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

Authors: Coșkun Yakar; Hazm Talab
Journal: Advances in Mathematical Physics
 (2021)

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## 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.


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## Body

## 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 Section2, 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

LetKℝn denotes all compact nonempty subsets of ℝn and Kcℝn denotes all compact and convex nonempty subsets of ℝn. The Hausdorff metric between any bounded sets A1 and A2 in ℝn is defined by
(1)DA1,A2=maxsupa2∈A2da2,A1,supa1∈A1da1,A2,where
(2)da2,A1=infda2,a1:a1∈A1.Each ofKℝn,D and Kcℝn,D forms a complete metric space. Kcℝn 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 ofD can be stated as follows:
(3)1DA1,A2=DA2,A1,2DA1+C,A2+C=DA1,A2,3DkA1,kA2=kDA1,A2,4DA1,A2≤DA1,C+DC,A2,for any A1,A2,C∈Kcℝn and k∈ℝ+, where A1+C denotes a1+c:a1∈A1,c∈Cand the scalar multiplication kA1 denotes ka1:a1∈A1. If k=−1, we get −A1=−1A1=−a1:a1∈A1.In general,A1+−A1≠0 (unless A1=a is a singleton). To overcome with this implication of Minkowski difference, i.e.,
(4)A1+−1A2=a1−a2:a1∈A1,a2∈A2.Hukuhara difference betweenA1,A2∈Kcℝn is introduced as follows:If there is aC∈Kcℝn where C+A2=A1, then Hukuhara difference exists and it is denoted by A1⊝A2, or simply A1−A2 when there is no confusion with Minkowski difference, i.e., A1⊝A2=C⟺C+A2=A1.Hukuhara difference has an important property, Minkowski difference does not have (in general), which isA1−A1=0, for any compact and convex nonempty subset A1 of ℝn.LetU:I⟶Kcℝn be a given multifunction, where I denotes a real-number interval. U is said to be Hukuhara differentiable at t0∈I, if we ensure the existence of DHUt0∈Kcℝn so
(5)limε⟶0+Ut0+ε−Ut0εandlimε⟶0+Ut0−Ut0−εε,both exist in Kcℝn and have the same value as DHUt0.The existence ofUt0+ε−Ut0 and Ut0−Ut0−ε, for a sufficiently small ε>0, is implicit in DHUt0 definition.By consideringKcℝn as a complete cone, and embedding it in an appropriate Banach space, with the properties of Bochner integral, it is seen that if
(6)Gt=Gt0+∫t0tFsds,t∈I,where F:I⟶Kcℝn is integrable in Bochner sense, then G is Hukuhara differentiable, i.e., DHGt exits, and DHGt=Ft holds almost everywhere on I.The Hukuhara integral ofF, over a compact set I⊂ℝ+, is defined as the integral of a continuous selector f of F over I.
(7)∫IFtdt=∫Iftdt.Considering the null elementθ of ℝn as a point set, let us define . on the space E=Ct0,∞,Kcℝn as
(8)U=sups∈t0,∞DUs,θhs<∞,where U∈E and h:t0,∞⟶ℝ+ is a continuous map. Then, E,. is a complete normed space.The operatorQ∈CE,E is called causal if U1s=U2s, for U1,U2∈E, and s∈t0,t implies
(9)QU1s=QU2s,s∈t0,t.Consider the following equations:(10)DHU=QUt,Ut0=U0,forU0∈Kcℝnandt≥t0≥0,(11)DHU=QUt,Uτ0=V0,forV0∈Kcℝnandt≥τ0≥0,(12)DHV=PVt,Vτ0=V0,forV0∈Kcℝnandt≥τ0,(13)DHW=SWt,Wτ0=V0−U0,forWτ0=W0∈Kcℝnandt≥τ0,where (10) and (11) are different in initial time and position, (12) is the perturbed form corresponding to the unperturbed system (11), and where Q,P,S:E⟶E are causal operators and Sρ=U∈Kcℝn:DU,0~<ρ<∞.A special case of (12) is
(14)PVt=QVt+RVt,where RVt is the perturbation term.Assume thatQ0~t≡0~ for t≥0, and assume the necessary smoothness of P, Q, and R to guarantee the existence and uniqueness of the solution Ut=Ut,t0,U0 of (10) through t0,U0 for t≥t0 and those of the solution Vt=Vt,τ0,V0 of (12) through τ0,V0 for t≥τ0, in addition to their continuous dependence on initial conditions.IfU∈C1J1,Kcℝn on J1=t0,t0+T1, then it is said to be a solution of (10) on J1 if (10) holds with this U for all t∈J1. If U,V,andW∈C1J2,Kcℝn on J2=t0,t0+T2, then these are said to be solutions of (11), (12), and (13) on J2 provided that they satisfy (11), (12), and (13) on J2, respectively.The continuously differentiability ofUt,Vt and Wt enables us to write
(15)Ut=U0+∫t0tDHUsds,U0∈Kcℝnandt∈J1,Ut=U0+∫τ0tDHUsds,U0∈Kcℝnandt∈J2,Vt=V0+∫τ0tDHVsds,V0∈Kcℝnandt∈J2,Wt=W0+∫τ0tDHWsds,W0∈Kcℝnandt∈J2.Thus, the corresponding equations with the initial value problems (IVPs) of (10), (11), (12), and (13) are the followings, respectively.
(16)Ut=U0+∫t0tQUssds,U0∈Kcℝnandt∈J1,(17)Ut=U0+∫τ0tQUssds,U0∈Kcℝnandt∈J2,(18)Vt=V0+∫τ0tPVssds,V0∈Kcℝnandt∈J2,(19)Wt=W0+∫τ0tSWssds,W0∈Kcℝnandt∈J2,where the integrals are the Hukuhara integrals.Note thatUt and also Ut, Vt, and Wt are solutions of (10) and (11), (12), and (13) if and only if they satisfy (16) on J1 and (17), (18), and (19) on J2, respectively.Furthermore, to resemble the behavioral properties of solution of (10) with these of solution of an ordinary correspondent one, we assume that U0=Z0+U0∗ to ensure the existence of the Hukuhara difference U0∗=U0−Z0. Accordingly, we have the solution Ut,t0,U0−Z0=Ut,t0,U0∗ and the corresponding equation
(20)DHU=QUt,Ut0=U0∗,forU0∗∈Kcℝnandt≥t0≥0.We also assume thatV0=S0+V0∗ so that the Hukuhara difference V0∗=V0−S0 exists. Consequently, we have the solution Vt,t0,V0−S0=Ut,t0,V0∗ and the corresponding equation
(21)DHU=QUt,Uτ0=V0∗,forV0∗∈Kcℝnandt≥τ0≥0,and the corresponding perturbed system of (21)
(22)DHV=PVt,Vτ0=V0∗,forV0∗∈Kcℝnandt≥τ0≥0.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.
LetU~t,τ0,U0=Ut−η,t0,U0, where Ut,t0,U0 solves (10) for t≥τ0, τ0=η+t0 and t0,τ0∈ℝ+. The solution Vt,τ0,V0 of (12) for t≥τ0,τ0∈ℝ+ with the initial conditions τ0,V0 is called
(S1) An ITD stable solution w.r.t. the solutionU~t,τ0,U0 if and only if for any given ε>0, we can designate two positive values δ1=δ1ε,τ0 and δ2=ε,τ0 which give us the following inequality:
