Tail convergences of pullback attractors for asymptotically converging multi-valued dynamical systems

Abstract

In this paper we study pullback attractors of multi-valued dynamical systems that are asymptotically convergent. It is shown that, under certain conditions, the components of the pullback attractor of a dynamical system can converge in time to those of the pullback attractor of the limiting dynamical system. Particular examples are asymptotically autonomous and asymptotically periodic pullback attractors. Different criteria theorems requiring different conditions are established and their applicability and advantages are highlighted.

Keywords

Pullback attractor asymptotic autonomous system multi-valued limiting equation

1. Introduction

Attractor theory is a useful tool to learn about the long-time behavior of a dynamical system, see for instance [9,14], etc. One of the attractors that has attracted much attention in last two decades is the so-called pullback attractor, which is a time-indexed family of compact sets, where each compact set attracts the solution trajectories at the corresponding time as the initial data is pulled back to minus infinity [9,12].

A pullback attractor gives rich information of the asymptotic dynamics of the dynamical system. On one hand, it depicts the limiting behavior of solution trajectories from the distant past to any observation time, and on the other hand, it has close relationship with other kinds of attractors. For instance, the component sets of a pullback attractor can be interpreted as the kernel sections of a uniform attractor which describes the limiting behavior of the system in the distant future [3,18], and can have an equivalent relationship with cocycle attractors [2,5,8], while cocycle attractor theory is known as a useful tool to study the asymptotic behavior of random dynamical systems [7,15,19].

Generally, the time-dependence of a pullback attractor is directly related to the non-autonomous characteristic of the dynamical system caused by the time-dependent forcing field. For instance, if the forcing field is periodic, then the pullback attractor as a set-valued mapping is also periodic with the same period, see Theorem 2.5. Hence, intuitively, if the non-autonomous forcing of a dynamical system becomes more and more autonomous in some sense, the non-autonomous nature of the pullback attractor should correspondingly become weaker and weaker. This leads to the study of asymptotically autonomous pullback attractors, e.g., [10,11,16].

Most recently, for single-valued dynamical systems Li et al. [13] showed that

Lemma 1.1 ([13]).

Let $A$ be the pullback attractor of a (single-valued) process U and let $A$ be the global attractor of a (single-valued) semigroup S. Suppose that

$A$ is forward compact, i.e., the union $\overline{⋃_{t ⩾ 0} A (t)}$ is compact;

for any ${x_{τ}}$ with ${lim}_{τ \to \infty} x_{τ} = x_{0}$ , $\begin{matrix} (1.1) & lim_{τ \to \infty} d_{X} (U (τ + t, τ, x_{τ}), S (t, x_{0})) = 0, \forall t ⩾ 0 . \end{matrix}$

Then

\begin{matrix} lim_{t \to \infty} dist (A (t), A) = 0 . \end{matrix}

Lemma 1.1 improves the corresponding results in [10] where the two conditions were assumed with more uniformity. However, it can still be hard to apply in applications since the forward compactness is usually technically difficult to verify especially in unsmoothing PDE problems. Hence, Cui [4] established the following alternative result using forward boundedness condition instead.

Lemma 1.2 ([4]).

Let $A$ be the pullback attractor of a (single-valued) process U and let $A$ be the global attractor of a (single-valued) semigroup S. Suppose that

$A$ is forward bounded, i.e., there is a bounded set B such that $⋃_{t ⩾ 0} A (t) \subset B$ ;

the following asymptotically autonomous condition holds $\begin{matrix} (1.2) & lim_{t \to \infty} sup_{x \in B} d_{X} (U (t + T, t, x), S (T, x)) = 0, \forall T > 0 . \end{matrix}$

Then

\begin{matrix} lim_{t \to \infty} dist (A (t), A) = 0 . \end{matrix}

In this paper, we generalize the above results in three directions: first, we study multi-valued dynamical systems rather than only single-valued ones; second, we consider the convergence of pullback attractors to pullback attractors which covers the particular case that pullback attractors converge to semigroup attractors; third, both forward and backward convergences are studied and, moreover, under equi-attraction condition we show that the convergences can be in the full Hausdorff metric sense.

It is worth noting that the difference between conditions (1.1) and (1.2) can be crucial in particular applications. In single-valued cases, condition (1.1) is equivalent to $\begin{matrix} lim_{t \to \infty} sup_{x \in E} d_{X} (U (t + T, t, x), S (T, x)) = 0, \forall E compact, T > 0, \end{matrix}$ so it is strictly weaker than (1.2). But the multi-valued functions defined on $R^{+} \times R$ $\begin{matrix} η (t, x) = \{\begin{matrix} [\frac{1}{t} - 1, \frac{1}{t} + 1], & x > 0; \\ 0, & x ⩽ 0, \end{matrix} and S (x) = \{\begin{matrix} [- 1, 1], & x > 0; \\ 0, & x ⩽ 0, \end{matrix} \end{matrix}$ show that it is not the case for multi-valued dynamical systems, since for any bounded $B \subset R$ , ${lim}_{t \to \infty} {sup}_{x \in B} dist (η (t, x), S (x)) = 0$ while ${lim}_{t \to \infty} dist (η (t, 1 / t), S (0)) = 1$ , i.e., condition (1.1) can fail under condition (1.2).

Inspired by the above observation, a scalar differential inclusion is studied as an example highlighting the advantages of Lemma 1.2 over Lemma 1.1. It is shown that the pullback attractor is asymptotically more and more periodic if so is the non-autonomous forcing, and is asymptotically autonomous if the forcing is asymptotically autonomous as well. Moreover, the equi-attraction of the pullback attractor is also proved, which makes the convergences in this application all valid in the full Hausdorff metric sense.

2. Pullback attractors of multi-valued processes

In this section we recall some preliminaries on pullback attractors of multi-valued dynamical systems, and then show that the pullback attractor of a periodic system is periodic.

Let $(X, d_{X})$ be a complete metric space and $P (X)$ the space of nonempty closed bounded subsets of X. Denote by $R_{⩾}^{2} : = {(t, τ) \in R^{2} : t ⩾ τ}$ and $R_{+} : = [0, \infty)$ . The Hausdorff semi-metric between nonempty sets in X is denoted by dist, i.e., $\begin{matrix} dist (A, B) : = sup_{a \in A} inf_{b \in B} d_{X} (a, b), \end{matrix}$ and denote the Hausdorff metric by ${dist}_{H} (A, B) : = max {dist (A, B), dist (B, A)}$ .

