EXISTENCE，UNIQUENESS AND STABILITY OF RANDOM IMPULSIVE FRACTIONAL DIFFERENTIAL E

A.Vinodkumar

Department of Mathematics，Amrita School of Engineerring，

Amrita Vishwa Vidyapeetham University，Coimbatore-641 112 Tamil Nadu，INDIA

E-mail∶vinod026@gmail.com；a vinodkumar@cb.amrita.edu

K.Malar Department of Mathematics，Erode Arts and Science College，Erode-638 009，Tamil Nadu，INDIA

E-mail∶malarganesaneac@gmail.com

M.Gowrisankar

Department of Mathematics，Annapoorana Engineering College，Salem-636 308，Tamil Nadu，INDIA

E-mail∶tamilangowri@gmail.com

P.Mohankumar

Department of Mathematics，A.C.T College of Engineering and Technology，Kancheepuram-603 107，Tamil Nadu，INDIA

E-mail∶profmohankumar@yahoo.co.in



EXISTENCE，UNIQUENESS AND STABILITY OF RANDOM IMPULSIVE FRACTIONAL DIFFERENTIAL EQUATIONS∗

A.Vinodkumar

Department of Mathematics，Amrita School of Engineerring，

Amrita Vishwa Vidyapeetham University，Coimbatore-641 112 Tamil Nadu，INDIA

E-mail∶vinod026@gmail.com；a vinodkumar@cb.amrita.edu

K.Malar Department of Mathematics，Erode Arts and Science College，Erode-638 009，Tamil Nadu，INDIA

E-mail∶malarganesaneac@gmail.com

M.Gowrisankar

Department of Mathematics，Annapoorana Engineering College，Salem-636 308，Tamil Nadu，INDIA

E-mail∶tamilangowri@gmail.com

P.Mohankumar

Department of Mathematics，A.C.T College of Engineering and Technology，Kancheepuram-603 107，Tamil Nadu，INDIA

E-mail∶profmohankumar@yahoo.co.in

In this article，we study the existence，uniqueness，stability through continuous dependence on initial conditions and Hyers-Ulam-Rassias stability results for random impulsive fractional differential systems by relaxing the linear growth conditions.Finally，we give examples to illustrate its applications.

Fractional differential systems；random impulses；stability；Hyers-Ulam stability；Hyers-Ulam-Rassias stability

2010 MR Subject Classification34A08；34A37；34K20

1　Introduction

Impulsive differential equations are a suitable mathematical model to simulate the evolution of large classes of real processes.These processes are subjected to short temporary perturbations.The duration of these perturbations is negligible compared to the duration of the whole process.These perturbations occur in the form of impulses（see［1，2］）.When the impulses exists at random，it affect the nature of the differential system.There are a few results discussed. Iwankievicz and Nielsen［3］investigated the dynamic response of non-linear systems to Poissondistributed random impulses.A.Anguraj and A.Vinodkumar studied the existence，uniqueness and stability of random impulsive semilinear differential systems［4］.Sanz-Serna and Stuart［5］first brought dissipative differential equations with random impulses and used Markov chains to simulate such systems.For further reading on random impulsive type differential equations，refer［6-12］and the references therein.

Recently，fractional differential equations（FDEs）and impulsive fractional differential equations（IFDEs）was proved to be a valuable tool in the modelling of many phenomena in various fields of science and engineering.Similarly，the stabilities like continuous dependence，Hyers-Ulam stability，Hyers-Ulam-Rassias stability，local stability，and Mittag-Leffler stability for FDEs and IFDEs attracted the attention of many mathematicians（see［13-19］and the references therein）.In［20］，the authors given the Ulam's type stability and data dependence for FDEs.JinRong Wang et al［21］studied stability of FDEs using fixed point theorem in a generalized complete metric space.In［22］，JinRong Wang et al studied Ulam's stability for the nonlinear IFDEs.Michal Feˇckan et al proved the concept and existence of solution for IFDEs［23］.For more details on FDE and its stability concepts，please see［24-34］.

Motivated by the above mentioned works，the main purpose of this article is to study random impulsive fractional differential systems.We relaxed the Lipschitz condition on the impulsive term and under our assumption it is enough to be bounded.To best of our knowledge，there is no paper that study the random impulsive type fractional differential equations. Furthermore，for Hyers-Ulam-Rassias stability，we utilize the technique from［20，22，24］and for relaxing linear growth conditions，we refer［35，36］.