(23)DVt,τ0,V0−Ut−η,t0,U0,0~<ε,whenever DV0−U0,0~<δ1 and dsτ0−t0,0<δ2 for t≥τ0.
(S2) An ITD uniformly stable solution w.r.t. the solutionU~t,τ0,U0 if both δ1,δ2 in (S1) are independent of τ0∈ℝ
(S3) An ITD attractive solution w.r.t. the solutionU~t,τ0,U0 if and only if for any ε,τ0>0, we can designate two positive values δ1=δ1τ0,δ2=δ2τ0 and a T=Tε,τ0 which give us the following inequality:
(24)DVt,τ0,V0−Ut−η,t0,U0,0~<ε,provided thatDV0−U0,0~<δ1 and dsτ0−t0,0<δ2 for t≥τ0+T
(S4) An ITD uniformly attractive solution w.r.t. the solutionU~t,τ0,U0 if and only if all δ1>0,δ2>0, and T>0 in (S3) are independent of τ0∈ℝ+
(S5) An ITD asymptotically stable solution w.r.t. the solutionU~t,τ0,U0 if and only if (S1) and (S3) hold all together, or equivalently if (S1) holds and there exist γ1τ0>0 and γ2τ0>0 so that
(25)limt⟶∞DVt,τ0,V0−Ut−η,t0,U0,0~=0,for any Vt,τ0,V0 and Ut−η,t0,U0 with
(26)DV0−U0,0~<γ1,anddsτ0−t0,0<γ2,fort≥τ0.
(S6) An ITD uniformly asymptotically stable solution w.r.t. the solutionU~t,τ0,U0 if and only if (S2) and (S4) hold all together, or equivalently if in addition to (S2) we can designate, for any given ε>0, two positive values γ1=γ1ε and γ2=γ2ε and a Tε>0 so
(27)DVt,τ0,V0−Ut−η,t0,U0,0~<ε,provided that DV0−U0,0~<γ1 and dsτ0−t0,0<γ2 for each t≥τ0+Tε; or if γ1 and γ2 in (S5) are independent of τ0∈ℝ+.
(S7) An ITD exponentially asymptotically stable solution w.r.t. the solutionU~t,τ0,U0 if and only if there exists a constant α>0 such that
(28)DVt,τ0,V0−Ut−η,t0,U0,0~≤DV0−U0,0~exp−αt−τ0,for t≥τ0∈ℝ+.Definition 2.
The functionσ∈C0,ρ,ℝ+ is said to be from the class K, or simply written as σ∈K, if and only if it associates zero to zero and σλ is strictly increasing in λ. If ρ=∞ and σλ⟶∞ as λ⟶∞, then it is said to be from the class K∞, or simply written as σ∈K∞.Definition 3.
LetL∈Cℝ+×Kcℝn,ℝ+,
(a)
The Dini derivatives ofL are defined as
(29)D+Lt,V=limε⟶0+sup1εLt+ε,V+εQVt−Lt,V,D−Lt,V=limε⟶0−inf1εLt+ε,V+εQVt−Lt,V,for t,V∈ℝ+×Kcℝn.(b)
The generalized (Dini-like) derivatives ofL are defined as
(30)D∗+Lt,V−U~=limε⟶0+sup1εLt+ε,V−U~+εPVt−Q~U~t−Lt,V−U~,D∗−Lt,V−U~=limε⟶0−inf1εLt+ε,V−U~+εPVt−Q~U~t−Lt,V−U~,where Vt,τ0,V0 solves (12) and U~t,τ0,U0=Ut−η,t0,U0 and Ut,t0,U0 solve (10) for t≥τ0,t0,τ0∈ℝ+, where τ0=η+t0.(c)
The generalized derivatives of a Lyapunov functionalLt,W∈Cℝ+×E,ℝ+ are defined as
(31)D+Lt,Vt−U~t=limε⟶0+sup1εLt+ε,Vt+ε,t,V−U~t+ε,t,U~−Lt,Vt−U~t,D−Lt,Vt−U~t=limε⟶0−inf1εLt+ε,Vt+ε,t,V−U~t+ε,t,U~−Lt,Vt−U~t,where Vt+ε,t,V solves (12) and U~t+ε,t,U~ solves (10). Furthermore, U~t=Ut−η, where Ut=Ut,t0,U0 solves (10) for t≥τ0.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(32)DHV=QVt,Vτ0=V0,forV0∈Kcℝnandt≥τ0,τ0∈ℝ+,where Q:E⟶E and Sρ=V∈Kcℝn:DV,0~<ρ<∞.LetVt,t0,V0 be any solution of (32) and assume that Q0~t=0~,t∈ℝ+ so that the null solution V≡0 solves (32) with the initial conditions t0,0.Definition 4.
The null solutionV=0 of (32) is called
(S1) A stable solution if and only if given anyε>0, τ0∈ℝ+, we can designate two positive functions δ1=δ1ε,τ0 and δ2=δ2ε,τ0, which are continuous in τ0 for each ε, and which give us
(33)DVt,τ0,V0,0~<ε,for t≥τ0, given that DV0,0~<δ1 and dsτ0,0<0.
(S2) A uniformly stable solution if additionallyδ1 and δ2 are independent of τ0
(S3) A quasiequiasymptotically stable solution if and only if given any positiveε,t0 we can designate δ0=δ0t0 and T=Tt0,ε so
(34)DVt,τ0,V0,0~<ε,for t≥t0+T provided that DV0,0~<δ0.
(S4) A quasiuniformly asymptotically stable solution if additionally each ofδ0 and T in (S3) is independent of t0
(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 valuesε,α,t0, we can designate T=Tt0,ε,α>0 which gives us
(35)DVt,τ0,V0,θ<ε,for t≥t0+T, provided that DV0,0~≤α.
(S8) A quasiuniformly asymptotically stable solution if and only if (S7) is verified withT being independent of t0 additionally
(S9) A completely stable solution if and only if (S1) holds and (S7) is verified given anyα,0≤α<∞
(S10) A uniformly completely stable solution if and only if (S2) holds and (S8) is verified for allα,0≤α<∞
(S11) an unstable solution if and only if (S1) fails to hold.Definition 5.
The solutionVt,τ0,V0 of (32) with the initial conditions τ0,V0 is called a stable solution w.r.t. the solution Vt,τ0,U0 of (32) for t≥t0≥0 if and only if for each ε>0, we can designate δ1=δ1ε,τ0 and δ2=δ2ε,τ0 which are positive and continuous in τ0 for each ε so
(36)DVt,τ0,V0−Vt,τ0,U0,0~<ε,provided that DV0−U0,0~<δ1 and dsτ0,0<δ2 for t≥τ0.Additionally, if each ofδ1 and δ2 is independent of τ0∈ℝ+, then the solution Vt,τ0,V0 of (32) is called a uniformly stable solution w.r.t. the solution Vt,τ0,U0.Remark 6.
LetVt,t0,V0 be a given solution of (32). We can use the following change of variable to study the stability of this solution.Fort≥t0, set
(37)Ωt,t0,V0−U0=Vt,t0,V0−Vt,t0,U0.Then,(38)DHΩt,t0,V0−U0=DHVt,t0,V0−DHVt,t0,U0,(39)DHΩt,t0,V0−U0=QΩt,t0,V0−U0t+QVt,t0,U0t−QVt,t0,U0t=QΩt,t0,V0−U0+Vt,t0,U0t−QVt,t0,U0t=Q~Ωt,t0,V0−U0t.IfV0−U0=0~, we can ascertain that Ωt,t0,V0−U0≡0~ solves the transformed SDE. Hence, it indicates that Q0~t=0~ for t≥t0≥0.SinceDHΩ=0~ and Ωt,t0,0~=0~, the solution Vt,t0,U0 of (32) corresponds to the identically null solution of DHΩ=Q~Ωt where
(40)Q~Ωt,t0,V0−U0t=QΩt,t0,V0−U0+Vt,t0,U0t−QVt,t0,U0t.Hence, without loss of generality, it is enough to consider the stability of the null solutionVt,t0,U0≡0~ of (32). Unfortunately, such procedure is not feasible for ITD stability analysis.