2.1. Preliminaries

A mapping $U : R_{⩾}^{2} \times X \to P (X)$ is called a multi-valued process if $U (t, τ, x) \subset U (t, s, U (s, τ, x))$ and $U (τ, τ, x) = x$ for all $t ⩾ s ⩾ τ$ and $x \in X$ . It is called strict if $U (t, τ, x) = U (t, s, U (s, τ, x))$ for all $t ⩾ s ⩾ τ$ and $x \in X$ .

Definition 2.1.
A family $A = {A (t)}_{t \in R}$ of compact sets is called the pullback attractor of a multi-valued process U if
it is pullback attracting, i.e., for every nonempty bounded set D in X, $\begin{matrix} lim_{τ \to - \infty} dist (U (t, τ, D), A (t)) = 0, \forall t \in R; \end{matrix}$

it is negatively semi-invariant, i.e., $A (t) \subset U (t, τ, A (τ))$ for all $t ⩾ τ$ ; and

it is minimal among families satisfying the above conditions.
Moreover, it is called strictly invariant if $A (t) = U (t, τ, A (τ))$ for all $t ⩾ τ$ .
Definition 2.2.
A family ${E (t)}_{t \in R}$ of nonempty sets is said to be
forward bounded/compact, if there exists a bounded/compact set B such that $\begin{matrix} ⋃_{t ⩾ 0} E (t) \subset B; \end{matrix}$

backward bounded/compact, if there exists a bounded/compact set K such that $\begin{matrix} ⋃_{t ⩽ 0} E (t) \subset K . \end{matrix}$

The following existence theorem is well-known.
Lemma 2.3 ([1, Lemma 2.5]).

Suppose that there exists a pullback attracting family of compact set ${K (t)}_{t \in R}$ and that the map $x \mapsto U (t, τ, x)$ has closed graph for all $t ⩾ τ$ . Then U has a pullback attractor $A = {A (t)}_{t \in R}$ given by $\begin{matrix} A (t) = \overline{⋃_{E \in P (X)} ω (t, E)}, \forall t \in R, \end{matrix}$ where $ω (t, B)$ is the pullback ω-limit set of B under U. Moreover, if U is strict and the pullback attractor $A$ is backward bounded, then $A$ is strictly invariant.

A special multi-valued process is the so-called multi-valued semigroup of which the dynamical behavior depends only on the elapsed time and not on the initial time. It is defined as a multi-valued mapping $S : R_{+} \times X \to P (X)$ satisfying $S (0, x) = x$ and $S (t + s, x) \subset S (t, S (s, x))$ for all $t, s \in R_{+}$ and $x \in X$ .

Definition 2.4.
A compact set $A \subset X$ is said to be the global attractor of multi-valued semigroup S if
it is globally attracting, i.e., for every nonempty bounded set D in X, $\begin{matrix} lim_{t \to \infty} dist (S (t, D), A) = 0; \end{matrix}$

it is negatively semi-invariant, i.e., $A \subset S (t, A)$ for all $t ⩾ 0$ ;

it is the minimal compact set satisfying the above conditions.

2.2. Periodicity of pullback attractors

In this part we prove that the pullback attractors of periodic processes are periodic. Similar results are also seen in Wang [17] where random pullback attractors are studied, but note that our result is not a particular case since, unlike in Wang [17], the pullback attractors here cannot pullback attract themselves in general.

Let $T > 0$ . A multi-valued process U is said to be periodic with period T if $\begin{matrix} U (t + T, τ + T, x) = U (t, τ, x), \forall t ⩾ τ, x \in X . \end{matrix}$ A pullback attractor $A$ is called periodic with period T if $A (t) = A (t + T)$ for all $t \in R$ .

Theorem 2.5.
Suppose that U is a multi-valued process with pullback attractor $A$ . If U is periodic with period T, then $A$ is periodic with period T.
Proof.
Define $\tilde{A} = {\tilde{A} (t)}_{t \in R}$ by $\tilde{A} (t) : = A (t + T)$ . We need to prove that $A = \tilde{A}$ .

Let B be any a nonempty bounded set. By the pullback attraction of $A$ we have $\begin{matrix} lim_{τ \to - \infty} dist (U (t, τ, B), \tilde{A} (t)) = lim_{τ \to - \infty} dist (U (t + T, τ + T, B), A (t + T)) = 0 . \end{matrix}$ Hence, $\tilde{A}$ is a compact non-autonomous set which is pullback attracting under U. By the minimal property of a pullback attractor we have $A (t) \subset \tilde{A} (t)$ .

In the same way, since $\begin{matrix} lim_{τ \to - \infty} dist (U (t + T, τ + T, B), A (t)) = lim_{τ \to - \infty} dist (U (t, τ, B), A (t)) = 0, \end{matrix}$ by the minimal property again we have $A (t) \supset A (t + T) = \tilde{A} (t)$ . □

3. First convergence theorems from compactness properties

In this section we study pullback attractors with forward or backward compactness properties. Both forward and backward convergences are studied.

3.1. Semi convergences in Hausdorff semi-metric sense

The following is a multi-valued generalization of Lemma 1.1.

Theorem 3.1 (Forward semi convergence).

Let U be a multi-valued process with pullback attractor $A = {A (t)}_{t \in R}$ and let S be a multi-valued semigroup with global attractor $A$ . Suppose that

for every sequence ${x_{τ}}$ with ${lim}_{τ \to \infty} x_{τ} = x_{0}$ , $\begin{matrix} (3.1) & lim_{τ \to \infty} dist (U (τ + t, τ, x_{τ}), S (t, x_{0})) = 0, \forall t ⩾ 0; \end{matrix}$

$A$ is forward compact.

Then

\begin{matrix} lim_{t \to \infty} dist (A (t), A) = 0 . \end{matrix}

Proof.
We prove the theorem by contradiction. Suppose that for some $δ > 0$ there exists a sequence $t_{n} \to \infty$ such that $dist (A (t_{n}), A) ⩾ δ$ for all $n \in N$ . Then by the compactness of $A$ there exists a sequence $x_{n} \in A (t_{n})$ such that $\begin{matrix} (3.2) & dist (x_{n}, A) = dist (A (t_{n}), A) ⩾ δ, \forall n \in N . \end{matrix}$

Since $A$ is forward compact, the set $B : = \overline{⋃_{t ⩾ 0} A (t)}$ is compact. By the forward attraction of $A$ under S, there exists a $T_{0} > 0$ such that $\begin{matrix} (3.3) & dist (S (T_{0}, B), A) < δ / 2 . \end{matrix}$ Besides, by the negative semi-invariance of $A$ , for every $x_{n}$ there is $b_{n} \in A (t_{n} - T_{0}) \subset B$ such that $\begin{matrix} x_{n} \in U (t_{n}, t_{n} - T_{0}, b_{n}), \end{matrix}$ and $b_{n} \to b_{0}$ as $n \to \infty$ for some $b_{0} \in B$ in a subsequence sense. Hence, by condition (3.1), there exists an $N = N (δ) > 0$ such that $\begin{matrix} (3.4) & dist (x_{N}, S (T_{0}, b_{0})) ⩽ dist (U (t_{N}, t_{N} - T_{0}, b_{N}), S (T_{0}, b_{0})) < δ / 2 . \end{matrix}$ Therefore, from (3.4) and (3.3) it follows that, $\begin{matrix} dist (x_{N}, A) ⩽ dist (x_{N}, S (T_{0}, b_{0})) + dist (S (T_{0}, b_{0}), A) < δ, \end{matrix}$ which contradicts (3.2). □

Analogously, we have the following backward semi convergence theorem. Theorem 3.2 (Backward semi convergence).