This article is organized as follows：In Section 2，we recall briefly the notations，definitions，lemmas and preliminaries，which are used throughout this article.In Section 3，we investigate the existence and uniqueness of solutions of random impulsive fractional differential systems by relaxing the linear growth condition.In Section 4，we study the stability through continuous dependence on initial conditions of random impulsive fractional differential systems.The Hyers Ulam stability and Hyers Ulam-Rassias stability of the solutions of random impulsive fractional differential systems is investigated in Section 5 and finally in Section 6，examples are given to illustrate our theoretical results.

2　Preliminaries

Let‖·‖denote Euclidean norm in ℜn.Let ℜnbe n-dimensional Euclidean space and Ωa nonempty set.Assume that τkis a random variable defined from Ω tofor all k=1，2，···，where 0＜dk＜+∞.Furthermore，assume that τkfollow Erlang distribution，where k=1，2，···and let τiand τjbe independent with each other as i 6=j for i，j=1，2，···. Let t0，T∈ℜ be two constants satisfying t0＜T.For the sake of simplicity，we denoteℜτ=［t0，T］.

Consider the fractional differential system with the random impulses of the following form

Let us denote｛Bt，t≥0｝as the simple counting process generated by｛ξn｝，that is，｛Bt≥n｝=｛ξn≤t｝，and denote Ftthe σ-algebra generated by｛Bt，t≥0｝.Then，（Ω，P，｛Ft｝）is a probability space.E（·）is the expectation with respect to the measure P.Let B be the Banach space of all square integrable and Ftadapted process ψ（t）mapping［t0，T］×ℜnintoℜnequipped with the norm，

Definition 2.1The fractional order integral of the function h∈L1（［a，b］，ℜn）of order α∈ℜ+is defined by

Definition 2.2The αthRiemann-Liouville derivative of a function h on the given interval［a，b］is defined by

here n=［α］+1 and［α］denotes the integer part of α.

Definition 2.3The Caputo derivative of order α for a function h on the given interval［a，b］is defined by

where n=［α］+1 and［α］denotes the integer part of α.

Definition 2.4For a given T∈（t0，+∞），a stochastic process｛x（t）∈B，t0≤t≤T｝is called a solution of equation（2.1）-（2.3）in（Ω，P，｛Ft｝），if

（i）x（t）∈B is Ft-adapted.

（ii）

where is the index function，that is，

3　Existence and Uniqueness

In this section，we discuss the existence and uniqueness of the solution of the system（2.1）-（2.3）.The following proofs are similar to the proofs in［35，36］.Furthermore，we need the following hypothesis：

（H1）The conditionis uniformly bounded if there is a constant C＞0 such that E

Lemma 3.1Assume that（H1）holds and for any b∈［t0，T），

If f satisfy the global Lipschitz condition in［t0，b］×ℜn，that is，there is a constant L＞0 such that for all ϑ，ˆϑ∈ℜnand t∈［t0，b］，

then there exists a unique continuous solution x（t）for equation（2.1）-（2.3）.

ProofAs the solution of（2.1）-（2.3）satisfies the integral equation（2.4）.We define the Picard sequence as

and

From（3.2），

Next，we will prove that，for n=1，2，···，

where

From（3.1）and（3.5），it is obvious that（3.6）holds for n=0，that is，f（s，x0（s））∈B（［t0，b］，ℜn）.Here，x1（t）defined by x0（t）and（3.4）is Ftmeasurable and continuous.

Now，applying Schwarz inequality to（3.4）with n=1，we have

For any t∈［t0，b］，

It leads to that，for any t∈［t0，b］，

So，（3.6）-（3.8）hold for n=1.Now，suppose that（3.6）-（3.8）hold for n=k.Then，for n=k+1，from（3.1），（3.5），and（3.8）with n=k，one obtains f（s，xk（s））∈B（［t0，b］，ℜn）. xk+1（t）defined by xk（t）and（3.4）is Ftmeasurable and continuous.

Furthermore，from（3.10）we have

That is，（3.6）-（3.8）hold for n=k+1.Hence，by induction，（3.6）-（3.8）hold for all n≥1. Now，for any m＞n≥0，we have from（3.11），

So，｛xn（t）｝is a Cauchy sequence，and｛xn（t）｝uniformly converges x（t）in the sense of L2.

