NON-INSTANTANEOUS IMPULSIVE FRACTIONAL DIFFERENTIAL EQUATIONS WITH STATE DEPENDENT DELAY AND PRACTICAL STABILITY?

Ravi AGARWAL

Department of Mathematics,Texas A&M University-Kingsville,Kingsville,TX 78363,USA Distinguished University Professor of Mathematics,Florida Institute of Technology,Melbourne,FL 32901,USA

E-mail:Ravi.Agarwal@tamuk.edu

Ricardo ALMEIDA

Center for Research and Development in Mathematics and Applications,Department of Mathematics,University of Aveiro,Portugal

E-mail:ricardo.almeida@ua.pt

Snezhana HRISTOVA

Department of Applied Mathematics and Modeling,University of Plovdiv Paisii Hilendarski,Plovdiv,Bulgaria

E-mail:snehri@gmail.com

Donal O’REGAN

School of Mathematics,Statistics and Applied Mathematics,National University of Ireland,Galway,Ireland

E-mail:donal.oregan@nuigalway.ie

Abstract Nonlinear delay Caputo fractional differential equations with non-instantaneous impulses are studied and we consider the general case of delay,depending on both the time and the state variable.The case when the lower limit of the Caputo fractional derivative is fixed at the initial time,and the case when the lower limit of the fractional derivative is changed at the end of each interval of action of the impulse are studied.Practical stability properties,based on the modi fied Razumikhin method are investigated.Several examples are given in this paper to illustrate the results.

Key words non-instantaneous impulses;Caputo fractional differential equations;practical stability

1 Introduction

Many evolution processes are characterized by abrupt state changes and these are modeled by impulsive differential equations.In the literature there are two popular types of impulses:instantaneous impulses(whose duration is short compared to the overall duration of the whole process)and non-instantaneous impulses(here the action starts at some points and remain active on a finite time interval).Additionally,when fractional derivatives with their memory property are involved in the equations,the impulses cause some problems connected with the lower limit of the fractional derivative.There are mainly two types of fractional differential equations with impulses in the literature:ones with fixed lower limit at the initial time and the second with a changeable lower limit at each each time of impulse.Caputo fractional differential equations with changeable lower limit at the impulsive time are studied in[16]and the explicit formulas for the solutions are given.Also,the non-instantaneous impulsive differential equations are natural generalizations of impulsive differential equations(see,for example,[3,6,7,12]).An overview of the main properties of the presence of non-instantaneous impulses to differential equations with ordinary derivatives as well as Caputo fractional derivatives is given in the book[2].

An important qualitative problem for differential equations is stability.Often Lyapunov functions and different modi fications of the Lyapunov direct method are applied to study stability properties of solutions([4,5]).The application of Lyapunov functions to fractional differential equations requires appropriate de finition of their derivatives among the solutions of the studied fractional equations.Note there many different types of stability de fined and used to various kinds of differential equations.In[14,15]the stability properties of fractional difference equations are investigated.One type of stability useful in real world problems is the so called practical stability problem,introduced by LaSalle and Lefschetz[11],and it considers the question of whether the system state evolves within certain subsets of the state-space.For example,an equilibrium point may not be stable in the sense of Lyapunov and yet the system response may be acceptable in the vicinity of this equilibrium.

In this paper we study nonlinear delay Caputo fractional differential equations with noninstantaneous impulses and we consider delays depending on both the time and the state variable.Some applications of state dependent delays are given in[8,13].We study two cases,one when the lower limit of the Caputo fractional derivative is fixed at the initial time,and the other when the lower limit of the fractional derivative is changed at the end of each interval of action of the impulse.Practical stability of the solutions is investigated and our arguments are based on the application of Lyapunov like functions and the modi fied Razumikhin method.We will need appropriate de finitions of the derivative of Lyapunov functions among the studied fractional equations.In our paper we use three different types of fractional derivatives of Lyapunov functions.Comparison results for nonlinear non-instantaneous impulsive fractional differential equations without any delay are used.Some sufficient conditions for practical stability and quasi practical stability are obtained.Also several examples are given to illustrate our results.