### 3.2. ITD Stability of SDE with Causal Operator

AssumeUt,τ0,V0 solves (11) and let U~t,τ0,U0=Ut−η,t0,U0 where Ut,t0,U0 solves (10) for t≥τ0≥0. Let us go through a similar procedure as in (39) by setting
(41)Ωt,τ0,V0−U0=Ut,τ0,V0−Ut−η,t0,U0,fort≥τ0.Then,(42)DHΩt,τ0,V0−U0=DHUt,τ0,V0−DHUt−η,t0,U0,DHΩt,τ0,V0−U0=Q~Ωt,τ0,V0−U0η;t.Considering that even whenV0=U0,Ωt,τ0,0~ is neither zero nor it solves the transformed differential system; also, Ut−η,t0,U0 is not the exactly null solution of DHΩ=Q~Ωη;t. 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.


## 3.1. Classical Notion of Stability of SDE with Causal Operator

Consider the following SDE with causal operator(32)DHV=QVt,Vτ0=V0,forV0∈Kcℝnandt≥τ0,τ0∈ℝ+,where Q:E⟶E and Sρ=V∈Kcℝn:DV,0~<ρ<∞.LetVt,t0,V0 be any solution of (32) and assume that Q0~t=0~,t∈ℝ+ so that the null solution V≡0 solves (32) with the initial conditions t0,0.Definition 4.
The null solutionV=0 of (32) is called
(S1) A stable solution if and only if given anyε>0, τ0∈ℝ+, we can designate two positive functions δ1=δ1ε,τ0 and δ2=δ2ε,τ0, which are continuous in τ0 for each ε, and which give us
(33)DVt,τ0,V0,0~<ε,for t≥τ0, given that DV0,0~<δ1 and dsτ0,0<0.
(S2) A uniformly stable solution if additionallyδ1 and δ2 are independent of τ0
(S3) A quasiequiasymptotically stable solution if and only if given any positiveε,t0 we can designate δ0=δ0t0 and T=Tt0,ε so
(34)DVt,τ0,V0,0~<ε,for t≥t0+T provided that DV0,0~<δ0.
(S4) A quasiuniformly asymptotically stable solution if additionally each ofδ0 and T in (S3) is independent of t0
(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 valuesε,α,t0, we can designate T=Tt0,ε,α>0 which gives us
(35)DVt,τ0,V0,θ<ε,for t≥t0+T, provided that DV0,0~≤α.
(S8) A quasiuniformly asymptotically stable solution if and only if (S7) is verified withT being independent of t0 additionally
(S9) A completely stable solution if and only if (S1) holds and (S7) is verified given anyα,0≤α<∞
(S10) A uniformly completely stable solution if and only if (S2) holds and (S8) is verified for allα,0≤α<∞
(S11) an unstable solution if and only if (S1) fails to hold.Definition 5.
The solutionVt,τ0,V0 of (32) with the initial conditions τ0,V0 is called a stable solution w.r.t. the solution Vt,τ0,U0 of (32) for t≥t0≥0 if and only if for each ε>0, we can designate δ1=δ1ε,τ0 and δ2=δ2ε,τ0 which are positive and continuous in τ0 for each ε so
(36)DVt,τ0,V0−Vt,τ0,U0,0~<ε,provided that DV0−U0,0~<δ1 and dsτ0,0<δ2 for t≥τ0.Additionally, if each ofδ1 and δ2 is independent of τ0∈ℝ+, then the solution Vt,τ0,V0 of (32) is called a uniformly stable solution w.r.t. the solution Vt,τ0,U0.Remark 6.
LetVt,t0,V0 be a given solution of (32). We can use the following change of variable to study the stability of this solution.Fort≥t0, set
(37)Ωt,t0,V0−U0=Vt,t0,V0−Vt,t0,U0.Then,(38)DHΩt,t0,V0−U0=DHVt,t0,V0−DHVt,t0,U0,(39)DHΩt,t0,V0−U0=QΩt,t0,V0−U0t+QVt,t0,U0t−QVt,t0,U0t=QΩt,t0,V0−U0+Vt,t0,U0t−QVt,t0,U0t=Q~Ωt,t0,V0−U0t.IfV0−U0=0~, we can ascertain that Ωt,t0,V0−U0≡0~ solves the transformed SDE. Hence, it indicates that Q0~t=0~ for t≥t0≥0.SinceDHΩ=0~ and Ωt,t0,0~=0~, the solution Vt,t0,U0 of (32) corresponds to the identically null solution of DHΩ=Q~Ωt where
(40)Q~Ωt,t0,V0−U0t=QΩt,t0,V0−U0+Vt,t0,U0t−QVt,t0,U0t.Hence, without loss of generality, it is enough to consider the stability of the null solutionVt,t0,U0≡0~ of (32). Unfortunately, such procedure is not feasible for ITD stability analysis.