On the other hand，（3.8）implies that｛xn（t）｝belongs to B（［t0，b］，ℜn）.Letting n→+∞and using the Fatou's lemma，we obtain

This implies that x（t）∈B（［t0，b］，ℜn）.Next，we will prove the uniqueness of x（t）.

Let x（t）and y（t）be two solutions with the initial values x0，that is，x（t0）=x0and y（t0）=y0.Then，

Hence，we proved that x（t）is a unique continuous solution for equation（2.1）-（2.3）.

Theorem 3.2Suppose that（H1）and Condition（3.1）hold.Furthermore，

for all t∈［t0，T］and φ，ψ∈B with‖φ‖∨‖ψ‖≤n and for n＞0，Kn＞0.Then，there must be a stopping time β=β（ω）∈（t0，T］such that system（2.1）-（2.3）has a unique maximal local solution x（t）for t∈［t0，β）（see［37］）.Furthermore，the global solution x（t）exists if Ex（t）＜∞for all t＜t0.

ProofApplying the standing truncation technique［35，36］and condition（3.13），we can define functions fnsuch that for‖x（t）‖≤n，

and fnsatisfy（3.2）for the given n.Then，by Lemma 3.1，the following equation has a unique

By the Gronwall inequality，we obtain

continuous solution，

Define a sequence of stopping time δnby

where we set inf⊘=∞as usual.Thus，we can get

and δnis nondecreasing in n.So，we can define δ=

The stopping time δ depends on b，that is，δ=δ（b，ω）.Taking β（ω）=，the maximal local solution is obtained.The existence of the global solution can be given by imitating Corollary 3.1 and Remark 3.4 in［35］.The proof is completed.

4　Stability

In this section，we study the stability of system（2.1）-（2.3）through the continuous dependence of solutions on initial condition.

Definition 4.1A solution x（t）of system（2.1）-（2.2）with initial value φ satisfing（2.3）is said to be stable in the mean square if for any ǫ＞0，there exists δ＞0 such that

whereˆx（t）is another solution of system（2.1）-（2.2）with initial valueˆφ defined in（2.3）.

Theorem 4.2Let x（t）and¯x（t）be solutions of system（2.1）-（2.3）with initial values x0and，respectively.If the assumptions of Lemma 3.1 are satisfied，then the solutions of system（2.1）-（2.3）are stable in the mean square.

ProofBy the assumptions，x and¯x are the two solutions of system（2.1）-（2.3）for t∈［t0，T］.Then，

Using the hypothesis（H1）and（3.2），we get

Therefore，

Now，given ǫ＞0，choose δ=∈ˆζsuch that E‖x0−¯x0‖2≤δ.Then，‖x−¯x‖2≤ǫ.

Thus，it is apparent that the difference between the solution x（t）and¯x（t）in the interval［t0，T］is small provided the change in initial point（t0，x0）as well as in the function f（t，x（t））do not exceed prescribed amounts.This completes the proof.

5　Ulam-Hyers Stability Results

In this section，we study the Ulam-Hyers stability of random impulsive fractional differential equation（2.1）-（2.3）.Let ǫ＞0 and φ：［t0，T］→ℜ+be a continuous function.We consider the following inequalities：

Definition 5.1System（2.1）-（2.3）is Ulam-Hyers stable in the mean square if there exists a real number κ＞0 such that for each ǫ＞0 and for each solution x∈B of inequality（5.1），there exists a solution y∈B of system（2.1）-（2.3）with

Definition 5.2System（2.1）-（2.3）is generalized Ulam-Hyers stable in the mean square if there exists a real number η∈B，η（0）=0 such that for each solution x∈B of inequality（5.1），there exists a solution y∈B of system（2.1）-（2.3）with

Definition 5.3System（2.1）-（2.3）is Ulam-Hyers-Rassias stable in the mean square with respect to（φ，µ）if there exists a real number ζ＞0 such that for each ǫ＞0 and for each solution x∈B of inequality（5.3），there exists a solution y∈B of system（2.1）-（2.3）with

Definition 5.4System（2.1）-（2.3）is generalized Ulam-Hyers-Rassias stable in the mean square with respect to（φ，µ）if there exists a real number ζ＞0 such that for each solution x∈B of inequality（5.2），there exists a solution y∈B of system（2.1）-（2.3）with

Remark 5.5It is clear that

1.Definition（5.1）⇒Definition（5.2）；

2.Definition（5.3）⇒Definition（5.4）；

3.Definition（5.3）for φ（t）=µ=1⇒Definition（5.1）.

