Stability of Set Differential Equations in Fréchet Spaces

Bao, Junyan; Chen, Wei; Wang, Peiguang

doi:https://doi.org/10.1155/2023/5134374

Advances in Mathematical Physics

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Research Article | Open Access

Volume 2023 | Article ID 5134374 | https://doi.org/10.1155/2023/5134374

Stability of Set Differential Equations in Fréchet Spaces

Junyan Bao,¹Wei Chen,¹and Peiguang Wang¹

Academic Editor: Manuel De Le n

Received10 Nov 2022

Revised07 Jan 2023

Accepted20 Mar 2023

Published31 Mar 2023

Abstract

In this paper, we investigate the stability of set differential equations in Fréchet space . Some comparison principles and stability criteria are established for set differential equations with the fact that every Fréchet space is a projective limit of Banach spaces.

1. Introduction

In recent years, set differential equations (SDEs for short) have attracted extensive attention due to its intrinsic advantages. In 1970, Blasi and Iervolino [1] first began the study of SDEs in semilinear metric spaces. Nowadays, the theory of SDEs has been developed into an independent subject area. There are a few results on the existence, stability, and other properties of solutions for various equations, such as SDEs [2–11], set functional differential equations [12–16], set integrodifferential equations [17–20], SDEs on time scales [21, 22], SDEs with causal operators [23–27], and others [15, 28–31], and references are given therein. Systematic development of set differential equations has been provided by Lakshmikantham et al. [32] and Martynyuk [33].

However, with the development of SDEs, when we further try to extend the results to the case of infinite dimensional locally convex spaces, namely, Fréchet spaces which are not Banach spaces, most of the previous definitions are no longer available in the new framework. Apart from this, the increasing problems modeled in the framework of non-Banach infinite dimensional spaces appear in modern analysis, differential geometry, and theoretical physics, also inspiring us to seek alternative methods to investigate SDEs. Thus, we will study SDEs within the Fréchet framework in this paper. In fact, several basic results have been obtained in this field. For example, Galanis et al. [34] generalized the basic notions about SDEs to a Fréchet space using the fact that it is a projective limit of Banach spaces. In the new framework, they obtained the properties of the Hausdorff metric and constructed the continuity, and Hukuhara differentiability of set-valued mappings, which lay the foundation for the study of SDEs. In [35], Galanis et al. investigated the existence of solutions of SDE in Fréchet space. Wang et al. [36] obtained the criterion of rapid convergence of solutions for SDEs.

The major purpose of this paper is to establish some comparison results and further attempt to study the stability of solutions of SDEs in Fréchet space employing Lyapunov-like functions and comparison principles. We recall some necessary background materials on the study of SDEs in Banach spaces and Fréchet ones in Section 2. Subsequently, in Section 3, we establish some comparison results of SDEs in Fréchet space. Finally, some types of stability properties of solutions of SDEs in Fréchet space are derived in Section 4.

2. Preliminaries

We first give some concepts related to Banach space [32, 34, 35] that will be used in the later discussion.

Let denote the collection of all nonempty, compact, and convex subsets of Banach space . For any two nonempty subsets and on , the Hausdorff metric is determined by the formula where denotes the norm in . Then, is a complete metric space.

For the space , if we define the natural algebraic operations of addition and nonnegative scalar multiplication, then is a semilinear metric space (see [32]).

For and , the Hausdorff metric (1) has the following properties:

For , we call the mapping is Hukuhara differentiable if there exists , such that where the differences are in the sense of the Hausdorff difference.

Moreover, if there exists a mapping , such that where the integrable is in the sense of Bochner, then a.e. on .

We denote the integral as follows

However, because the topology of is derived from a set of seminorms rather than a single norm, the application of the classical definition of Hausdorff distance in is impossible. Therefore, the above methodology is no longer available if the space we dealt with is not Banach space , but Fréchet space . In order to overcome the above shortcomings, Galanis et al. [34, 35] proposed a set of structures suitable for generalized locally convex topological vector spaces on Fréchet spaces. The main difference is that the classical Hausdorff metric is replaced by a group of semimetrics in a Fréchet space.

Let be a Fréchet space defined by a sequence of seminorms, and the sequence be increasing. Then, can be seen as a projective limit of Banach spaces , where denotes the completion of the quotient , , and are the connecting morphisms the bracket stands for the corresponding equivalence class, and is a continuous mapping (see [37, 38]).

In this case, the space can be viewed as a projective limit space with a corresponding structure to s: where the mapping . Each is continuous concerning the topologies induced by the Hausdorff metrics on and . Therefore, every element of can be realized as where are the canonical projections of to .

With the above definition, we can revise the notion of the Hausdorff metric in Fréchet space as follows:

Definition 1 (see [34]). Let be a Fréchet space which is defined by a group of seminorms. For , we call -Hausdorff metric between and . From the perspective of classical conceptions of Banach factors, the connection below exists: if , , , then where the revised -Hausdorff metric also satisfies the corresponding properties (2).

Let the set-valued mapping can be seen as a projective limit of the corresponding mappings in the Banach factors ; that is, , in which , denotes the canonical projections of the limit .

Proposition 2 (see [35]). The mapping is Hukuhara differentiable at if and only if every is Hukuhara differentiable at and

3. Comparison Results of SDEs in Fréchet Space

In this section, we consider the IVP for SDEs in the Fréchet space with the framework of where , .

The mapping is called a solution of SDEs (12) on ; that is

Then, we associate the IVP (12) with the integral equations

In our further discussion, we will use the comparison differential equation where , .

Theorem 3. Suppose that the following conditions are satisfied:
B1: the projective limit ,
B2: , ,
B3: the scalar differential equation (15) has a maximal solution existing on
Then, provided , where and are solutions of SDEs (12) on .