## 3.2. ITD Stability of SDE with Causal Operator

AssumeUt,τ0,V0 solves (11) and let U~t,τ0,U0=Ut−η,t0,U0 where Ut,t0,U0 solves (10) for t≥τ0≥0. Let us go through a similar procedure as in (39) by setting
(41)Ωt,τ0,V0−U0=Ut,τ0,V0−Ut−η,t0,U0,fort≥τ0.Then,(42)DHΩt,τ0,V0−U0=DHUt,τ0,V0−DHUt−η,t0,U0,DHΩt,τ0,V0−U0=Q~Ωt,τ0,V0−U0η;t.Considering that even whenV0=U0,Ωt,τ0,0~ is neither zero nor it solves the transformed differential system; also, Ut−η,t0,U0 is not the exactly null solution of DHΩ=Q~Ωη;t. 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 Vt,τ0,V0 of (12) w.r.t. U~t=Ut−η,t0,U0 where Ut,t0,U0 solves (10).Theorem 7.
Let(i)
Lt,W∈Cℝ+×Kcℝn,ℝ+ be locally Lipschitzian in W; i.e.,
(43)Lt,W1−Lt,W2≤KDW1−W2,0~.(ii)
Fort,V,t,U~∈ℝ+×Kcℝn and G∈Cℝ+×ℝ+,ℝ+, it is considered
(44)D∗+Lt,V−U~≤Gt,Lt,Vt−U~t,t≥τ0,where
(45)U∈E1=Ω∈Kcℝn:Ls,Ωs≤Lt,Ωt,t0≤s≤t,and
(46)D∗+Lt,Vt−U~t=limε⟶0+sup1εLt+ε,Vt−U~t+εPVt−Q~U~t−Lt,Vt−U~t.(iii)
rt=rt,τ0,z0 is the maximal solution for
(47)z′=Gt,z,zτ0=z0≥0,fort≥τ0,existing on τ0,∞.And letU~t=Ut−η,t0,U0,τ0=η+t0, where Ut=t,t0,U0 solves (10) and Vt=Vt,τ0,V0 solves (12) for t≥τ0, and assume Lτ0,V0−U0≤z0; then,
(48)Lt,Vt−U~t≤rt,fort≥τ0.Proof.
Let(49)mt=Lt,Vt−U~t,fort≥τ0.
Fort≥τ0, let us show the inequality mt≤rt.
As a matter of fact,(50)U~τ0=Uτ0−η,t0,U0,τ0=η+t0.
Substituting the last statement inmτ0 together with the theorem’s assumptions gives us
(51)mτ0=Lτ0,Vτ0−U~τ0=Lτ0,V0−Uτ0−η,t0,U0=Lτ0,V0−Ut0,t0,U0=Lτ0,V0−U0≤z0.
Letε>0 be sufficiently small, and let us consider
(52)z′=Gt,z+ε,zτ0=z0+ε,fort≥τ0,whose we ensure the existence of its solutions zt,ε=zt,τ0,z0,ε up to rt.
Let us conclude this proof by showing(53)mt<zt,ε,fort≥τ0.
To do so, let us suppose on the contrary that there is at∗≥τ0 where
(54)mt<zt,ε,fort∗>t≥τ0,mt∗=zt∗,ε.
Thus, it follows that(55)D+mt∗≥m′t∗=z′t∗,ε=Gt∗,zt∗,ε+ε.
Taking into consideration the proposition regardingG and since Gt,z≥0, we conclude that the solutions zt,ε are nondecreasing in t.
Asms=Ls,Vs−U~s for s≥τ0, we obtain
(56)Ls,Vs−U~s≤zt∗,ε,fort∗≥s≥τ0.
Consequently,V−U~∈E1.
For sufficiently small enoughδ>0, we have
(57)mt+δ−mt=Lt+δ,Vt+δ−U~t+δ−Lt,Vt−U~t≤Lt+δ,Vt+δ−U~t+δ−Lt+δ,Vt−U~t+δPVt−Q~U~t+Lt+δ,Vt−U~t+δPVt−Q~U~t−Lt,Vt−U~t≤KDVt+δ−Vt−U~t+δ−U~t−δPVt−Q~U~t,0~+Lt+δ,Vt−U~t+δPVt−Q~U~t−Lt,Vt−U~t.
Hence, with considering the assumptions in (i) regarding the locally Lipschitzity ofLt,W in W, it is seen
(58)mt+δ−mt≤Kε1δ−ε2δ+Lt+δ,Vt−U~t+δPVt−Q~U~t−Lt,Vt−U~t,where ε1 and ε2stand for errors.This gives us the following estimation regarding the Dini derivative ofmt(59)D+mt≤limδ⟶0+sup1δKε1δ−ε2δ+limδ⟶0+sup1δLt+δ,Vt−U~t+δPVt−QU~~t−Lt,Vt−U~t≤Gt,Lt,Vt−U~t=Gt,mt,forτ0≤t≤t∗<∞.Sincelimδ⟶0ε1δ−ε2δ/δ=0, when t=t∗, we have
(60)D+mt∗≤D∗+Lt∗,Vt∗−U~t∗≤Gt∗,Lt,Vt∗−U~t∗=Gt∗,mt∗<Gt∗,zt∗,ε+ε,which contradicts (55).Consequently, we havemt<zt,ε, which in its turn, as ε⟶0, gives us the desired estimation
(61)mt≤rt,τ0,z0,fort≥τ0.☐Corollary 8.
Assume thatL in Theorem 7 satisfies the aforementioned conditions in addition to that Gt,z≡0 and
(62)U∈E1=Ω∈Kcℝn:Ls,Ωs≤Lt,Ωt,t0≤s≤t.
Then,Ut and Vt solve (10) and (11), respectively.
Equivalently, forτ0≤t∗≤t∗∗<∞, we have
(63)Lt∗∗,Vt∗∗−U~t∗∗<Lt∗,Vt∗−U~t∗.Theorem 9.
Let(i)
Lt,W∈Cℝ+×Kcℝn,ℝ+ be locally Lipschitzian in W; i.e.,
(64)Lt,W1−Lt,W2≤KDW1−W2,0~.(ii)
Fort,V,t,U~∈ℝ+×Kcℝn and G∈Cℝ+×ℝ+,ℝ+, we have
(65)AtD∗+Lt,Vt−U~t+Lt,Vt−U~tA′t≤Gt,Lt,Vt−U~tAt,for t≥τ0 and U∈EA where
(66)EA=Ω∈Kcℝn:Ls,ΩsAs≤Lt,ΩtAt,t0≤s≤t,and
(67)D∗+Lt,Vt−U~t=limε⟶0+sup1εLt+ε,Vt−U~t+εPVt−Q~U~t−Lt,Vt−U~t.(iii)
rt=rt,τ0,z0 is the maximal solution of
(68)z′=Gt,z,zτ0=z0≥0,fort≥τ0,existing on τ0,∞.
And letU~t=Ut−η,t0,U0,τ0=η+t0, where Ut=Ut,t0,U0 solves (10) and Vt=Vt,τ0,V0 solves (12) for t≥τ0, and assume
(69)Aτ0Lτ0,V0−U0≤z0.
Then,(70)Lt,Vt−U~tAt≤rt,fort≥τ0.Proof.
Let(71)Mt,Vt−U~t=Lt,Vt−U~tAt,
fort≥τ0, and assume Ut∈EA. Then, in view of the theorem’s assumptions, we obtain
