Stability of Perturbed Set Differential Equations Involving Causal Operators in Regard to Their Unperturbed Ones considering Difference in Initial Conditions

Yakar, Coșkun; Talab, Hazm

doi:https://doi.org/10.1155/2021/9794959

Advances in Mathematical Physics

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 9794959 | https://doi.org/10.1155/2021/9794959

Stability of Perturbed Set Differential Equations Involving Causal Operators in Regard to Their Unperturbed Ones considering Difference in Initial Conditions

Coșkun Yakar¹and Hazm Talab¹

Academic Editor: Maria L. Gandarias

Received23 Jun 2021

Accepted21 Jul 2021

Published02 Aug 2021

Abstract

We investigate the stability of solutions of perturbed set differential equations with causal operators in regard to their corresponding unperturbed ones considering the difference in initial conditions (time and position) by utilizing Lyapunov functions and Lyapunov functionals.

1. Introduction

Set differential equations (SDEs) have received a lot of attention in recent decades, emerging as an independent discipline. The unifying approach of the SDEs [1–4] is one of their significant points, which is also considered as an advantage. It has been observed that SDEs are generalized forms of vector and nonlinear ordinary differential equations (ODEs) [5–7] and that ODEs can be considered as special cases of SDEs while studying its stability properties in a semilinear metric space. Moreover, SDEs can play an important role in examining multivalued differential equations and inclusions [2] and also in fuzzy equations [8]. Furthermore, causal operators [9–11] encompass a wide range of ODEs and integral [12] and integro-differential equations [13]. Thus, SDEs involving causal operators [14] give a comprehensive form of diverse classes of equations as they embrace the aforementioned special cases of differential equations.

On other hand, stability analysis [12, 14–21] can be useful for determining the qualitative properties, which in its turn leads to a more perceptive behavioral look of the differential equation’s solutions, even when they are unknown explicitly [5, 6].

In real-life problems, the solutions of differential equations may differ in initial conditions (time or position). To deal with such cases, initial time difference (ITD) stability analysis [15–17, 22–27] compares the behavioral properties of the solutions considering the change in initial conditions [22, 24–27].

Such generalized approach of stability plays an important role in the qualitative theory of differential equations in analysing the stability and other behavioral properties of its solutions using a more realistic manner that considers the difference in time and position in the initial state.

Diverse forms of ITD stability have been investigated in analysing the solutions for many forms of differential equations (such as, ordinary, fractional, and fuzzy).

In this work, we study ITD stability results for SDEs involving causal operators, by employing Lyapunov functions and functionals. We give sufficient conditions to ITD stability and ITD asymptotic stability.

We laid the foundations in Section 2, by investigating the basic definitions and results regarding ITD stability and stability of null solution of SDE with causal operator. In Section 3, we present the difficulties we face when we try to infer ITD stability properties from those of null solution, by comparing both classical and ITD notions of stability of the solution of the SDE with causal operator. In Section 4, we establish the comparison theorems for initial time difference that resolve the complications regarding the classical notion stability. In Section 5, we present an approach that resolve the difficulties allowing us to infer stability properties for solution of the perturbed form of SDE involving causal operator corresponding to the unperturbed one, by using a suitable comparison system.

2. Preliminaries

Let denotes all compact nonempty subsets of and denotes all compact and convex nonempty subsets of The Hausdorff metric between any bounded sets and in is defined by where

Each of and forms a complete metric space. with ordinary addition and nonnegative scalar multiplication is a semilinear metric space that can be regarded as a cone in an appropriate Banach space.

Some properties of can be stated as follows: for any and , where denotes and the scalar multiplication denotes . If we get .

In general, (unless is a singleton). To overcome with this implication of Minkowski difference, i.e.,

Hukuhara difference between is introduced as follows:

If there is a where , then Hukuhara difference exists and it is denoted by , or simply when there is no confusion with Minkowski difference, i.e.,

Hukuhara difference has an important property, Minkowski difference does not have (in general), which is , for any compact and convex nonempty subset of .

Let be a given multifunction, where denotes a real-number interval. is said to be Hukuhara differentiable at , if we ensure the existence of so both exist in and have the same value as

The existence of and for a sufficiently small , is implicit in definition.

