Global Attractor for Second-Order Nonlinear Evolution Differential Inclusions

Abstract

In this paper, we address the model of global attractor formulated in the form of evolution differential inclusions with second order in Banach spaces. Firstly, based on the fixed point theorem, the existence result of mild solutions is deduced. Then, by implementing the measure of noncompactness, the existence of global attractor associated with -semiflow is validated. Finally, a concrete application of the main result is demonstrated to enhance the practical signification.

1. Introduction

Consider a Banach space . We attempt to handle the following problem:where is a closed, linear, and densely defined operator from to that generates a strongly continuous cosine family ; is measurable and upper semicontinuous in terms of both first-order and second-order variables, and it also satisfies the natural growth condition.

The nonlinear evolution differential inclusions in Banach spaces have drawn more and more attention of a large number of experts and scholars during the past decades, see [1]. At present, there are substantial outcomes focusing on the existence of solutions for differential equations with second or higher order [2, 3]. For instance, semilinear evolution differential inclusions were explored by Cardinali and Rubbioni in [4]. Liu, Li, and Migórski et al. discussed the problems of one and second evolution differential variational inequalities, and they attained several existence results about the mild solution, see [5, 6] and references therein.

It is known that global attractor has played an important role in studying the asymptotic feature of solutions for diverse types of differential systems. Melnik, Valero, and Tran et al. were devoted to researching on global attractor of multivalued semiflows and differential inclusions, see [7–13] and references therein. In [10], the authors studied the global attractor of the following functional differential inclusions:where represents the state function and describes the history of the state function, that is, for . They proved the existences of solutions of system (2) and global attractor by applying noncompactness measure. Global attractor is a gradual state of solution set, and it is also an important research content of a dynamic system. Compared with the first-order differential inclusions, the cosine family generated by operator is more complex and more meaningful. There are relatively few studies on this type of second-order differential inclusions, so it is more important to study the attractor of the second-order differential inclusions.

The first aim of the work is to explore the existence of solutions in terms of system (1) in infinite dimensional spaces. Secondly, we are devoted to show the existence of a global attractor for system (1). To our knowledge, there is currently a research gap in the aforementioned problem. However, it is not easy to establish a connection between mild solution and -semiflows for second-order differential inclusions.

The work is structured as follows. Section 2 presents several relevant definitions involving noncompactness measures and -semiflows with its global attractor. In Section 3, several sufficient conditions with respect to the mild solutions are invoked. In Section 4, the presence of a global attractor of system (1) is examined in Banach spaces. Finally, a concrete application is presented to emphasize the magnitude of the main results.

2. Preliminaries

This section recalls some fundamental definitions and properties of nonlinear analysis, for more details, see [14]. The symbol (resp., ) represents the set of real (resp., positive real) numbers. The strong and weak convergences of to are denoted by and , respectively.

Consider a Banach space whose dual space is denoted by . Let with and let be the Banach space of all (Bochner) integrable functions on endowed with the norm . The notations mentioned as follows will be employed:

Definition 1. Suppose that and are Banach spaces. A multivalued mapping is referred to as follows:(1)Upper semicontinuous (u.s.c), provided that, for an arbitrary closed set in , is closed in .(2)Weakly upper semicontinuous (weakly u.s.c), if for arbitrary weakly closed set in , is closed in .(3)Closed, if its graph is closed in .(4)Compact, if is a relatively compact subset of for every bounded set .A weakly upper semicontinuous mapping with a weakly compact convex value has good properties. The following conclusion can be found in [15].

Lemma 1. Suppose that and are Banach spaces and is a mapping. Then, is weakly u.s.c if only and if for any sequence of satisfying and , it implies up to a subsequence.
In the subsequent arguments, we consider the presence of mild solutions about system (1) from the viewpoint of evolution differential inclusions with second order. We only need to deal with the following differential equation:where .

Definition 2. Assume that is the space of all bounded linear operators defined from to . A mapping is called a strongly continuous cosine family, provided that ( is an identity operator) and for every .
Given a strongly continuous cosine family with . Then, its generator is an operator formulated aswhere the domain . By the definition of generator , it is known that is closed, linear, and densely defined over , see [2]. The sine operator , which is closely relevant to the cosine operator , is expressed by

Definition 3. A mapping is referred to as a mild solution of system (6), provided that there is an with andGiven an arbitrary bounded set in , the Hausdorff measure of noncompactness (MNC, for short) is given byLet . We say that is integrably bounded, if any element of satisfies that for a.e. with some . An integrably bounded is said to be semicompact, provided that is relatively compact for a.e.
The norm of product spaces is represented byIt is straightforward to see that, for any bounded set ,where is measure of noncompactness on .