Let U be a multi-valued process with pullback attractor $A = {A (t)}_{t \in R}$ and let S be a multi-valued semigroup with global attractor $A$ . Suppose that

for any ${x_{t}}$ with ${lim}_{τ \to - \infty} x_{τ} = x_{0}$ , $\begin{matrix} (3.5) & lim_{t \to - \infty} dist (U (t, t - T, x_{t}), S (T, x_{0})) = 0, \forall T \in R^{+}; \end{matrix}$

$A$ is backward compact.

Then

\begin{matrix} lim_{t \to - \infty} dist (A (t), A) = 0 . \end{matrix}

3.2. Full convergences in Hausdorff metric sense

In this section, based on the main ideas of Theorems 3.1 and 3.2 we study convergences in the full Hausdorff metric sense. To this end, an equi-attraction condition will be assumed.

Definition 3.3.
The pullback attractor $A$ of process U is said to be equi-attracting under U if for any nonempty bounded set B $\begin{matrix} (3.6) & sup_{t \in R} dist (U (t, t - τ, B), A (t)) \to 0, as τ \to \infty . \end{matrix}$

Note that property (3.6) is equivalent to $\begin{matrix} sup_{τ \in R} dist (U (τ + t, τ, B), A (τ + t)) \to 0, as t \to \infty . \end{matrix}$

We begin with the following propositions for convergences of pullback attractors towards pullback attractors.
Proposition 3.4.
Let U and $U_{\infty}$ be multi-valued processes with pullback attractors $A = {A (t)}_{t \in R}$ and $A_{\infty} = {A_{\infty} (t)}_{t \in R}$ , respectively. Suppose that
for any sequence ${x_{τ}}$ with ${lim}_{τ \to \infty} x_{τ} = x_{0}$ , $\begin{matrix} (3.7) & lim_{τ \to \infty} {dist}_{H} (U (τ + t, τ, x_{τ}), U_{\infty} (τ + t, τ, x_{0})) = 0, \forall t ⩾ 0; \end{matrix}$

the two attractors $A$ and $A_{\infty}$ are both forward compact;

$A$ and $A_{\infty}$ are equi-attracting under U and $U_{\infty}$ , respectively.
Then $A$ and $A_{\infty}$ are asymptotically identical in the distant future, i.e., $\begin{matrix} lim_{t \to \infty} {dist}_{H} (A (t), A_{\infty} (t)) = 0 . \end{matrix}$
Proof.
We first prove by contradiction that $\begin{matrix} (3.8) & lim_{t \to \infty} dist (A (t), A_{\infty} (t)) = 0 . \end{matrix}$ Suppose that for some $δ > 0$ there exists a sequence $t_{n} \to \infty$ such that $\begin{matrix} dist (A (t_{n}), A_{\infty} (t_{n})) ⩾ δ, \forall n \in N . \end{matrix}$ Then by the compactness of $A$ there exists a sequence $x_{n} \in A (t_{n})$ such that $\begin{matrix} (3.9) & dist (x_{n}, A_{\infty} (t_{n})) = dist (A (t_{n}), A_{\infty} (t_{n})) ⩾ δ, \forall n \in N . \end{matrix}$

Let $B : = \overline{⋃_{t ⩾ 0} A (t)}$ . By the equi-attraction of $A_{\infty}$ , there is an $M > 0$ such that $\begin{matrix} (3.10) & sup_{n \in N} dist (U_{\infty} (t_{n}, t_{n} - M, B), A_{\infty} (t_{n})) < δ / 2 . \end{matrix}$ Since $A$ is negatively semi-invariant, for each $x_{n}$ there is $b_{n} \in A (t_{n} - M) \subset B$ such that $\begin{matrix} x_{n} \in U (t_{n}, t_{n} - M, b_{n}) . \end{matrix}$ By the compactness of B, $b_{n} \to b_{0}$ for some $b_{0} \in B$ as $n \to \infty$ in a subsequence sense, which along condition (3.7) implies that there exists an $N \in N$ such that $\begin{matrix} (3.11) & \begin{matrix} dist (x_{N}, U_{\infty} (t_{N}, t_{N} - M, b_{0})) \\ ⩽ dist (U (t_{N}, t_{N} - M, b_{N}), U_{\infty} (t_{N}, t_{N} - M, b_{0})) < δ / 2 . \end{matrix} \end{matrix}$ Therefore, from (3.10) and (3.11) it follows that $\begin{matrix} dist (x_{N}, A_{\infty} (t_{N})) & ⩽ dist (x_{N}, U_{\infty} (t_{N}, t_{N} - M, b_{0})) \\ + dist (U_{\infty} (t_{N}, t_{N} - M, b_{0}), A_{\infty} (t_{N})) < δ, \end{matrix}$ which contradicts (3.9). Hence, convergence (3.8) holds.

Exchanging the roles played by $A$ and $A_{\infty}$ we have ${lim}_{t \to \infty} dist (A_{\infty} (t), A (t)) = 0$ . □

By an analogous proof to Proposition 3.4 we have
Proposition 3.5.
Let U and $U_{\infty}$ be multi-valued processes with pullback attractors $A = {A (t)}_{t \in R}$ and $A_{\infty} = {A_{\infty} (t)}_{t \in R}$ , respectively. Suppose that
for any sequence ${x_{t}}$ with ${lim}_{t \to - \infty} x_{t} = x_{0}$ , $\begin{matrix} (3.12) & lim_{t \to - \infty} {dist}_{H} (U (t, t - T, x_{t}), U_{\infty} (t, t - T, x_{0})) = 0, \forall T ⩾ 0; \end{matrix}$

the two attractors $A$ and $A_{\infty}$ are both backward compact;

$A$ and $A_{\infty}$ are equi-attracting under U and $U_{\infty}$ , respectively.
Then $A$ and $A_{\infty}$ are asymptotically identical in the distant past, i.e., $\begin{matrix} lim_{t \to - \infty} {dist}_{H} (A (t), A_{\infty} (t)) = 0 . \end{matrix}$

As corollaries of Propositions 3.4 and 3.5, we have the following two theorems. Theorem 3.6 (Forward full convergence).