(72)D+Mt,Vt−U~t≤AtD∗+Lt,Vt−U~t+Lt,Vt−U~tA′t≤Gt,Lt,Vt−U~tAt=Gt,Mt,Vt−U~t.
The utilization of Theorem7, in its turns, gives us the desired estimation
(73)Lt,Vt−U~tAt=Mt,Vt−U~t≤rt,for t≥τ0.☐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 solutionUt=Ut,t0,U0 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 solutionUt=Ut,t0,U0 of (10), using Lyapunov functions, and assuming the existence and uniqueness of this solution for t≥τ0.Theorem 10.
Let(i)
Lt,W∈Cℝ+×Sρ,ℝ+ be be locally Lipschitzian in W; i.e.,
(74)Lt,W1−Lt,W2≤KDW1−W2,0~.(ii)
Fort,V,t,U~∈ℝ+×Sρ and for t≥τ0, and U∈E1(75)D∗+Lt,V−U~≤0,where
(76)D∗+Lt,Vt−U~t=limε⟶0+sup1εLt+ε,Vt−U~t+εPVt−Q~U~t−Lt,Vt−U~t.(iii)
Lt,W is positive definite and decrescent on ℝ+×Sρ; i.e., there exists two functions a,b from the class K such that
(77)aDW,0~≤Lt,W≤bDW,0~,t,W∈ℝ+×Sρ.
Then, the solutionVt,τ0,V0 of (12) is ITD stable w.r.t. the solution U~t,τ0,U0=Ut−η,t0,U0of (10), where Ut,t0,U0 solves (10), for t≥τ0.Proof.
Given a sufficiently smallε>0 and t≥τ0, let us designate δ1ε,τ0>0 and δ2ε,τ0>0, with the aim that bδ<aε holds, where
(78)δε,τ0=maxδ1ε,τ0,δ2ε,τ0>0.
Then, with thisδ, we shall prove the stability of the solution Ut,t0,U0 of (10) for t≥t0.
If we suppose the opposite, then there would exist solutionsVt,τ0,V0 of (12) and U~t,τ0,U0 of (10), on τ0,∞, and t∗∗>t∗>τ0 so
(79)DVt∗−U~t∗,0~=δ,DVt∗∗−U~t∗∗,0~=ε,δ≤DVt−U~t,0~≤ε,fort∗≤t≤t∗∗.
By the assumption (ii) and Corollary8 we obtain the following estimation
(80)Lt∗∗,Vt∗∗−U~t∗∗≤Lt∗,Vt∗−U~t∗,fort∗≤t≤t∗∗<∞.
Hence, employing the previous assumptions, especially (79), (80), and (iii), together with the choice of δ, gives us that
(81)aε=aDVt∗∗−U~t∗∗,0~≤Lt∗∗,Vt∗∗−U~t∗∗≤Lt∗,Vt∗−U~t∗≤bDVt∗−U~t∗,0~=bδ<aε,which is a contradiction.
Therefore, we conclude the stability of the solutionUt,t0,U0 of (10) for t≥t0. Hence, the solution Vt,τ0,V0 of (12) is ITD stable w.r.t. the solution U~t,τ0,U0=Ut−η,t0,U0of (10), assuming Ut,t0,U0 solves (10), for t≥τ0.
If additionally bothδ1=δ1ε,τ0 and δ2=δ2ε,τ0 are independent of τ0, then the solution Ut,t0,U0 of (10) is uniformly stable for t≥t0.☐Theorem 11.
Let(i)
Lt,W∈Cℝ+,Sρ,ℝ+ be locally Lipschitzian in W; i.e.,
(82)Lt,W1−Lt,W2≤KDW1−W2,0~.(ii)
Fort,V,t,U~∈ℝ+×Sρ and for t≥τ0, and U∈EA where
(83)EA=Ω∈Kcℝn:Ls,ΩsAs≤Lt,ΩtAt,t0≤s≤t,the following inequality holds
(84)D∗+Lt,Vt−U~tAt+Lt,Vt−U~tA′t≤0,where
(85)D∗+Lt,Vt−U~t=limε⟶0+sup1εLt+ε,Vt−U~t+εPVt−Q~U~t−Lt,Vt−U~t.Additionally,Aτ0=1,A′t is continuous for t≥τ0,At≥1, and limt⟶∞At=∞, and(iii)
Lt,W is positive definite and decrescent on ℝ+×Sρ; i.e., there exists two functions a,b from the class K such that
(86)aDW,0~≤Lt,W≤bDW,0~,t,W∈ℝ+×Sρ.
Then, the solutionVt,τ0,V0 of (12) is ITD asymptotically stable w.r.t. the solution U~t,τ0,U0=Ut−η,t0,U0 of (10) where Ut,t0,U0 solves (10), for t≥τ0.Proof.
Theorem9 yields
(87)Lt,Vt−U~tAt≤Lτ0,V0−U0fort≥τ0.
Furthermore, by Theorem10, we have the stability of the solution Ut,t0,U0 of (10) for t≥t0. Therefore, it is sufficient to prove its quasiasymptotic stability property. To do this, let ε=ρ so that
(88)δ0=δρ,τ0=maxδ1ρ,τ0,δ2ρ,τ0>0.
ChooseDV0−U0,0~<δ0 and dsτ0−t0,0<δ2. Then, by using (ii), (iii), and (87), and since limt⟶∞At=∞, i.e., limt⟶∞A−1t=0, it follows that given an ε>0 and τ0≥0, a positive T=Tε,τ0 can be designated in order to satisfy
(89)aDVt−U~t,0~≤Lt,Vt−U~t≤Lτ0,V0−U0A−1t≤bDV0−U0,0~A−1t<bδ0A−1t<ε,given that DV0−U0,0~<δ0for t≥τ0+T.
Therefore, the solutionUt,t0,U0 of (10) is a quasiasymptotically stable solution for t≥t0. Hence, the solution Vt,τ0,V0 of (12) is ITD asymptotically stable w.r.t. the solution U~t,τ0,U0=Ut−η,t0,U0 of (10), or the solution Ut,t0,U0 of (10) is an asymptotically stable solution for t≥t0.
If additionallyδ1ε,τ0,δ2ε,τ0 and Tε,τ0 are all independent of τ0, then the solution Ut,t0,U0 of (10) is uniformly asymptotically stable for t≥t0.☐