The main contributions in this paper could be summarized:

-non-linear Caputo fractional differential equations with non-instantaneous impulses and general delay depending on both the time and the state variable are set up in both cases:the case of a fractional derivative with fixed lower limit at the initial time and the case of a fractional derivative with changed lower limit at the end of each interval of acting of the impulse;

-practical and quasi practical stability for studied equations are de fined;

-both the initial conditions and the impulsive conditions are set up in appropriate way;

-three types of the derivatives of the applied Lyapunov functions among the solutions of the studied equation are used:Caputo fractional derivative;Dini fractional derivative and Caputo fractional Dini derivative;

-fractional modi fication of Razumikhin method is suggested and applied in the case of any of the mentioned above three types of derivatives of Lyapunov functions;

-several sufficient conditions for practical stability are obtained for both types of the fractional derivatives-fi xed lower limit as well as changeable one.

2 Preliminaries

The intervals(t,s),i=0,1,2,···,k,will be the domain of the fractional differential equations,while in the intervals(s,t),i=1,2,···,k,the impulsive conditions are given.

In the paper we will use the Caputo fractional derivative of order q∈(0,1)for a function m∈C([t,t+T],R),and is given by

and Riemann-Liouville fractional derivative of order q∈(0,1)

and Gr¨unwald?Letnikov fractional derivative

The point tis the lower limit of the fractional derivative.Note that,for vector valued functions,the Caputo fractional derivative is taken component-wise.

Consider the space PCof all piecewise continuous functions φ:[?r,0]→Rwith finite number of points of discontinuity τ∈(?r,0)at which

endowed with the norm

where‖.‖is a norm in R.

In the case of the presence of any kind of impulses to the fractional differential equations,the memory of the fractional derivative leads to considering two different types of equations:

-fractional derivative with fixed lower limit at the initial time(NIFrDDE):

where the function f:R×R×PC→R;

-fractional derivative with changed lower limit at the end of each interval of acting of the impulse(NIFrDDE):

Remark 2.1

The functions Φare called impulsive functions and the intervals(s,t],k=1,2,···are called intervals of non-instantaneous impulses.

We introduce the following assumptions:

Remark 2.2

Any of the assumptions A2.1 or A2.2 guarantee the delay in the argument of the unknown function in(2.2),i.e.,the function ρ determines the state-dependent delay.

Example 2.3

The function ρ(t,u)=t?cos(u)satis fies the assumptions A2.1 and A2.2 with r=1,i.e.,t?1≤t?cos(u)≤t.

If any of the assumptions A2.1 or A2.1 is satis fied,then ψ∈PCand ρ(t,ψ)?t∈[?r,0].

Remark 2.5

Note that,for the NIFrDDE(2.1),the functions f and ρ have to be de fined for all t≥0,in spite of the fact they only appear in the fractional differential equation(see assumptions A1.1 and A2.1.With respect to the NIFrDDE(2.2),both functions are de fined only on the intervals of fractional differential equations(see assumptions A1.2 and A2.2).

Let J?Rbe a given interval.We will use the following classes of functions

3 De finition of Practical Stability and Lyapunov Functions

We will consider the cases of fractional differential equations with non-instantaneous impulses(2.1)and(2.2).Following the classical concept of the idea of practical stability(see[10]),we will give a de finition for various types of practical stability of the zero solution of NIFrDDE(2.1)(respectively(2.2)).In the de finition below,we denote by x(t;t,φ)any solution of the IVP for NIFrDDE(2.1)(respectively(2.2)).

De finition 3.1

The zero solution of the system of NIFrDDE(2.1)(respectively(2.2))is said to be

In connection with our stability study,we will use Lyapunov type functions:

De finition 3.2

([2]) Let α<β≤∞be given numbers and??Rbe a given set.We say that the function V:[α?r,β]×?→Rbelongs to the class Λ([α?r.β],?)if

In this paper we will use three main type derivatives of Lyapunov functions V∈Λ([t?r,t+T),?),0

-Dini fractional derivative given t∈(t,s]∩[t,t+T),k=0,1,2,···,

Note that,if condition(A2.2)is satis fied,then ρ(t,ψ(s))?t∈[?r,0]and ψis well de fined.

-Caputo fractional Dini derivative given t∈(t,s]∩[t,t+T),k=0,1,2,···,

where φ,ψ∈PC([?r,0],?).

Expression(3.3)is equivalent to

Remark 3.3

The relation between the Dini fractional derivative de fined by(3.2)and the Caputo fractional Dini derivative de fined by(3.4)is given by

Example 3.4

Let n=1 and V(t,x)=m(t)x,where m∈C(R,R).