Proof. By the fact of the mapping as a projective limit, Equations (12) can be reduced to a system of SDEs on the Banach spaces : where denotes the projection of .
From condition B2, (10), and the previously stated Property 2.1, we have According to the comparison principle [32] for SDEs in Banach space with the framework of , if and are solutions of (16) through on , then providing that .
In addition, by considering the projective limits of , we can obtain that means is a solution of (12) through on . In the same way, we can claim that is a solution of (12) through . Thus providing .
This proves the claimed estimation of Theorem 3.
Next, we give a comparison result under weaker assumptions.

Theorem 4. Suppose that the conditions B1 and B3 are satisfied in Theorem 3, and Then, the conclusion of Theorem 3 holds.

Proof. By condition B1, the inequality (21) can be transformed into the inequality on Banach spaces: where , .
Using the properties (2) and the inequality (21), we can obtain the inequality (17) hold. The rest argument is similar to the proof of Theorem 3. We omit here.
After that, we give a comparison theorem via Lyapunov-like functions which can be used to the discussion of the stability theory of Lyapunov in Fréchet space.

Theorem 5. Suppose that , , and.
B4: , is a positive constant
B5:
Then, provided .

Proof. By Theorem 3, equations (12) can be transformed into the following SDEs on the Banach spaces : where and . Meanwhile, we have where .
Similarly, we have namely, If is any solution of SDEs (12) existing on and satisfies , then, we obtain Moreover, by considering the projective limits of , we have Then, is a solution of (12) on . Furthermore, we obtain This proves the claimed estimation of Theorem 5.

4. Stability Criteria

In this section, we give the stability criteria via Lyapunov functions in Fréchet spaces.

Firstly, we give the following sets for convenience.

Definition 6. The trivial solution of (12) is said to be
C1: equistable, if for each and , there exists a , such that where is the solution of SDEs (12)
C2: uniformly stable, if the in (C1) is independent of
C3: equiattractive, if for each and , there exist and , such that C4: uniformly attractive, if the and in C3 are independent of
C5: equiasymptotically stable, if C1 and C3 hold simultaneously
C6: uniformly asymptotically stable, if C2 and C4 hold simultaneously

Theorem 7. Suppose the following conditions are satisfied:
S1: and for such that S2: for , there exist and satisfying Then, the trivial solution of SDEs (12) is equistable.

Proof. From the previous section, we know that equation (12) can be reduced to a system of SDEs on the Banach spaces by considering the mapping as a projective limit. Using the known conditions, we obtain where and . Noticing that , we have where and , .
Furthermore, we obtain Similar to the proof of stability criteria in Theorem 3.4.1 [32], we know that the trivial solution of (36) is equistable. Thus, for , , there exists a , such that that is, for each , there exists a , such that provided . Therefore, the trivial solution of (33) is equistable.
The proof is complete.

Similar to the proof process of Theorem 7, we can obtain the following theorems.

Theorem 8. Suppose that the conditions of Theorem 7 hold, except that inequality (34) can be strengthened to Then, the trivial solution of SDEs (12) is equiasymptotic stable.

Theorem 9. Suppose that the following conditions are satisfied:
S3: for any , , and , such that (33) and (34) hold;
S4: there exist satisfying Then, the trivial solution of SDEs (12) is uniformly stable.

Theorem 10. Suppose that the conditions of Theorem 9 are satisfied except that the condition (S3) of Theorem 9 is strengthened to where . Then, the trivial solution of SDEs (12) is uniformly asymptotic stable.

Finally, we shall present the stability of SDEs in Fréchet space via the comparison principle which will unite the stability relations between SDEs and the comparison differential equation (15).

Theorem 11. Suppose that the conditions B3 in Theorem 3 and S4 in Theorem 9 are satisfied, and
S5: for , and , such that Then, the stable (uniform) stability properties of the trivial solution of (15) imply the corresponding equistable (uniform) stability of the trivial solution of SDEs (12).

Proof. (i)Firstly, we prove the equistable of SDEs (12). Assume that the scalar differential equation (15) is stable. Hence, for the given and , there exists a , such that Choosing and satisfying .
Letting . Then, for any solution of SDEs (12), we claim that , . If it is not true, then there exists a solution of (12) with and a such that By Theorem 5, we obtain Then, From (47) and (48), we get which is a contradiction. This proves that the trivial solution of SDE (12) is equistable. (ii)Secondly, we prove the equiasymptotic stability of (12). Suppose that the solution of the scalar differential equation (15) is asymptotic stable. Then, the trivial solution of SDEs (12) is equistable. From , the trivial solution of comparison equation (15) is equistable. Therefore, we only need to prove that SDE (12) is equiattractiveConsidering the attractivity of the trivial solution of (15), we take and choose . By setting , then for the given , there exist and , such that Choosing and such that (47) holds, that is .
Let , then using the conditions (S4) in Theorem 9, (51) and the result of Theorem 9, we obtain that is, the trivial solution of (12) is equiattractive. Furthermore, it is equiasymptotic stable. (iii)Finally, the proof of the uniform stability is analogous to the proof of equistability, and what difference is only to choose the independent of . The proof of the uniformly asymptotic stability is considered in the same way. Here, we omit the details.

Remark 12. Another way to prove the above Theorem is that, firstly, we transform the known conditions in Fréchet space into Banach space by the realization of the projective limit; then, we prove the stability results in Banach spaces; then, by the relation , where and , we can obtain the corresponding results.

Data Availability

No underlying data was collected or produced in this study.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

All authors completed the paper together. All authors read and approved the final manuscript.

Acknowledgments

This paper is supported by the National Natural Science Foundation of China (12171135, 11771115).

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Copyright © 2023 Junyan Bao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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