Let U be a multi-valued process with pullback attractor $A = {A (t)}_{t \in R}$ and let S be a multi-valued semigroup with global attractor $A$ . Suppose that

for every sequence ${x_{τ}}$ with ${lim}_{τ \to \infty} x_{τ} = x_{0}$ , $\begin{matrix} (3.13) & lim_{τ \to \infty} {dist}_{H} (U (τ + t, τ, x_{τ}), S (t, x_{0})) = 0, \forall t ⩾ 0; \end{matrix}$

$A$ is forward compact;

$A$ is equi-attracting.

Then the global attractor

A

is the ω-limit set of the pullback attractor

A

, i.e.,

\begin{matrix} lim_{t \to \infty} {dist}_{H} (A (t), A) = 0 . \end{matrix}

Proof.
Define $U_{\infty} (t, s, x) : = S (t - s, x)$ for every $(t, s) \in R_{⩾}^{2}$ and $x \in X$ , then $U_{\infty}$ is a multi-valued process with pullback attractor $A_{\infty}$ with $A_{\infty} (t) \equiv A$ . Hence, by Proposition 3.4, the theorem follows. □
Theorem 3.7 (Backward full convergence).

Let U be a multi-valued process with pullback attractor $A = {A (t)}_{t \in R}$ and let S be a multi-valued semigroup with global attractor $A$ . Suppose that

for any ${x_{t}}$ with ${lim}_{τ \to - \infty} x_{τ} = x_{0}$ , $\begin{matrix} (3.14) & lim_{t \to - \infty} {dist}_{H} (U (t, t - T, x_{t}), S (T, x_{0})) = 0, \forall T \in R^{+}; \end{matrix}$

$A$ is backward compact;

$A$ is equi-attracting.

Then the global attractor

A

is the α-limit set of the pullback attractor

A

, i.e.,

\begin{matrix} lim_{t \to - \infty} {dist}_{H} (A (t), A) = 0 . \end{matrix}

4. Alternative convergence theorems from boundedness properties

Since compactness conditions of attractors are usually technically difficult to establish in applications, by the spirit of Cui [4], using boundedness conditions instead of the compactness conditions, we establish some alternative criteria results.

4.1. Convergences without equi-attraction assumption

We first show results that do not rely on the equi-attraction property, including the full backward convergence which in Theorem 3.7 required the equi-attraction assumption.

Theorem 4.1 (Forward semi convergence).

Let U be a multi-valued process with pullback attractor $A = {A (t)}_{t \in R}$ and let S be a multi-valued semigroup with global attractor $A$ . Suppose that

$A$ is forward bounded, i.e., there is a bounded set B such that $⋃_{t ⩾ 0} A (t) \subset B$ ;

the following asymptotically autonomous condition holds $\begin{matrix} (4.1) & lim_{t \to \infty} sup_{x \in B} dist (U (t + T, t, x), S (T, x)) = 0, \forall T > 0 . \end{matrix}$

Then

\begin{matrix} (4.2) & lim_{t \to \infty} dist (A (t), A) = 0 . \end{matrix}

Proof.
If it were not the case, then there would exist $δ > 0$ and $t_{n} \to \infty$ such that $\begin{matrix} dist (A (t_{n}), A) > δ, \forall n \in N . \end{matrix}$ Since B is attracted by $A$ under S, there exists a $T > 0$ such that $\begin{matrix} (4.3) & dist (S (T, B), A) < δ / 2 . \end{matrix}$ By the compactness of the attractor $A$ , for any $n \in N$ there exists an $x_{n} \in A (t_{n})$ such that $\begin{matrix} (4.4) & dist (x_{n}, A) = dist (A (t_{n}), A) > δ . \end{matrix}$ In addition, by the negative semi-invariance and forward boundedness of $A$ , for each $x_{n}$ there exists $y_{n} \in A (t_{n} - T) \subset B$ such that $x_{n} \in U (t_{n}, t_{n} - T, y_{n})$ , which along with (4.4) implies $\begin{matrix} (4.5) & dist (U (t_{n}, t_{n} - T, y_{n}), A) ⩾ dist (x_{n}, A) > δ, \forall n \in N . \end{matrix}$ On the other hand, by condition (4.1), there exists an $N = N (δ) > 0$ such that $\begin{matrix} dist (U (t_{N}, t_{N} - T, y_{N}), S (T, y_{N})) < δ / 2, \end{matrix}$ so, by (4.5), $\begin{matrix} dist (S (T, y_{N}), A) & ⩾ dist (U (t_{N}, t_{N} - T, y_{N}), A) \\ - dist (U (t_{N}, t_{N} - T, y_{N}), S (T, y_{N})) > δ / 2, \end{matrix}$ which contradicts (4.3). □
Remark 4.2.
Slightly modifying the proof as in [6, Theorem 2] one sees that the condition (4.1) in Theorem 4.1 can be slightly weakened to the following condition $\begin{matrix} lim_{t \to \infty} dist (U (t + T, t, B), S (T, B)) = 0, \forall T > 0 . \end{matrix}$

Now we turn to the backward case. We shall see that, without the semi-uniform attraction property as previously in Theorem 3.7, the convergence can still be in the Hausdorff metric sense, not only in the semi-metric sense as the forward convergence. To this end, we begin again with a more general convergence, that of pullback attractors to pullback attractors.
Proposition 4.3.
Let $A = {A (t)}_{t \in R}$ and $A_{\infty} = {A_{\infty} (t)}_{t \in R}$ be pullback attractors of multi-valued processes U and $U_{\infty}$ , respectively. Suppose that
$A$ is backward bounded, i.e. there is a bounded set B such that $⋃_{t ⩽ 0} A (t) \subset B$ ;

the following convergence holds $\begin{matrix} (4.6) & sup_{x \in B, τ \in R^{+}} dist (U (t, t - τ, x), U_{\infty} (t, t - τ, x)) \to 0, as t \to - \infty . \end{matrix}$
Then $\begin{matrix} lim_{t \to - \infty} dist (A (t), A_{\infty} (t)) = 0 . \end{matrix}$ If, moreover, $A_{\infty}$ is also backward bounded and $\begin{matrix} (4.7) & sup_{x \in B, τ \in R^{+}} {dist}_{H} (U (t, t - τ, x), U_{\infty} (t, t - τ, x)) \to 0, as t \to - \infty, \end{matrix}$ then the two attractors $A$ and $A_{\infty}$ are asymptotically identical in distant past, i.e., $\begin{matrix} lim_{t \to - \infty} {dist}_{H} (A (t), A_{\infty} (t)) = 0 . \end{matrix}$
Proof.
We first prove the first part by contradiction. Suppose that for some $δ > 0$ there exists a sequence $t_{n} \to \infty$ such that $dist (A (- t_{n}), A_{\infty} (- t_{n})) ⩾ δ$ for all $n \in N$ . Then by the compactness of $A$ there exists a sequence $x_{n} \in A (- t_{n})$ such that $\begin{matrix} (4.8) & dist (x_{n}, A_{\infty} (- t_{n})) = dist (A (- t_{n}), A_{\infty} (- t_{n})) ⩾ δ, \forall n \in N . \end{matrix}$