Case 1

Caputo fractional derivative:

where x(t)=x(t;t,φ),t≥tis a solution of(2.1).

Case 2

Dini fractional derivative.

where ψ∈PC([?r,0],R).

Case 3

Caputo fractional Dini derivative:Let ψ∈PC([?r,0],R).Then

4 Main Results About Practical Stability

4.1 Fixed lower limit of the fractional derivative

Consider the initial value problem for the nonlinear system of non-instantaneous impulsive Caputo fractional differential equations with state dependent delay(2.1).We will study practical stability by the fractional extension of the Razumikhin method.In[5],some stability results for delay fractional differential equations(no impulses of any kind)are obtained,by applying the Caputo fractional derivative of the Lyapunov function and the generalized Razumikhin condition

Remark 4.1

Note that condition(4.1)is restrictive,but it is necessary because of the application of Caputo fractional derivative of the Lyapunov function and the fractional derivative with fixed lower limit in(2.1).

We will give sufficient conditions for practical stability of the zero solution of NFrDDE(2.1)by applying the Caputo fractional derivative of the Lyapunov function.

Theorem 4.2

(Practical stability for the Caputo fractional derivative) Let assumptions A1.1,A2.1,A3.1,and A4 be satis fied.Assume that there exist a number t∈[0,s)and a continuously differentiable Lyapunov function V∈Λ([t?r,∞),R),with V(t,0)=0,such that

Proof

Let x(t)=x(t;t,φ)be a solution of NIFrDDE(2.1),with‖φ‖<λ.De fine the function

The function v is nondecreasing.According to condition(i),the inequalities

hold for s∈[?r,0],i.e.,v(t)∈S.We will prove that

Assume that(4.2)is not true.

Case 1

There exists a natural number p such that v(t)=v(t)∈S,for t∈[t,s],but v(t)>v(t),for t∈(s,s+ε],where ε>0 is a small enough number.Then,given t∈(s,s+ε],we get v(t)>vand V(s,x(s?0))∈S.According to condition(iii)we have

According to the assumption,we get v(t)=0,for t∈[t,T],and v(t)>0,for t∈(T,T+ε].Then,for any t∈(T,T+ε],we obtain

a contradiction.This proves(4.2).

From(4.2)and condition(i),we get

4.2 Fractional equations with changed lower limit of the derivative

Consider the initial value problem for a nonlinear system of non-instantaneous impulsive Caputo fractional differential equations with state dependent delay(2.2).First we give comparison results(known in the literature)by Lyapunov functions for systems of fractional differential equations with state dependent delays(no impulses)

where 0<Θ≤∞.

Lemma 4.3

([1]Caputo fractional Dini derivative) Assume that

1.The function x(t)=x(t;a,φ)∈?,??Ris a solution of(4.4),de fined for t∈[a,a+Θ],Θ>0.

3.The function V∈Λ([a?r,a+Θ],?)and,for any point t∈[a,a+Θ]such that

the inequality

holds.

Lemma 4.4

([1]Dini fractional derivative) Assume the conditions of Lemma 4.3 are satis fied,where inequality(4.5)is replaced by

We study the practical stability using the following scalar comparison differential equation with non-instantaneous impulses(NIFrDE):

We introduce the following assumptions:

H2

For all natural numbers k,the functions Ψ∈C([s,t]×R,R)are such that Ψ(t,0)=0,for t∈[s,t],and Ψ(t,u)≤Ψ(t,v),for u≤v,t∈[s,t].

H3

There exists a positive number K such that,for any k=1,2,···,the inequality|Ψ(s,u)|We will study the connection between the practical stability properties of the system NIFrDDE(2.2)and the practical stability properties of the scalar NIFrDE(4.7).

Theorem 4.5

(for the Caputo fractional Dini derivative) Let the following conditions be satis fied:

1.Assumptions A1.2,A2,A3.2,A4 and H1–H3 are ful filled.