Definition 4. Presume that is a Hausdorff measure of noncompactness and . A multivalued mapping is referred to as -condensing, ifThe open (resp., closed) ball with centrum 0 and radius is denoted by (resp., ). The fixed point theorem used in our main results is introduced in [16] as follows.

Lemma 2. Define two operators and by coupling with the following properties:(1) is a contraction mapping with a single value and a coefficient (2) is compact and u.s.cThen, each of the following conclusions holds:(i)There exists an element such that for some (ii)The operator inclusion admits a solution in Next, we introduce related theories involving global attractors and -semiflow, see [8, 10, 11]. Suppose that is a nontrivial subgroup of the additive group of and .

Definition 5. A mapping is said to be a multivalued flow (-flow, in short) if the following conditions are guaranteed:(1) for every (2) for any pair and with

Remark 1. A mapping is referred to as an -semiflow, if conditions (1) and (2) are satisfied for any .
A mapping is said to be a strict -semiflow if for every , , we have . A map is called a trajectory of -semiflow with respect to the initial condition , if for all .

Definition 6. Let be an -semiflow. A bounded set is referred to as an absorbing set for , provided that, for every bounded , holds for some , where .
For a given and , we define . The metric excess of over is given as . It is well known that the Hausdorff distance between and can be represented in the following way:

Definition 7. Let be an -semiflow. A set is said to be a global attractor of , provided that the following conditions are satisfied:(1) attracts any bounded set in , namely, as (2) is negatively semi-invariant, namely, , for all We now give an existence result about a global attractor for a given -semiflow , see [11].

Lemma 3. Assume that a mapping satisfies the following conditions:(i)For every , is u.s.c. with closed values(ii) has an absorbing set(iii) is asymptotically upper semicompact, i.e., for every bounded with bounded for some positive value , any sequence is relatively compact as Then, admits a compact global attractor in . In particular, is invariant if it is a strict -semiflow, i.e.,

3. The Existence Result of Mild Solutions

This section is based on the idea of [3] for nonlinear evolution hemivariational inequalities with second order, which is adapted for evolution differential inclusions in our context. We will handle problem (6) under the following basic assumptions: (A) The operator is a generator of a cosine operator and for some and . Moreover, is compact for arbitrary . (F) The multivalued mapping is u.s.c., and there are and with

By combining the uniform boundedness principle with the assumption , it is inferred that (resp., ) is uniformly bounded on subject to some upper-bound (resp., ). We set .

In the sequel, let a multivalued operator be

It is well known that has a measurable selection for every and a.e. under the assumption , which means that is nonempty and well-defined.

We also construct a multivalued operator by

Set by

Lemma 4. Assume that and hold. Then, for an arbitrary sequence with semicompactness, the sequence is relatively compact. In particular, if , then .

Proof. It is shown by analogizing the proof of Theorem 2 of [4].
Aiming to validate the presence of solutions for system (1), we now clarify a property of the multivalued operator , which is inspired by Lemma 5 of [3].

Lemma 5. Assume that and hold. Then, the multivalued mapping is compact and u.s.c. with convex and compact values.

Proof. Note that the operator has convex values for any due to the convexity of . The verification process will be divided into four steps as follows: Step 1: is a bounded operator on . Given and , from and Hölder inequality, it follows that Hence, is a bounded subset of , i.e., the operator is a bounded operator. Step 2: is equicontinuous. Presume that , is sufficiently small, then From the continuity of the sine operator , we get easily that the above inequality is independent of and it converges to 0 as . This implies the equicontinuity of in . Step 3: is relatively compact in for a.e. . Firstly, the compactness of is trivially checked. Then, fix . For any and , there exists with For every , we define It can be seen that the compactness of with the boundedness of together assures that is relatively compact in . Furthermore, we have Thereby, is totally bounded, and thus it is relatively compact in . To this end, by applying Ascoli–Arzelà theorem and taking account of Steps 1–2, it can be verified that the mapping is compact. Step 4: has a closed graph. In fact, let , and . Choose any sequence such that . From , we conclude that the sequence is integrable bounded. Hence, is weakly compact in . It is rational to presume that . From Lemma 4, we haveBesides, due to Lemma 1. Therefore, has a closed graph.

Theorem 1. Presume that and hold. Then, there is at least one mild solution related to system (6) for each pair of initial values .