### 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)
Lt,W∈Cℝ+×E,ℝ+ and satisfies the following estimation
(90)D+Lt,Vt−U~t≤−cDVt−U~t,0~,c∈K.(ii)
Lt,W is positive definite and decrescent on ℝ+×E; i.e., there exists two functions a,b from the class K such that
(91)aDW,0~≤Lt,W≤bDW,0~,t,W∈ℝ+×E.
Then, the solutionVt,τ0,V0 of (12) is ITD uniformly asymptotically stable w.r.t. the solution U~t,τ0,U0=Ut−η,t0,U0 of (10) where Ut,t0,U0 solves (10), for t≥τ0.Proof.
Given a sufficiently smallε>0 and t≥τ0, let us designate δ1ε,τ0=δ1ε>0 and δ2ε,τ0=δ2ε>0 so
(92)bδ<aε,where δε=maxδ1ε,δ2ε>0.
Then, with thisδ and ε, we shall prove the stability of the solution Ut,t0,U0 of (10) for t≥t0.
If we suppose the opposite, then there would exist solutionsVt,τ0,V0 of (12) and U~t,τ0,U0 of (10) for t≥τ0, and t1>τ0 so
(93)DVt1,τ0,V0−U~t1,τ0,U0,0~=ε,DVt,τ0,V0−U~t,τ0,U0,0~≤ε,τ0≤t≤t1.
Then, the assumption (ii) and Corollary8 give us the following estimate
(94)Lt,Vt,τ0,V0−U~t,τ0,U0≤Lτ0,V0−U0,for τ0≤t≤t1<∞.
Hence, in view of (92), (93), (94), and (ii), in addition to the assumptions on δ, we have the following statement
(95)aε=aDVt1,τ0,V0−U~t1,τ0,U0,0~≤Lt1,Vt1,τ0,V0−U~t1,τ0,U0≤Lτ0,V0−U0≤bDV0−U0,0~=bδ<aε,which is a contradiction. This yields that the solution Ut,t0,U0 of the unperturbed differential system (10) is uniformly stable for t≥t0.To complete the proof, let us prove the uniformly asymptotic stability. Assumeε=ρ and let δ0=δρ such that DV0−U0,0~<δ0,dsτ0−t0,0<δ0 gives us
(96)DVt,τ0,V0−Ut−η,t0,U0,0~<ρfort≥τ0.Taking into consideration the uniform stability, it follows the existence of sucht∗ that satisfies for τ0<t∗<τ0+T,
(97)DV0−U0,0~<δ0andDVt∗,τ0,V0−U~t∗,τ0,U0,0~<δε,with δ=δε>0 corresponds to ε>0 for uniform stability.If we assume that it is not true, let(98)δ≤DVt,τ0,V0−U~t,τ0,U0,0~,t∈τ0,τ0+T.Then, in view of (i), it leads to(99)D+Lt,Vt,τ0,V0−U~t,τ0,U0≤−cDVt,τ0,V0−U~t,τ0,U0,0~.Hence, fort∈τ0,τ0+T, we obtain
(100)Lt,Vt,τ0,V0−U~t,τ0,U0≤Lτ0,V0−U0−∫τ0tcVs,τ0,V0−U~s,τ0,U0ds.Sinceb∈K, i.e., b associates zero to zero and bλ is strictly increasing in λ, and considering DV0−U0,0~<δ0 we obtain that
(101)bDV0−U0,0~≤bδ0.By substituting eacht by τ0+T in the inequality (100) and by choosing
(102)T=1+bδ0cδ,with the above assumptions, we have
(103)0≤Lτ0+T,Vτ0+T,τ0,V0−U~τ0+T,τ0,U0≤Lτ0,V0−U0−∫τ0τ0+TcVs,τ0,V0−U~s,τ0,U0ds≤bDV0−U0,0~−∫τ0τ0+TcVs,τ0,V0−U~s,τ0,U0ds≤bδ0−cδT=bδ0−cδ1+bδ0cδ=bδ0−cδ−bδ0=−cδ<0,giving us a contradiction. Hence, there exists a t∗>τ0 such that
(104)DVt∗,τ0,V0−Ut∗−η,t0,U0,0~<δ,which in its turn, by the stability property, gives us that
(105)DVt,τ0,V0−Ut−η,t0,U0,0~<ε,for t≥τ0+T, given that
(106)DV0−U0,0~<δanddsτ0−t0,0<δ.☐