2.There exist a point t∈[0,s),a function V∈Λ([t?r,∞),R),and

(i)the inequality

holds,where a,b∈K,A=b(K),and K is the number de fined in condition(H3);

(ii)for any functions ψ,φ∈PCsuch that‖ψ‖∈Sand‖φ‖∈S,and for any point t∈(t,s),k=0,1,2,···,such that V(t+τ,ψ(τ))≤V(t,ψ(0)),for all τ∈[?r,0],the inequality

holds;

(iii)for any k=1,2,···,the inequality

holds.

3.There exists a constant λ,with 0<λ

Then,the zero solution of(2.2)is practically stable w.r.t.(λ,A).

Consider the solution uof(4.7)with the initial value u.According to condition 3,the inequality

for τ∈[?r,0].According to condition 2(ii)of Theorem 4.2,we get

with φ(0)=x(t),i.e.,condition 3 of Lemma 4.3 is satis fied with a=t,G(t,u)=g(t,u)and φ(0)=x(t),Θ=s?t.

We will prove that

For t=twe get

Assume(4.9)is not true and let t=inf{t>t:V(t,x(t))≥b(A)}.

Case 1

Suppose that t∈(t,s),for some non-negative integer p.Then,the function x is continuous at tand V(t,x(t))Case 1.1

Assume that p=0.From condition 2(i)and the choice of the initial function,we have

i.e.,x(t)∈S,for t∈[t?r,t].Also,

This contradiction proves the assumption is not true.

Case 1.2

Assume now that p≥1.From condition 2(i)we have

This contradiction proves the assumption is not true.

Case 2

Suppose that t∈(s,t),for some natural number p.From condition 2(i)it follows that

i.e.,x(t)∈S.Then,from condition 2(iii),we get

From condition(H3),using the inequality V(s?0,x(s?0))

Case 3

Finally,suppose that t=s,for some natural number p.

Case 3.1

Let V(t,x(t))Case 3.2

Let V(t,x(t))

From(4.9)and condition 3(i)we obtain the claim in Theorem 4.5.

Remark 4.6

Note that,in Theorem 4.5,the condition in(ii)is similar to the Razumikhin condition and it is not as restrictive as the condition used in Theorem 4.2(we note the type of the fractional derivatives used in(2.1)and(2.2)).

Theorem 4.7

(for the Dini fractional derivative) Let the conditions of Theorem 4.5 be satis fied but replace condition 2(ii)by:

2(ii)for any function ψ∈PCwith‖ψ‖∈S,and any point t∈(t,s),k=0,1,2,···,such that V(t+τ,ψ(τ))≤V(t,ψ(0)),for τ∈[?r,0],the inequality

holds.Then,the zero solution of(2.2)is practically stable w.r.t.(λ,A).

The proof of Theorem 4.7 is similar to that in Theorem 4.5 where instead of Lemma 4.3 we apply Lemma 4.4.

Theorem 4.8

Let the following conditions be satis fied:

1.Assumptions A1.2,A2,A3.2,A4 and H1–H3 are ful filled.

2.There exist a point t∈[0,s)and a function V∈Λ([t?r,∞),R)such that

(i)the inequality

holds,where a,b∈K;

(ii)for any function ψ∈PCand any point t∈(t,s),k=0,1,2,···,such that V(t+τ,ψ(τ))≤V(t,ψ(0)),for τ∈[?r,0],the inequality

holds;

(iii)for any k=1,2,···,the inequality

holds.

3.There exist positive constants λ,T with 0<λ

Then,the zero solution of(2.2)is practically quasi stable w.r.t.(λ,B,T).

Proof

Choose the initial function φ∈PCwith‖φ‖<λ,and consider the solution x(t)=x(t;t,φ)of system(2.2)for the initial time tde fined in condition 2.Let

From the choice of the initial function φ and the properties of the function b,applying condition 2(i),we get

Consider the solution u(t)=u(t;t,u)of(4.7).Therefore,the function u satis fies

for t≥t+T.

In the case of the application of the Dini fractional derivative we obtain:

Theorem 4.9

Let the conditions of Theorem 4.8 be satis fied with replacing the condition 2(ii)by:

2(ii)for any function ψ∈PCwith‖ψ‖∈S,and any point t∈(t,s),k=0,1,2,···,such that V(t+τ,ψ(τ))≤V(t,ψ(0)),for τ∈[?r,0],the inequality

holds.

Then,the zero solution of(2.2)is practically quasi stable w.r.t.(λ,B,T).