Proof. Construct a multivalued mapping byFor convenience, we write , where is given by (17) and is defined asObviously, is a fixed point of if only and if it is a mild solution of system (6). According to Lemma 2, we need to prove that (i) of Lemma 2 is not satisfied.
In fact, suppose that with and there exists withThis yields thatwhere . Employing Gronwall inequality, one getsThis impliesFurthermore, we writeObviously, is open in . Utilizing Lemma 5, we know that is compact and u.s.c. We also see easily that is a contraction mapping with single value and coefficient . Therefore, according to the choice of , none of with ensures that with some .
Applying Lemma 3, one knows that has a fixed point due to all the conditions being fulfilled. Hence, there is a mild solution for system (6).
Next, let us discuss some features of the solution set. For every , we say that is a truncation operator form to if for any , we have . Putwhere . Obviously, and . Here, stands for the solution set of the fixed points of system (6).

Lemma 6. Let and all the assumptions of Theorem 1 be fulfilled. Then, is relatively compact in for arbitrary sequence in . Especially, for every , is compact in .

Proof. It is verified by analogizing the proof of Lemma 3.7 of [8].
The -semiflow associated with system (1) is given as follows:By applying a similar discussion in [11], we can obtain that is a strict -semiflow, i.e., for arbitrary and , it holds that

Lemma 7. Assume that all the hypotheses of Theorem 1 are fulfilled. For all , is u.s.c. with compact values.

Proof. It is verified by analogizing the proof of Lemma 3.8 of [8].

4. The Existence Result of Global Attractors

In the sequel, we will present a severe validation for the existence of a global attractor of -semiflow induced by system (1). The conditions and () are replaced by () and () as follows, respectively: () The cosine and sine functions are exponentially stable with exponent , that is, there exist and with () The multivalued mapping is u.s.c. and satisfies the following conditions:

(1)There exists and , ensuring that(2)There exists ensuring that, for every bounded in ,

Given , for convenience, the translation operator is defined by . The following condensing property of is an essential result to reveal that is asymptotically upper semicompact.

Lemma 8. Suppose that and are satisfied. Then, there are and , guaranteeing that, for all and all bounded , it is inferred thatwhenever

Proof. Let , be bounded in . Denote , we obtainTherefore, we getHence,Applying the Gronwall inequality, we haveEquivalently,Note that , we getwhere .
Taking and , we conclude the proof.

Lemma 9. Presume that and are fulfilled. Then, is asymptotically upper semicompact.

Proof. Let and be a bounded subset of and be a collection consisted of sequences PutWe claim that by using reduction. Assume that there exists such thatwhere is the same as in Lemma 8. Choose as described in Lemma 8; then, there exists such that for any . Let . Then,By taking such that , we obtainThis is a contradiction.

Lemma 10. Presume that and are fulfilled. If , then an absorbing set exists for -semiflow .

Proof. Let and be bounded in . Then, for each and , there is withfor some . The combination of Hlder inequality, and the part (1) of givesPutting and , we obtainThe employment of Gronwall inequality yieldsThis impliesThe above inequality holds subject to the ball with the radius asMoreover, it is actually an absorbing set in terms of .

Theorem 2. Assume that and are fulfilled. Then, the -semiflow induced by system (1) has a compact global attractor, if the following holds:

Proof. By Lemmas 7, 9, and 10, we obtain that all the conditions of Lemma 3 are satisfied. Therefore, the -semiflow admits a compact global attractor.

5. An Illustrative Example

In this section, a concrete example is illustrated for the application of evolution partial different equations.

Take an open set which is bounded in and denote its boundary by . The following system is worthy of survey:where are continuous functions. This system describes a classical wave equation with a free bounded issue [1].

Put . The multivalued mapping is described as

Then, (57) can be reformulated asin which , , and .

Suppose that there exists and positive constants such that

Note that the mapping has convex, closed, and compact values. In addition,

By the continuity of and , we can obtain that there is a closed graph for . Furthermore, by exploiting Lebesgue dominated convergence theorem, one can verify the quasicompactness of . Therefore, is u.s.c., and is testified. If is exponentially stable and compact, then and is also satisfied. This indicates the presence of a global attractor, which belongs to the -semiflow induced by (57).

Data Availability

The data used to support the findings of this study are available within the article.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant nos. 11761011 and 71862003), Guangxi Natural Science Foundation (Grant no. 2020GXNSFAA297010), Guangxi College Young and Middle-aged Teachers Basic Ability Promotion Project (Grant no. 2019KY0669) and Collaborative Innovation Center for Integration of Terrestrial and Marine Economies.