Since $A$ is negatively semi-invariant, for every $m, n \in N$ there is $b_{n, m} \in A (- t_{n} - m) \subset B$ such that $\begin{matrix} x_{n} \in U (- t_{n}, - t_{n} - m, b_{n, m}), \forall n, m \in N . \end{matrix}$ Hence, by condition (4.6), there exists an $N = N (δ) > 0$ such that for all $m \in N$ $\begin{matrix} (4.9) & \begin{matrix} dist (x_{N}, U_{\infty} (- t_{N}, - t_{N} - m, b_{N, m})) \\ ⩽ dist (U (- t_{N}, - t_{N} - m, b_{N, m}), U_{\infty} (- t_{N}, - t_{N} - m, b_{N, m})) \\ ⩽ sup_{x \in B, τ \in R^{+}} dist (U (- t_{N}, - t_{N} - τ, x), U_{\infty} (- t_{N}, - t_{N} - τ, x)) < δ / 2 . \end{matrix} \end{matrix}$ In addition, since ${b_{n, m}} \subset B$ is pullback attracted by $A_{\infty}$ under $U_{\infty}$ , there is an $M = M (N, δ) > 0$ such that $\begin{matrix} (4.10) & \begin{matrix} dist (U_{\infty} (- t_{N}, - t_{N} - m, b_{N, m}), A_{\infty} (- t_{N})) \\ ⩽ dist (U_{\infty} (- t_{N}, - t_{N} - m, B), A_{\infty} (- t_{N})) < δ / 2, \forall m ⩾ M . \end{matrix} \end{matrix}$ Therefore, from (4.9) and (4.10) it follows that, for all $m ⩾ M$ , $\begin{array}{l} dist (x_{N}, A_{\infty} (- t_{N})) & ⩽ dist (x_{N}, U_{\infty} (- t_{N}, - t_{N} - m, b_{N, m})) \\ + dist (U_{\infty} (- t_{N}, - t_{N} - m, b_{N, m}), A_{\infty} (- t_{N})) < δ, \end{array}$ which contradicts (4.8).

In case of $A_{\infty}$ being also backward bounded with (4.7), exchanging the roles played by $A$ and $A_{\infty}$ we have ${lim}_{t \to - \infty} dist (A_{\infty} (t), A (t)) = 0$ . □

As a corollary of Proposition 4.3, we have
Theorem 4.4 (Backward full convergence).

Suppose that U is a multi-valued process with pullback attractor $A = {A (t)}_{t \in R}$ and that S is a multi-valued semigroup with global attractor $A$ . If

$A$ is backward bounded, i.e. there is a bounded set B such that $⋃_{t ⩽ 0} A (t) \subset B$ ;

the following backward asymptotically autonomous condition holds $\begin{matrix} (4.11) & sup_{x \in B, τ \in R^{+}} dist (U (t, t - τ, x), S (τ, x)) \to 0, as t \to - \infty, \end{matrix}$

then

\begin{matrix} lim_{t \to - \infty} dist (A (t), A) = 0 . \end{matrix}

If, moreover,

\begin{matrix} (4.12) & sup_{x \in B, τ \in R^{+}} {dist}_{H} (U (t, t - τ, x), S (τ, x)) \to 0, as t \to - \infty, \end{matrix}

then the global attractor

A

is the α-limit set of the pullback attractor

A

, i.e.,

\begin{matrix} lim_{t \to - \infty} {dist}_{H} (A (t), A) = 0 . \end{matrix}

Note that in Theorem 4.4 the full backward convergence does not require the equi-attraction property, which can be regarded as an advantage over Theorem 3.7.

4.2. Full forward convergences under the equi-attraction assumption

With the equi-attraction property, we have the following result on full forward convergence analogously to Proposition 3.4.

Proposition 4.5.
Let U and $U_{\infty}$ be multi-valued processes with pullback attractors $A = {A (t)}_{t \in R}$ and $A_{\infty} = {A_{\infty} (t)}_{t \in R}$ , respectively. Suppose that
the following condition holds for every nonempty bounded set B $\begin{matrix} (4.13) & lim_{t \to \infty} sup_{x \in B} dist (U (t + T, t, x), U_{\infty} (t + T, t, x)) = 0, \forall T > 0; \end{matrix}$

$A$ and $A_{\infty}$ are both forward bounded;

$A$ and $A_{\infty}$ are both equi-attracting.
Then $A$ and $A_{\infty}$ are asymptotically identical in the distant future, i.e., $\begin{matrix} lim_{t \to \infty} {dist}_{H} (A (t), A_{\infty} (t)) = 0 . \end{matrix}$
Proof.
The proof can be done similarly to Proposition 3.4, and thus is omitted. □

As a corollary of Proposition 4.5, we have the following theorem. Theorem 4.6 (Forward full convergence).

Let U be a multi-valued process with a pullback attractor $A = {A (t)}_{t \in R}$ and let S be a multi-valued semigroup with a global attractor $A$ . Suppose that

the following condition holds for every nonempty bounded set B $\begin{matrix} (4.14) & lim_{t \to \infty} sup_{x \in B} {dist}_{H} (U (t + T, t, x), S (T, x)) = 0, \forall T > 0; \end{matrix}$

$A$ is forward bounded;

$A$ is equi-attracting under U.

Then the semigroup attractor

A

is the ω-limit set of the pullback attractor, i.e.,

\begin{matrix} lim_{t \to \infty} {dist}_{H} (A (t), A) = 0 . \end{matrix}

5. Applications to a scalar differential inclusion

Compared with Theorem 3.1, Theorem 4.1 has an evident advantage that it requires only a forward boundedness condition rather than a forward compactness condition. This is an important improvement for infinite-dimensional dynamical systems even in single-valued cases; an example of a reaction-diffusion equation is given in [4].