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

Let us present sufficient conditions for stability of the solutionUt=Ut,t0,U0 of (10), using Lyapunov functions, and assuming the existence and uniqueness of this solution for t≥τ0.Theorem 10.
Let(i)
Lt,W∈Cℝ+×Sρ,ℝ+ be be locally Lipschitzian in W; i.e.,
(74)Lt,W1−Lt,W2≤KDW1−W2,0~.(ii)
Fort,V,t,U~∈ℝ+×Sρ and for t≥τ0, and U∈E1(75)D∗+Lt,V−U~≤0,where
(76)D∗+Lt,Vt−U~t=limε⟶0+sup1εLt+ε,Vt−U~t+εPVt−Q~U~t−Lt,Vt−U~t.(iii)
Lt,W is positive definite and decrescent on ℝ+×Sρ; i.e., there exists two functions a,b from the class K such that
(77)aDW,0~≤Lt,W≤bDW,0~,t,W∈ℝ+×Sρ.
Then, the solutionVt,τ0,V0 of (12) is ITD stable w.r.t. the solution U~t,τ0,U0=Ut−η,t0,U0of (10), where Ut,t0,U0 solves (10), for t≥τ0.Proof.
Given a sufficiently smallε>0 and t≥τ0, let us designate δ1ε,τ0>0 and δ2ε,τ0>0, with the aim that bδ<aε holds, where
(78)δε,τ0=maxδ1ε,τ0,δ2ε,τ0>0.
Then, with thisδ, we shall prove the stability of the solution Ut,t0,U0 of (10) for t≥t0.
If we suppose the opposite, then there would exist solutionsVt,τ0,V0 of (12) and U~t,τ0,U0 of (10), on τ0,∞, and t∗∗>t∗>τ0 so
(79)DVt∗−U~t∗,0~=δ,DVt∗∗−U~t∗∗,0~=ε,δ≤DVt−U~t,0~≤ε,fort∗≤t≤t∗∗.
By the assumption (ii) and Corollary8 we obtain the following estimation
(80)Lt∗∗,Vt∗∗−U~t∗∗≤Lt∗,Vt∗−U~t∗,fort∗≤t≤t∗∗<∞.
Hence, employing the previous assumptions, especially (79), (80), and (iii), together with the choice of δ, gives us that
(81)aε=aDVt∗∗−U~t∗∗,0~≤Lt∗∗,Vt∗∗−U~t∗∗≤Lt∗,Vt∗−U~t∗≤bDVt∗−U~t∗,0~=bδ<aε,which is a contradiction.
Therefore, we conclude the stability of the solutionUt,t0,U0 of (10) for t≥t0. Hence, the solution Vt,τ0,V0 of (12) is ITD stable w.r.t. the solution U~t,τ0,U0=Ut−η,t0,U0of (10), assuming Ut,t0,U0 solves (10), for t≥τ0.
If additionally bothδ1=δ1ε,τ0 and δ2=δ2ε,τ0 are independent of τ0, then the solution Ut,t0,U0 of (10) is uniformly stable for t≥t0.☐Theorem 11.
Let(i)
Lt,W∈Cℝ+,Sρ,ℝ+ be locally Lipschitzian in W; i.e.,
(82)Lt,W1−Lt,W2≤KDW1−W2,0~.(ii)
Fort,V,t,U~∈ℝ+×Sρ and for t≥τ0, and U∈EA where
(83)EA=Ω∈Kcℝn:Ls,ΩsAs≤Lt,ΩtAt,t0≤s≤t,the following inequality holds
(84)D∗+Lt,Vt−U~tAt+Lt,Vt−U~tA′t≤0,where
(85)D∗+Lt,Vt−U~t=limε⟶0+sup1εLt+ε,Vt−U~t+εPVt−Q~U~t−Lt,Vt−U~t.Additionally,Aτ0=1,A′t is continuous for t≥τ0,At≥1, and limt⟶∞At=∞, and(iii)
Lt,W is positive definite and decrescent on ℝ+×Sρ; i.e., there exists two functions a,b from the class K such that
(86)aDW,0~≤Lt,W≤bDW,0~,t,W∈ℝ+×Sρ.
Then, the solutionVt,τ0,V0 of (12) is ITD asymptotically stable w.r.t. the solution U~t,τ0,U0=Ut−η,t0,U0 of (10) where Ut,t0,U0 solves (10), for t≥τ0.Proof.
Theorem9 yields
(87)Lt,Vt−U~tAt≤Lτ0,V0−U0fort≥τ0.
Furthermore, by Theorem10, we have the stability of the solution Ut,t0,U0 of (10) for t≥t0. Therefore, it is sufficient to prove its quasiasymptotic stability property. To do this, let ε=ρ so that
(88)δ0=δρ,τ0=maxδ1ρ,τ0,δ2ρ,τ0>0.
ChooseDV0−U0,0~<δ0 and dsτ0−t0,0<δ2. Then, by using (ii), (iii), and (87), and since limt⟶∞At=∞, i.e., limt⟶∞A−1t=0, it follows that given an ε>0 and τ0≥0, a positive T=Tε,τ0 can be designated in order to satisfy
(89)aDVt−U~t,0~≤Lt,Vt−U~t≤Lτ0,V0−U0A−1t≤bDV0−U0,0~A−1t<bδ0A−1t<ε,given that DV0−U0,0~<δ0for t≥τ0+T.
Therefore, the solutionUt,t0,U0 of (10) is a quasiasymptotically stable solution for t≥t0. Hence, the solution Vt,τ0,V0 of (12) is ITD asymptotically stable w.r.t. the solution U~t,τ0,U0=Ut−η,t0,U0 of (10), or the solution Ut,t0,U0 of (10) is an asymptotically stable solution for t≥t0.
If additionallyδ1ε,τ0,δ2ε,τ0 and Tε,τ0 are all independent of τ0, then the solution Ut,t0,U0 of (10) is uniformly asymptotically stable for t≥t0.☐


