently, there exists such that
Hence, we deduce from (7.13) and (7.14) that for any fixed and ,
Let , , be a sequence of closed intervals of . Without loss of generality, we assume that with . By (3.3) and assumptions (H1)–(H4) and (H6)–(H7), we can deduce that for all ,
Noticing that for any , , hence (7.16) and (H7) imply that for any , we can find a such that for all with , we have for all ,
Then by the Arzelà–Ascoli theorem, and the technique of diagonal subsequence in , there exist a subsequence of (still denote ) such that
for any compact interval . For any fixed , since is arbitrary and , , we can define a function
Then it is clear that for any fixed , there exist a subsequence of (still denote ) such that
and .
Since is a solution of Eqs (7.1) and (7.2) with , we have
Since and are continuous with respect to v (see Lemma 11), then by (7.18) and Lebesgue’s theorem we deduce that for any ,
(7.18) implies that there exists a constant such that
Noticing that , . Hence for any , there exists a such that
On the other hand, it follows from that for sufficiently large n, we have
By (7.22) and (7.23), we get that for all n sufficiently large,
Then by (3.3), (7.18), (7.20) and (7.24), we have for any ,
As is arbitrary, we get the inequality
Since is arbitrary, (7.25) holds true for all . Recall , and , we see that is a -complete orbit of . By the definition of , we have for each , and thus the proof is finished. □
From Lemma 20 and the argument of contradiction, we obtain the following upper semicontinuity of the pullback attractor.
Assume (H1)–(H4), (H6)–(H7) and (
7.3
)–(
7.5
) hold. Letbe the pullback attractor for the multi-valued processgenerated by Eqs (
7.1
) and (
7.2
), and letgiven in Lemma
19
be the pullback attractor for the multi-valued process. Then for any,whereis the Hausdorff semidistance between two nonempty subsets of.
Upper semicontinuity of pullback attractors for finite-dimensional delay approximation systems
In this section, we use the pullback attractors of finite truncated ODEs with delays to consider the approximation of the pullback attractors for the following first order delay lattice dynamical system
with the initial condition
For each positive integer , we investigate the following truncated ODEs with delays:
with the initial functions
where the functions , , and () are exactly chosen as the same ones as in (3.1). We rewrite Eqs (8.3) and (8.4) in the vector form as
with the initial condition
where , , , , , , , and
We consider Eqs (8.3) and (8.4) in the state space . Denote by the class of all families such that , for some , where denotes the set of all functions satisfying (4.13), denotes the family of all nonempty closed subsets of and denotes the closed ball in centered at zero with radius . Obviously, is inclusion-closed. Analogous to the arguments in Lemmas 11–13, Theorems 15 and 17, we obtain the following results.
Let (H1)–(H7) and (
4.1
)–(
4.2
) hold. Then we can define a multi-valued processonby settingand the multi-valued processpossesses a closed-pullback absorbing setin, whereis defined byand
Let (H1)–(H7) and (
4.1
)–(
4.2
) hold. Then for any, anyand every, there existandsuch that for any solutionsatisfies
Let (H1)–(H7) and (
4.1
)–(
4.2
) hold. Then the multi-valued processassociated to Eqs (
8.3
) and (
8.4
) possesses a unique pullback attractorin.
We use to denote the extension of such that for . In this sense, can be regarded as a natural embedding into , and we get
Hence, we also use to denote in the following.
In order to establish the upper semicontinuity of pullback attractors when the infinite delay lattice system is approached by the finite delay lattice system, we will need the following lemma.
Assume (H1)–(H7) and (
4.1
)–(
4.2
) hold. Letand, respectively, be the pullback attractors for the multi-valued processesandgenerated by Eqs (
8.3
)–(
8.4
) and (
8.1
)–(
8.2
), and let,, then there exists a subsequenceofandsuch that
Let be a -complete orbit of with . Then by Theorem 6 and Remark 25, we see that for any , , . Let , , be a sequence of closed intervals of . Since is a Hilbert space, arguing as in the proof of Lemma 20, in view of Theorem 15, Remark 25, the Arzelà–Ascoli theorem and the technique of diagonal subsequence in , we obtain that there exist a subsequence of (still denote ) and such that
for any compact interval . For any fixed , we can define a function , . It follows from (8.6) that there exist a subsequence of (still denote ) such that
and .
Noticing that is a solution of Eqs (8.3) and (8.4) with . Hence,
By (3.3) and (8.6), we obtain that for any ,
as and by , we have
By slightly modifying the proof of Lemma 11 and Theorem 15, in view of (8.6), (H2)–(H4) and (H6), we deduce that for any , there exists such that for all n sufficiently large,
and in the similar way, we get
Hence, (8.6) and (8.8)–(8.14) imply that for any ,
Since and are arbitrary, in view of , and , we deduce that v is a -complete orbit of generated by Eqs (8.1) and (8.2). By Theorems 6 and 17, we have for each , then the conclusion follows immediately from (8.7). □
By Lemma 26 and the argument of contradiction, we obtain that the pullback attractors of finite-dimensional delay approximation systems converge to the pullback attractor of the original system.
Assume (H1)–(H7) and (
4.1
)–(
4.2
) hold. Letand, respectively, be the pullback attractors for the multi-valued processesandgenerated by Eqs (
8.3
)–(
8.4
) and Eqs (
8.1
)–(
8.2
). Then for any,whereis the Hausdorff semidistance between two nonempty subsets of.
Footnotes
Acknowledgements
This work was supported by NSF of China under Grant No. 11571153, the Fundamental Research Funds for the Central Universities under Grant No. lzujbky-2016-100, the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry, and Creative Experimental Project of National Undergraduate Students under Grant No. 201310730085.
The authors wish to express their thanks to the reviewer for his/her very careful reading of the paper, giving valuable comments and suggestions. The author also thanks the editors for their kind help.
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