In this section, by a scalar application we highlight another advantage that condition (4.1) has over (3.1). The main idea is that a multi-valued system can behave very differently at different points, in which case it is much convenient to use the convergence (4.1) with the initial data being the same.

We consider the following scalar differential inclusion $\begin{matrix} (5.1) & \{\begin{matrix} \frac{d u}{d t} + λ u \in b (t) H_{0} (u), t ⩾ s, \\ u (s) = u_{s}, \end{matrix} \end{matrix}$ where $λ > 0$ , $H_{0}$ is the Heaviside function given by $\begin{matrix} H_{0} (u) = \{\begin{matrix} - 1, & if u < 0, \\ [- 1, 1], & if u = 0, \\ 1, & if u > 0, \end{matrix} \end{matrix}$ which is the subdifferential of the absolute value $| u |$ , and $b : R \to R^{+}$ is a continuous function satisfying $\begin{matrix} 0 < b_{0} ⩽ b (t) ⩽ b_{1} . \end{matrix}$

Definition 5.1.
The function $u \in C ([s, + \infty), R)$ is called a solution of (5.1) if $d u / d t \in L_{loc}^{\infty} ([s, + \infty), R)$ and there exists $h \in L_{loc}^{\infty} ([s, + \infty), R)$ such that $h (t) \in H_{0} (u (t))$ , for a.e. $t > s$ , and $\begin{matrix} (5.2) & \frac{d u}{d t} + λ u = b (t) h (t) for a.e. t > s . \end{matrix}$

It is clear that the problem (5.1) possesses at least one solution for every initial data [1]. Moreover, the following lemma shows that the uniqueness of solutions fails only when the initial data is zero.
Lemma 5.2 ([1]).

For any $u_{s} \neq 0$ the problem ( 5.1 ) has the unique solution given by $\begin{matrix} u (t) = e^{- λ (t - s)} u_{s} + \int_{s}^{t} e^{- λ (t - τ)} b (τ) d τ . \end{matrix}$ For $u_{s} = 0$ , there are infinitely many solutions given by $\begin{array}{l} u_{\infty} \equiv 0, \\ u_{r}^{+} (t) = \{\begin{matrix} 0, & s ⩽ t ⩽ r, \\ \int_{r}^{t} e^{- λ (t - τ)} b (τ) d τ, & t > r, \end{matrix} \\ u_{r}^{-} (t) = \{\begin{matrix} 0, & s ⩽ t ⩽ r, \\ - \int_{r}^{t} e^{- λ (t - τ)} b (τ) d τ, & t > r, \end{matrix} \end{array}$ where $r ⩾ s$ is arbitrary, and these are the only possible solutions.

The recent work by Caraballo et al. [1] shows that the scalar problem (5.1) generates a strict multi-valued process, which possesses a compact pullback attractor.

Lemma 5.3 ([1]).

The mapping $(t, s, u_{s}) \mapsto U (t, s, u_{s})$ given by $\begin{matrix} U (t, s, u_{s}) & = {y \in R : there exists a continuous function η \in C ([s, + \infty), R) \\ such that y = η (t) and u_{s} = η (s)} \end{matrix}$ is a strict multi-valued process which has a closed graph. Moreover, the process U has a unique pullback attractor $A = {A (t)}_{t \in R}$ .

In this paper, making use of our theoretical results we study the asymptotic behavior of the tails of the pullback attractor under some conditions of $b (t)$ . To begin with, we show that the pullback attractor $A$ is both backward and forward bounded, and is equi-attracting under U.

Proposition 5.4.
The pullback attractor $A$ is both forward and backward bounded.
Proof.
By (5.2), any solution u of the problem (5.1) satisfies $\begin{matrix} \frac{1}{2} \frac{d}{d t} | u |^{2} + λ | u |^{2} = b (t) h (t) u ⩽ \frac{1}{2 λ} {| b (t) |}^{2} + \frac{λ}{2} | u |^{2} . \end{matrix}$ By Gronwall’s lemma we have $\begin{matrix} {| u (t, s, u_{s}) |}^{2} ⩽ e^{- λ (t - s)} u_{s} + \frac{1}{λ} \int_{s}^{t} e^{- λ (t - τ)} {| b (τ) |}^{2} d τ . \end{matrix}$ Therefore, the non-autonomous set $B = {B (t)}_{t \in R}$ , given by $\begin{matrix} B (t) = {η \in R : | η |^{2} ⩽ 1 + \frac{1}{λ} \int_{s}^{t} e^{- λ (t - τ)} {| b (τ) |}^{2} d τ}, \end{matrix}$ is a pullback absorbing set of the process U. Moreover, $| B (t) | ⩽ 1 + \frac{b_{1}^{2}}{λ^{2}}$ for all $t \in R$ since $| b (τ) | ⩽ b_{1}$ . Hence, $B$ is both forward and backward bounded, and thereby so is the pullback attractor $A$ . □
Proposition 5.5.
The pullback attractor $A$ is equi-attracting under process U.
Proof.
Given a nonempty bounded set $K \subset R$ , without loss of generality we assume that it can be divided into three parts $K = K_{1} \cup {0} \cup K_{2}$ , where $K_{1}$ contains all positive elements of K and $K_{2}$ contains all negative ones. We conclude the lemma by proving that $\begin{matrix} (5.3) & lim_{t \to \infty} sup_{τ \in R} dist (U (t + τ, τ, K), A (t + τ)) = 0 \end{matrix}$ holds for $K = K_{1}, {0}, K_{2}$ , respectively.

Consider the bounded function $ξ_{M} : R \to [b_{0} / λ, b_{1} / λ]$ given by $\begin{matrix} ξ_{M} (t) : = \int_{- \infty}^{t} e^{- λ (t - s)} b (s) d s, \forall t \in R . \end{matrix}$ By the spirit of [1, Theorem 3.10], the mapping $ξ_{M} (\cdot)$ is a bounded complete trajectory of U satisfying $\begin{matrix} U (t, s, ξ_{M} (s)) = ξ_{M} (t), \forall t ⩾ s, \end{matrix}$ which implies that $ξ_{M} (t) \in A (t)$ for all $t \in R$ . Hence, by Lemma 5.2, since U is single-valued on $K_{1}$ , we have $\begin{array}{l} dist (U (t + τ, τ, K_{1}), A (t + τ)) & ⩽ sup_{x \in K_{1}} dist (U (t + τ, τ, x), ξ_{M} (t + τ)) \\ = sup_{x \in K_{1}} | e^{- λ t} x - \int_{- \infty}^{τ} e^{- λ (t + τ - s)} b (s) d s | \\ ⩽ e^{- λ t} (| K_{1} | + \int_{- \infty}^{τ} e^{- λ (τ - s)} b_{1} d s) \\ = e^{- λ t} (| K_{1} | + b_{1} / λ) \to 0, as t \to \infty, \end{array}$ uniformly for all $τ \in R$ . Therefore, the equi-attraction (5.3) holds for $K_{1}$ .