The proof of Theorem 4.9 is similar to that in Theorem 4.8,where instead of Lemma 4.3,we apply Lemma 4.4.

Remark 4.10

Note that condition(H3)of Theorems 4.5,4.8,and 4.9,could be replaced by the condition:

For all k=1,2,···,the functions Ψsatisfy Ψ(t,u)≤u,for t∈[s,t]and u∈R.

Remark 4.11

The point tin the conditions of all the above Theorems is from the interval[0,s)but one can modify the proofs so it can be from any interval[t,s),k=1,2,···.

4.3 Some examples

We will consider several particular examples and apply our results to illustrate the practical stability properties.

Example 4.12

(constant delay) Let s=2k+1,t=2k,for k=0,1,2,···,and r=1.Consider the IVP for the nonlinear system of non-instantaneous impulsive fractional differential equations with constant delay:

where x,y∈R,a,b∈(?1,1)are given constants.In this particular case,

and the conditions A2.1 and A2.2 are satis fied.Therefore,

For any t∈(2k+1,2k+2],k=0,1,2,···,x,y∈R,we have

Consider the IVP for the scalar fractional differential equation with non-instantaneous impulses

It has a solution(see Figure 1)

Figure 1 Example 4.12.Graph of the solution of(4.13)for different initial values

then

Let,for example,q=0.5 and the initial functions φ(s)=φ(s)=0.2(1+sin(s)),s∈[?1,0],i.e.,

The solution of the system(4.11)with these particular initial functions φ(s)=φ(s)=0.2(1+sin(s)),s∈[?1,0]is shown in Figure 2.

Figure 2 Example 4.12.Graph of the solution of(4.11)

Figure 3 Example 4.12.Graph of the solution of(4.11)for large impulsive functions

Now,let change the impulsive functions in(4.11)to other ones,for example,let the impulsive conditions of the system(4.11)be changed by

Note the impulsive functions?tx and ty do not satisfy the conditions of Theorems 4.5,4.7,and 4.8,and as can be seen from Figure 3,the system(4.11)is not practically stable.Thus,condition(iii)is not only sufficient but also it is necessary to assure the practical stability for the system.

Example 4.13

(variable time delay) Let s=2k+1,t=2k for k=0,1,2,···,and r=1.Consider the initial value problem for the nonlinear system of non-instantaneous impulsive fractional differential equations with time variable delay:

where x,y∈R.

In this partial case,

and the assumptions A2.1 and A2.2 are satis fied,with r=1(see Figure 4).

Figure 4 Example 4.13.Graph of the delay in(4.16)

Therefore,

and

Let V(t,x,y)=1.5(x+2y).Similar to Example 4.12 and(4.12)and(4.13),we prove the validity of Conditions 2(ii)and 2(iii)of Theorem 4.5 and the comparison scalar equation is also(4.14).According to Theorem 4.5,the solution of the system(4.16)is practically stable.

Let for example q=0.5 and the initial functions φ(s)=φ(s)=0.2(1+sin(s)),s∈[?1,0],i.e.,

The solution of the system(4.16)for these particular initial functions is shown in Figure 5.

Figure 5 Example 4.13.Graph of the solution of(4.16)with time variable delay

Figure 6 Example 4.13.Graph of the solution of(4.16)with large impulsive functions

Similar to Example 4.12,we change the impulsive functions in(4.16)by othr ones.For example,let the impulsive conditions of the system(4.16)be changed by

The impulsive functions?tx and ty do not satisfy the conditions of Theorems 4.5,4.7,and 4.8,and as can be seen from Figure 6 the system(4.16)is not practically stable.Therefore,condition(iii)for the impulsive functions is not only sufficient but also it is necessary to assure the practical stability for the system.

Example 4.14

(state dependent delay) Let s=2k+1,t=2k for k=0,1,2,···,and r=1.Consider the initial value problem for the nonlinear system of non-instantaneous impulsive fractional differential equations with time variable delay

where x,y∈R,r>0 is a small constant,a,b∈(?1,1)are given constants,

Then,the assumptions A2.1 and A2.2 are satis fied.Therefore,

Similar as in Example 4.12,applying inequality(4.19),we get inequalities(4.12),(4.13)and the comparison scalar equation(4.14).According to Theorem 4.5,the solution of the system(4.18)is practically stable.

Acta Mathematica Scientia(English Series)2021年5期

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