By considering as a complete cone, and embedding it in an appropriate Banach space, with the properties of Bochner integral, it is seen that if where is integrable in Bochner sense, then is Hukuhara differentiable, i.e., exits, and holds almost everywhere on .

The Hukuhara integral of , over a compact set , is defined as the integral of a continuous selector of over .

Considering the null element of as a point set, let us define on the space as where and is a continuous map. Then, is a complete normed space.

The operator is called causal if , for , and implies

Consider the following equations: where (10) and (11) are different in initial time and position, (12) is the perturbed form corresponding to the unperturbed system (11), and where are causal operators and

A special case of (12) is where is the perturbation term.

Assume that for , and assume the necessary smoothness of , , and to guarantee the existence and uniqueness of the solution of (10) through for and those of the solution of (12) through for , in addition to their continuous dependence on initial conditions.

If on , then it is said to be a solution of (10) on if (10) holds with this for all . If on , then these are said to be solutions of (11), (12), and (13) on provided that they satisfy (11), (12), and (13) on , respectively.

The continuously differentiability of and enables us to write

Thus, the corresponding equations with the initial value problems (IVPs) of (10), (11), (12), and (13) are the followings, respectively. where the integrals are the Hukuhara integrals.

Note that and also , , and are solutions of (10) and (11), (12), and (13) if and only if they satisfy (16) on and (17), (18), and (19) on , respectively.

Furthermore, to resemble the behavioral properties of solution of (10) with these of solution of an ordinary correspondent one, we assume that to ensure the existence of the Hukuhara difference . Accordingly, we have the solution and the corresponding equation

We also assume that so that the Hukuhara difference exists. Consequently, we have the solution and the corresponding equation and the corresponding perturbed system of (21)

If we ensure the existence of aforementioned Hukuhara differences and there is no ambiguity in the context, we may omit the asterisk symbols in (20), (21), and (22) and continue with the familiar notation as in (10), (11), and (12).

Before establishing the comparison theorems and criteria for ITD stability for SDEs involving causal operators, let us present the following basic definitions for ITD stability.

Definition 1. Let where solves (10) for , and . The solution of (12) for with the initial conditions is called
(S1) An ITD stable solution w.r.t. the solution if and only if for any given , we can designate two positive values and which give us the following inequality: whenever and for
(S2) An ITD uniformly stable solution w.r.t. the solution if both in (S1) are independent of
(S3) An ITD attractive solution w.r.t. the solution if and only if for any , we can designate two positive values and a which give us the following inequality: provided that and for
(S4) An ITD uniformly attractive solution w.r.t. the solution if and only if all , and in (S3) are independent of
(S5) An ITD asymptotically stable solution w.r.t. the solution if and only if (S1) and (S3) hold all together, or equivalently if (S1) holds and there exist and so that for any and with (S6) An ITD uniformly asymptotically stable solution w.r.t. the solution if and only if (S2) and (S4) hold all together, or equivalently if in addition to (S2) we can designate, for any given , two positive values and and a so provided that and for each or if and in (S5) are independent of
(S7) An ITD exponentially asymptotically stable solution w.r.t. the solution if and only if there exists a constant such that for

Definition 2. The function is said to be from the class , or simply written as , if and only if it associates zero to zero and is strictly increasing in . If and as , then it is said to be from the class , or simply written as .

Definition 3. Let , (a)The Dini derivatives of are defined as for (b)The generalized (Dini-like) derivatives of are defined as where solves (12) and and solve (10) for where (c)The generalized derivatives of a Lyapunov functional are defined as where solves (12) and solves (10). Furthermore, , where solves (10) for

In the following section, let us present a comparative analysis on how we can infer the stability properties from those of the null solution of the SDE with causal operator in the classical sense of stability, whereas there is incompatibility in using the same manner in the context of ITD stability.