## 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)
Lt,W∈Cℝ+×E,ℝ+ and satisfies the following estimation
(90)D+Lt,Vt−U~t≤−cDVt−U~t,0~,c∈K.(ii)
Lt,W is positive definite and decrescent on ℝ+×E; i.e., there exists two functions a,b from the class K such that
(91)aDW,0~≤Lt,W≤bDW,0~,t,W∈ℝ+×E.
Then, the solutionVt,τ0,V0 of (12) is ITD uniformly asymptotically stable w.r.t. the solution U~t,τ0,U0=Ut−η,t0,U0 of (10) where Ut,t0,U0 solves (10), for t≥τ0.Proof.
Given a sufficiently smallε>0 and t≥τ0, let us designate δ1ε,τ0=δ1ε>0 and δ2ε,τ0=δ2ε>0 so
(92)bδ<aε,where δε=maxδ1ε,δ2ε>0.
Then, with thisδ and ε, we shall prove the stability of the solution Ut,t0,U0 of (10) for t≥t0.
If we suppose the opposite, then there would exist solutionsVt,τ0,V0 of (12) and U~t,τ0,U0 of (10) for t≥τ0, and t1>τ0 so
(93)DVt1,τ0,V0−U~t1,τ0,U0,0~=ε,DVt,τ0,V0−U~t,τ0,U0,0~≤ε,τ0≤t≤t1.
Then, the assumption (ii) and Corollary8 give us the following estimate
(94)Lt,Vt,τ0,V0−U~t,τ0,U0≤Lτ0,V0−U0,for τ0≤t≤t1<∞.
Hence, in view of (92), (93), (94), and (ii), in addition to the assumptions on δ, we have the following statement
(95)aε=aDVt1,τ0,V0−U~t1,τ0,U0,0~≤Lt1,Vt1,τ0,V0−U~t1,τ0,U0≤Lτ0,V0−U0≤bDV0−U0,0~=bδ<aε,which is a contradiction. This yields that the solution Ut,t0,U0 of the unperturbed differential system (10) is uniformly stable for t≥t0.To complete the proof, let us prove the uniformly asymptotic stability. Assumeε=ρ and let δ0=δρ such that DV0−U0,0~<δ0,dsτ0−t0,0<δ0 gives us
(96)DVt,τ0,V0−Ut−η,t0,U0,0~<ρfort≥τ0.Taking into consideration the uniform stability, it follows the existence of sucht∗ that satisfies for τ0<t∗<τ0+T,
(97)DV0−U0,0~<δ0andDVt∗,τ0,V0−U~t∗,τ0,U0,0~<δε,with δ=δε>0 corresponds to ε>0 for uniform stability.If we assume that it is not true, let(98)δ≤DVt,τ0,V0−U~t,τ0,U0,0~,t∈τ0,τ0+T.Then, in view of (i), it leads to(99)D+Lt,Vt,τ0,V0−U~t,τ0,U0≤−cDVt,τ0,V0−U~t,τ0,U0,0~.Hence, fort∈τ0,τ0+T, we obtain
(100)Lt,Vt,τ0,V0−U~t,τ0,U0≤Lτ0,V0−U0−∫τ0tcVs,τ0,V0−U~s,τ0,U0ds.Sinceb∈K, i.e., b associates zero to zero and bλ is strictly increasing in λ, and considering DV0−U0,0~<δ0 we obtain that
(101)bDV0−U0,0~≤bδ0.By substituting eacht by τ0+T in the inequality (100) and by choosing
(102)T=1+bδ0cδ,with the above assumptions, we have
(103)0≤Lτ0+T,Vτ0+T,τ0,V0−U~τ0+T,τ0,U0≤Lτ0,V0−U0−∫τ0τ0+TcVs,τ0,V0−U~s,τ0,U0ds≤bDV0−U0,0~−∫τ0τ0+TcVs,τ0,V0−U~s,τ0,U0ds≤bδ0−cδT=bδ0−cδ1+bδ0cδ=bδ0−cδ−bδ0=−cδ<0,giving us a contradiction. Hence, there exists a t∗>τ0 such that
(104)DVt∗,τ0,V0−Ut∗−η,t0,U0,0~<δ,which in its turn, by the stability property, gives us that
(105)DVt,τ0,V0−Ut−η,t0,U0,0~<ε,for t≥τ0+T, given that
(106)DV0−U0,0~<δanddsτ0−t0,0<δ.☐


## 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.



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*Source: 9794959-2021-08-02.xml*