The case of $K_{2}$ is similar to $K_{1}$ with $- ξ_{M} (\cdot)$ instead of $ξ_{M} (\cdot)$ .

Now we consider the case $K = {0}$ . Since the multi-valued process U is strict and from Proposition 5.4 the pullback attractor $A$ is backward bounded, by Lemma 2.3, $A$ is strictly invariant, that is, $U (t + τ, τ, A (τ)) = A (t + τ)$ for every $t ⩾ 0$ and $τ \in R$ . Therefore, since $ξ (t) \equiv 0$ belongs to $A (t)$ for all $t \in R$ , we have $\begin{matrix} dist (U (t + τ, τ, 0), A (t + τ)) ⩽ dist (U (t + τ, τ, A (τ)), A (t + τ)) \equiv 0, \end{matrix}$ for all $t ⩾ 0$ and $τ \in R$ . Hence, the equi-attraction (5.3) holds for ${0}$ . □

5.1. Convergences of asymptotically periodic pullback attractors

Let $p (t)$ be a continuous periodic function, and consider the following periodic version of (5.1) $\begin{matrix} (5.4) & \frac{d u}{d t} + λ u \in p (t) H_{0} (u) . \end{matrix}$ Clearly, the multi-valued process $U_{per}$ corresponding to (5.4) has a pullback attractor $A_{per} = {A_{per} (t)}_{t \in R}$ . Moreover, by Propositions 5.4 and 5.5 and Theorem 2.5 we have

Theorem 5.6.
The pullback attractor $A_{per}$ is both forward bounded and backward bounded, and is equi-attracting under $U_{per}$ . Moreover, it is periodic.

In the following, we study the convergences of the pullback attractor $A$ of (5.1) towards the periodic pullback attractor $A_{per}$ of (5.4). Theorem 5.7 (Forward convergence).

Let ${lim}_{t \to \infty} | b (t) - p (t) | = 0$ . Then the two attractors $A$ and $A_{per}$ are asymptotically identical in the distant future with respect to the full Hausdorff metric, i.e., $\begin{matrix} lim_{t \to \infty} {dist}_{H} (A (t), A_{per} (t)) = 0 . \end{matrix}$

Proof.
By Proposition 4.5 and Theorem 5.6, it suffices to prove $\begin{matrix} (5.5) & lim_{t \to \infty} sup_{x \in X} {dist}_{H} (U (t + T, t, x), U_{per} (t + T, t, x)) = 0, \forall T > 0 . \end{matrix}$ By assumption, for any $ϵ > 0$ there exists a $T_{} > 0$ such that $| b (τ) - p (τ) | < ϵ$ for all $τ ⩾ T_{}$ . Hence, if $x \neq 0$ , then, by Lemma 5.2, $\begin{array}{l} {dist}_{H} (U (t + T, t, x), U_{per} (t + T, t, x)) \\ = | \int_{t}^{t + T} e^{- λ (t + T - τ)} (b (τ) - p (τ)) d τ | \\ ⩽ \int_{t}^{t + T} | b (τ) - p (τ) | d τ < T ϵ, \forall t ⩾ T_{}; \end{array}$ if $x = 0$ , then, by Lemma 5.2 again, $\begin{array}{l} {dist}_{H} (U (t + T, t, 0), U_{per} (t + T, t, 0)) \\ ⩽ sup_{r \in [t, t + T)} \int_{r}^{t + T} e^{- λ (t + T - τ)} | b (τ) - p (τ) | d τ \\ ⩽ sup_{r \in [t, t + T)} \int_{r}^{t + T} | b (τ) - p (τ) | d τ \\ ⩽ \int_{t}^{t + T} | b (τ) - p (τ) | d τ < T ϵ, \forall t ⩾ T_{} . \end{array}$ Therefore, (5.5) holds as desired. □
Theorem 5.8 (Backward convergence).

Let ${lim}_{t \to - \infty} | b (t) - p (t) | = 0$ . Then the two attractors $A$ and $A_{per}$ are asymptotically identical in the distant past with respect to the full Hausdorff metric, i.e., $\begin{matrix} lim_{t \to - \infty} {dist}_{H} (A (t), A_{per} (t)) = 0 . \end{matrix}$

Proof.
By Proposition 4.3 it suffices to prove that $\begin{matrix} (5.6) & sup_{x \in X, τ \in R^{+}} {dist}_{H} (U (t, t - τ, x), U_{per} (t, t - τ, x)) \to 0, as t \to - \infty . \end{matrix}$ By assumption, for any $ϵ > 0$ there exists a $T_{} < 0$ such that $| b (s) - p (s) | < ϵ$ for all $s ⩽ T_{}$ . Hence, if $x \neq 0$ , then, by Lemma 5.2, $\begin{array}{l} sup_{τ \in R^{+}} {dist}_{H} (U (t, t - τ, x), U_{per} (t, t - τ, x)) \\ = | \int_{t - τ}^{t} e^{- λ (t - s)} (b (s) - p (s)) d s | \\ ⩽ \int_{- \infty}^{t} e^{- λ (t - s)} ϵ d s = \frac{ϵ}{λ}, \forall t ⩽ T_{}; \end{array}$ if $x = 0$ , then, by Lemma 5.2 again, $\begin{array}{l} sup_{τ \in R^{+}} {dist}_{H} (U (t, t - τ, 0), U_{per} (t, t - τ, 0)) \\ ⩽ sup_{τ \in R^{+}} (sup_{r \in [t - τ, t)} \int_{r}^{t} e^{- λ (t - s)} | b (s) - p (s) | d s) \\ ⩽ sup_{r \in (- \infty, t)} \int_{r}^{t} e^{- λ (t - s)} ϵ d s = \frac{ϵ}{λ} \end{array}$ for all $t ⩽ T_{}$ . Therefore, (5.6) holds as desired. □

5.2. Convergences of asymptotically autonomous pullback attractors

Let $\bar{b} \in [b_{0}, b_{1}]$ be a constant, and consider the autonomous version of (5.1) $\begin{matrix} (5.7) & \frac{d u}{d t} + λ u \in \bar{b} H_{0} (u) . \end{matrix}$ Clearly, the multi-valued semigroup S generated by (5.7) has a global attractor $A$ .