3. Classical vs. ITD Stability of SDE with Causal Operator

3.1. Classical Notion of Stability of SDE with Causal Operator

Consider the following SDE with causal operator where and

Let be any solution of (32) and assume that so that the null solution solves (32) with the initial conditions

Definition 4. The null solution of (32) is called
(S1) A stable solution if and only if given any , , we can designate two positive functions and , which are continuous in for each , and which give us for , given that and
(S2) A uniformly stable solution if additionally and are independent of
(S3) A quasiequiasymptotically stable solution if and only if given any positive we can designate and so for provided that .
(S4) A quasiuniformly asymptotically stable solution if additionally each of and in (S3) is independent of
(S5) An equiasymptotically stable solution if and only if both (S1) and (S3) are verified
(S6) A uniformly asymptotically stable solution if and only if both (S2) and (S4) are verified
(S7) A quasiequiasymptotically stable solution if and only if given any positive values , we can designate which gives us for , provided that .
(S8) A quasiuniformly asymptotically stable solution if and only if (S7) is verified with being independent of additionally
(S9) A completely stable solution if and only if (S1) holds and (S7) is verified given any
(S10) A uniformly completely stable solution if and only if (S2) holds and (S8) is verified for all
(S11) an unstable solution if and only if (S1) fails to hold.

Definition 5. The solution of (32) with the initial conditions is called a stable solution w.r.t. the solution of (32) for if and only if for each , we can designate and which are positive and continuous in for each so provided that and for

Additionally, if each of and is independent of , then the solution of (32) is called a uniformly stable solution w.r.t. the solution

Remark 6. Let be a given solution of (32). We can use the following change of variable to study the stability of this solution.

For , set

Then,

If , we can ascertain that solves the transformed SDE. Hence, it indicates that for

Since and , the solution of (32) corresponds to the identically null solution of where

Hence, without loss of generality, it is enough to consider the stability of the null solution of (32). Unfortunately, such procedure is not feasible for ITD stability analysis.

3.2. ITD Stability of SDE with Causal Operator

Assume solves (11) and let where solves (10) for Let us go through a similar procedure as in (39) by setting

Then,

Considering that even when is neither zero nor it solves the transformed differential system; also, is not the exactly null solution of Consequently, it is not an option to infer ITD stability properties using stability properties of the null solution.

This motivates us to search for other different approaches in studying such behavioral properties.

4. Comparison Theorems for ITD Stability of SDEs with Causal Operators

Based on our previous analysis and other’s studies [5, 7, 10], due to the differences between the ITD notion of stability and the classical one, it is seen that the behavioral analysis of the zero solution cannot be utilized in ITD stability. In what follows, we present an approach that resolves those complications and enables us to infer the stability properties of the solution of (12) w.r.t. where solves (10).

Theorem 7. Let (i) be locally Lipschitzian in ; i.e., (ii)For and , it is considered where and (iii) is the maximal solution for existing on

And let , where solves (10) and solves (12) for , and assume ; then,

Proof. Let For , let us show the inequality .
As a matter of fact, Substituting the last statement in together with the theorem’s assumptions gives us Let be sufficiently small, and let us consider whose we ensure the existence of its solutions up to
Let us conclude this proof by showing To do so, let us suppose on the contrary that there is a where Thus, it follows that Taking into consideration the proposition regarding and since , we conclude that the solutions are nondecreasing in
As for , we obtain Consequently,
For sufficiently small enough , we have Hence, with considering the assumptions in (i) regarding the locally Lipschitzity of in , it is seen where and stand for errors.

This gives us the following estimation regarding the Dini derivative of

for

Since , when , we have which contradicts (55).

Consequently, we have , which in its turn, as , gives us the desired estimation ☐

Corollary 8. Assume that in Theorem 7 satisfies the aforementioned conditions in addition to that and Then, and solve (10) and (11), respectively.
Equivalently, for , we have

Theorem 9. Let (i) be locally Lipschitzian in ; i.e., (ii)For and , we have for and where and (iii) is the maximal solution of existing on And let , where solves (10) and solves (12) for , and assume Then,

Proof. Let for , and assume . Then, in view of the theorem’s assumptions, we obtain The utilization of Theorem 7, in its turns, gives us the desired estimation for ☐

After presenting the previous comparison theorems for ITD stability of SDE with causal operators, we can now employ them in proofing the following theorems regarding ITD stability, ITD asymptotic stability, and ITD uniformly asymptotic stability of the solution of the SDE (10).

5. ITD Stability of SDEs with Causal Operators

5.1. ITD Stability of SDEs with Causal Operators via Lyapunov Functions

Let us present sufficient conditions for stability of the solution of (10), using Lyapunov functions, and assuming the existence and uniqueness of this solution for .