Remark 5.6A function x∈B is a solution of inequality（5.3）if and only if there exists a function h∈B and the sequence hk，k=1，2，···（which depend on x）such that

（i）E‖h（t）‖2≤ǫφ（t），t∈［t0，T］and E‖hk‖2≤ǫµ，k=1，2，···；

（ii）cx（t）=f（t，x（t））+h（t），t 6=ξk，t≥t0；

（iii）x（ξk）=bk（τk）x）+hk，k=1，2，···.

One can have similar remarks for inequalities（5.1）and（5.2）.

Remark 5.7Letting 0＜α＜1，if x∈B is a solution of inequality（5.3），then x is a solution of the following integral inequality

From Remark 5.6，we have

Then，

Therefore，

We have similar remarks for the solutions of inequalities（5.1）and（5.2）.Now，we give the main results in this section，that is，Ulam-Hyers-Rassias results.

Theorem 5.8If the assumptions of Theorem 3.2 are satisfied and there exists λ＞0 such that

where φ：［t0，T］→ℜ+is a continuous nondecreasing function.Then，system（2.1）-（2.3）is Ulam-Hyers-Rassias stable in the mean square.

ProofLet x∈B be a solution of inequality（5.3）.By Theorem 3.2，there exists a unique solution y of the random impulsive fractional differential system

Then，we have

By differential inequality（5.3），we have

Hence，for each t∈［t0，T］，we have

Thus，system（2.1）-（2.3）is Ulam-Hyers-Rassias stable in the mean square.Hence，the proof is completed.

Remark 5.91.Under the assumption of Theorem 5.8，we consider system（2.1）-（2.3）and inequality（5.1）.One can repeat the same process to verify that system（2.1）-（2.3）is Ulam-Hyers stable in the mean square.

2.Under the assumption of Theorem 5.8，we consider system（2.1）-（2.3）and inequality（5.3）.One can repeat the same process to verify that system（2.1）-（2.3）is generalized Ulam-Hyers-Rassias stable in the mean square.

6　Examples

6.1Example

In this section，we provide with an example to illustrate our main result.Consider the following random impulsive fractional differential system，

where α∈L2（［t0，b］，ℜ）and β∈B（［t0，b］，ℜ），for any b∈［t0，∞）.

It is easy to check that f（t，x（t））=α（t）+β（t）x（t）−x3（t）satisfy all the conditions of Lemma 3.1.In contrast，it also satisfies the hypothesis（H1）.Therefore，Lemma 3.1 yields that（6.1）with initial condition x（t0）=x0has a unique continuous non-continuable solution x（t）.

From all the above facts，in view of Theorem 3.2，we conclude that problem（6.1）has a unique solution on［t0，∞）.

The next results are consequences of Theorems 4.2 and 5.8 respectively.

Proposition 6.1If the assumptions of Lemma 3.1 are satisfied，then the solution of system（6.1）is stable in the mean square.

Proposition 6.2If the assumptions of Theorem 3.2 are satisfied and there exists λ＞0 such that

where φ：［t0，T］→ℜ+is a continuous nondecreasing function.Then，the system（6.1）is Ulam-Hyers-Rassias stable in the mean square.

6.2Example

We consider an interest rate model discussed with random impulsive differential equations in［9］.We generalize that model to fractional differential equation.The time when interest rate is adjusted is a random variable.However，interest rate is a constant during two neighboring adjusted times.Thus，the interest rate x（t）can be modeled by the random impulsive differential equation of the following form，

where 0＜α≤1 and B=ℜ.Let｛ξk｝denote the times that interest rate is adjusted，which are a series of random variable following Erlang distribution.Ikis some pending function of x（ξk）.When α=1，it represent the model in［9］.

Let us consider the inequality

Let u∈B be a solution of inequality（6.3）.Then，there exists a function h∈B such that

In equation（6.4），integrating we obtain

Let us take the unique solution x（t）of（6.2）given by

Then，we have

If there exists λ＞0 such that

then equation（6.2）is generalized Ulam-Hyers-Rassias stable on［t0，T］with respect to（φ，µ）.

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