In the following, we study the convergences of the pullback attractor $A$ of (5.1) towards the global attractor $A$ of (5.7).

Theorem 5.9 (Forward convergence).

Let ${lim}_{t \to \infty} | b (t) - \bar{b} | = 0$ . Then the semigroup attractor $A$ is the ω-limit set of the pullback attractor $A$ , i.e., $\begin{matrix} lim_{t \to \infty} {dist}_{H} (A (t), A) = 0 . \end{matrix}$

Proof.
Since the pullback attractor $A$ is forward bounded by Proposition 5.4 and equi-attracting under U by Proposition 5.5, by Theorem 4.6 it suffices to prove that $\begin{matrix} (5.8) & lim_{t \to \infty} sup_{x \in X} {dist}_{H} (U (t + T, t, x), S (T, x)) = 0, \forall T > 0 . \end{matrix}$ By assumption, for any $ϵ > 0$ there exists a $T_{} > 0$ such that $| b (τ) - \bar{b} | < ϵ$ for all $τ ⩾ T_{}$ . Hence, if $x \neq 0$ , then, by Lemma 5.2, $\begin{array}{l} {dist}_{H} (U (t + T, t, x), S (T, x)) & = | \int_{t}^{t + T} e^{- λ (t + T - τ)} (b (τ) - \bar{b}) d τ | \\ ⩽ \int_{t}^{t + T} | b (τ) - \bar{b} | d τ < T ϵ, \forall t ⩾ T_{}; \end{array}$ if $x = 0$ , then, by Lemma 5.2 again, $\begin{array}{l} {dist}_{H} (U (t + T, t, 0), S (T, 0)) & ⩽ sup_{r \in [t, t + T)} \int_{r}^{t + T} e^{- λ (t + T - τ)} | b (τ) - \bar{b} | d τ \\ ⩽ sup_{r \in [t, t + T)} \int_{r}^{t + T} | b (τ) - \bar{b} | d τ \\ ⩽ \int_{t}^{t + T} | b (τ) - \bar{b} | d τ < T ϵ, \forall t ⩾ T_{} . \end{array}$ Therefore, (5.8) holds as desired. □
Theorem 5.10 (Backward convergence).

Let ${lim}_{t \to - \infty} | b (t) - \bar{b} | = 0$ . Then the semigroup attractor $A$ is the α-limit set of the pullback attractor $A$ , i.e., $\begin{matrix} lim_{t \to - \infty} {dist}_{H} (A (t), A) = 0 . \end{matrix}$

Proof.

Since by Proposition 5.4 the pullback attractor $A$ is backward bounded, by Theorem 4.4, it suffices to prove that $\begin{matrix} (5.9) & sup_{x \in X, τ \in R^{+}} {dist}_{H} (U (t, t - τ, x), S (τ, x)) \to 0, as t \to - \infty . \end{matrix}$ By assumption, for any $ϵ > 0$ there exists a $T_{*} < 0$ such that $| b (s) - \bar{b} | < ϵ$ for all $s ⩽ T_{*}$ . Hence, if $x \neq 0$ , then, by Lemma 5.2, $\begin{array}{l} sup_{τ \in R^{+}} {dist}_{H} (U (t, t - τ, x), S (τ, x)) & = | \int_{t - τ}^{t} e^{- λ (t - s)} (b (s) - \bar{b}) d s | \\ ⩽ \int_{- \infty}^{t} e^{- λ (t - s)} ϵ d s = \frac{ϵ}{λ}, \forall t ⩽ T_{*}; \end{array}$ if $x = 0$ , then, by Lemma 5.2 again, $\begin{array}{l} sup_{τ \in R^{+}} {dist}_{H} (U (t, t - τ, 0), S (τ, 0)) & ⩽ sup_{τ \in R^{+}} (sup_{r \in [t - τ, t)} \int_{r}^{t} e^{- λ (t - s)} | b (s) - \bar{b} | d s) \\ ⩽ sup_{r \in (- \infty, t)} \int_{r}^{t} e^{- λ (t - s)} ϵ d s = \frac{ϵ}{λ}, \forall t ⩽ T_{*} . \end{array}$ Therefore, (5.9) holds as desired. □

Remark 5.11.

By Theorems 4.4 and 4.6 we have proved the full convergences of the pullback attractor to the semigroup attractor. This can be regarded as an example highlighting the advantages of Theorems 4.4 and 4.6 over first Theorems 3.6 and 3.7. Indeed, by Lemma 5.2 we have, for any $n \in N$ , $\begin{array}{l} {dist}_{H} (U (n + 1, n, 1 / n), S (1, 0)) & ⩾ | U (n + 1, n, 1 / n) - 0 | \\ = \frac{e^{- λ}}{n} + \int_{n}^{n + 1} e^{- λ (n + 1 - τ)} b (τ) d τ \\ ⩾ \int_{n}^{n + 1} e^{- λ (n + 1 - τ)} b_{0} d τ ⩾ \frac{b_{0}}{λ} (1 - e^{- λ}), \end{array}$ so condition (3.13) fails and Theorem 3.6 cannot be applied in this application. In comparison, condition (4.14) in Theorem 4.6 turns out to be easy to verify, see Theorem 5.9.

Footnotes

Acknowledgements

H. Cui was funded by NSFC grant 11801195, and P.E. Kloeden was partially supported by the Chinese NSF grant 11571125.

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Zhao, Random dynamics of stochastic p-Laplacian equations on

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Tail convergences of pullback attractors for asymptotically converging multi-valued dynamical systems

Abstract

Keywords

1. Introduction

Lemma 1.1 ([13]).

Lemma 1.2 ([4]).

2. Pullback attractors of multi-valued processes

2.1. Preliminaries

3.1. Semi convergences in Hausdorff semi-metric sense

Theorem 3.1 (Forward semi convergence).

3.2. Full convergences in Hausdorff metric sense

Proof. Define U ∞ ( t , s , x ) : = S ( t − s , x ) for every ( t , s ) ∈ R ⩾ 2 and x ∈ X , then U ∞ is a multi-valued process with pullback attractor A ∞ with A ∞ ( t ) ≡ A . Hence, by Proposition 3.4, the theorem follows. □ Theorem 3.7 (Backward full convergence).

4. Alternative convergence theorems from boundedness properties

4.1. Convergences without equi-attraction assumption

Theorem 4.1 (Forward semi convergence).

4.2. Full forward convergences under the equi-attraction assumption

5. Applications to a scalar differential inclusion

Lemma 5.3 ([1]).

Theorem 5.9 (Forward convergence).

Footnotes

Acknowledgements

References