Theorem 10. Let (i) be be locally Lipschitzian in ; i.e., (ii)For and for , and where (iii) is positive definite and decrescent on ; i.e., there exists two functions from the class such that Then, the solution of (12) is ITD stable w.r.t. the solution of (10), where solves (10), for .

Proof. Given a sufficiently small and , let us designate and , with the aim that holds, where Then, with this , we shall prove the stability of the solution of (10) for .
If we suppose the opposite, then there would exist solutions of (12) and of (10), on , and so By the assumption (ii) and Corollary 8 we obtain the following estimation Hence, employing the previous assumptions, especially (79), (80), and (iii), together with the choice of , gives us that which is a contradiction.
Therefore, we conclude the stability of the solution of (10) for Hence, the solution of (12) is ITD stable w.r.t. the solution of (10), assuming solves (10), for .
If additionally both and are independent of , then the solution of (10) is uniformly stable for .☐

Theorem 11. Let (i) be locally Lipschitzian in ; i.e., (ii)For and for , and where the following inequality holds where Additionally, is continuous for and , and(iii) is positive definite and decrescent on ; i.e., there exists two functions from the class such that Then, the solution of (12) is ITD asymptotically stable w.r.t. the solution of (10) where solves (10), for

Proof. Theorem 9 yields Furthermore, by Theorem 10, we have the stability of the solution of (10) for Therefore, it is sufficient to prove its quasiasymptotic stability property. To do this, let so that Choose and . Then, by using (ii), (iii), and (87), and since , i.e., , it follows that given an and , a positive can be designated in order to satisfy given that for .
Therefore, the solution of (10) is a quasiasymptotically stable solution for Hence, the solution of (12) is ITD asymptotically stable w.r.t. the solution of (10), or the solution of (10) is an asymptotically stable solution for .
If additionally and are all independent of , then the solution of (10) is uniformly asymptotically stable for .☐

5.2. ITD Stability of SDEs with Causal Operators via Lyapunov Functionals

In the following, we present sufficient conditions for ITD uniformly asymptotic stability for SDEs with causal operators, in correspondence to Lyapunov second method, by using Lyapunov functional.

Theorem 12. Let (i) and satisfies the following estimation (ii) is positive definite and decrescent on ; i.e., there exists two functions from the class such that Then, the solution of (12) is ITD uniformly asymptotically stable w.r.t. the solution of (10) where solves (10), for .

Proof. Given a sufficiently small and , let us designate and so where .
Then, with this and , we shall prove the stability of the solution of (10) for
If we suppose the opposite, then there would exist solutions of (12) and of (10) for , and so Then, the assumption (ii) and Corollary 8 give us the following estimate for
Hence, in view of (92), (93), (94), and (ii), in addition to the assumptions on , we have the following statement which is a contradiction. This yields that the solution of the unperturbed differential system (10) is uniformly stable for

To complete the proof, let us prove the uniformly asymptotic stability. Assume and let such that gives us

Taking into consideration the uniform stability, it follows the existence of such that satisfies for , with corresponds to for uniform stability.

If we assume that it is not true, let

Then, in view of (i), it leads to

Hence, for , we obtain

Since , i.e., associates zero to zero and is strictly increasing in , and considering we obtain that

By substituting each by in the inequality (100) and by choosing with the above assumptions, we have giving us a contradiction. Hence, there exists a such that which in its turn, by the stability property, gives us that for , given that ☐

6. Conclusion

Despite the obstacles we faced when trying to infer ITD stability properties from those ones concerning the classical notion of stability of the null solution, using a change-of-variable approach, we managed to resolve those difficulties via comparison theorems for ITD stability of SDEs with causal operators, that take into consideration the change in initial conditions regarding time and position.

In this manuscript, Lyapunov functions and Lyapunov functionals are utilized to predict ITD stability, ITD asymptotic stability, and ITD uniformly asymptotic stability of the solutions of perturbed forms of SDEs involving causal operators corresponding to unperturbed systems, in light of the classical stability properties of the trivial solution of suitable comparison systems.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors would like to thank the Presidency for Turks Abroad and Related Communities (YTB) of Republic of Turkey Ministry of Culture and Tourism, and Gebze Technical University, for their support.

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