Texas A&M University Kingsville, Kingsville, TX 78363, USA 2,4 Department of Applied Mathematics and Modeling

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1 International Journal of Pure and Applied Mathematics Volume 109 No , 9-28 ISSN: printed version; ISSN: on-line version url: doi: /ijpam.v109i1.3 PAijpam.eu P-MOMENT EXPONENTIAL STABILITY OF DIFFERENTIAL EQUATIONS WITH RANDOM NONINSTANTANEOUS IMPULSES AND THE ERLANG DISTRIBUTION R. Agarwal 1, S. Hristova 2, D. O Regan 3, P. Kopanov 4 1 Department of Mathematics Texas A&M University Kingsville, Kingsville, TX 78363, USA 2,4 Department of Applied Mathematics and Modeling Plovdiv University Tzar Asen 24, 4000 Plovdiv, BULGARIA 2 snehri@gmail.com 3 School of Mathematics, Statistics and Applied Mathematics National University of Ireland Galway, IRELAND Abstract: In some real world phenomena a process may change instantaneously at uncertain moments and act non instantaneously on finite intervals. In modeling such processes it is necessarily to combine deterministic differential equations with random variables at the moments of impulses. The presence of randomness in the jump condition changes the solutions of differential equations significantly. The study combines methods of deterministic differential equations and probability theory. In this paper we study nonlinear differential equations subject to impulses occurring at random moments. Inspired by queuing theory and the distribution for the waiting time, we study the case of Erlang distributed random variables at the moments of impulses. The p-moment exponential stability of the trivial solution is defined and Lyapunov functions are applied to obtain sufficient conditions. Some examples are given to illustrate the results. AMS Subject Classification: 34A37, 34F05, 34K20, 37B25 Key Words: random noninstantaneous impulses, Erlang distribution, p-moment exponential stability Received: August 1, 2016 Published: September 1, 2016 c 2016 Academic Publications, Ltd. url:

2 10 R. Agarwal, S. Hristova, D. O Regan, P. Kopanov In Memory of Professor Drumi Bainov July 2, 1933 July 1, Introduction In some real world phenomena a process may change instantaneously at some moments. In modeling such processes one uses impulsive differential equations see, for example, the books[5], [6], [11] and the cited references therein. In the case when the process has instantaneous changes at uncertain moments which act non instantaneously on finite intervals one combines ideas in differential equations and probability theory. When there is uncertainty in the behavior of the state of the investigated process an appropriate model is usually a stochastic differential equation where one or more of the terms in the differential equation are stochastic processes, and this usually results with the solution being a stochastic process [14], [15], [16], [17]. Sometimes the impulsive action starts at an random point and remains active on a finite time interval. These type of impulses are called noninstantaneous. Recently results concerning deterministic noninstantaneous impulses were obtained for differential equations [2], [8], [13], delay integro-differential equations [9], abstract differential equations [10], and fractional differential equations [1], [12]. Differential equations with instantaneously acting impulses at random times were studied in [4], [18] but there are some inaccuracies there in the mixing properties of deterministic variables and random variables, and inaccuracies in the convergence of a sequence of real numbers to a random variable. In this paper we study nonlinear differential equations subject to impulses starting abruptly at some random points and their action continue on intervals with a given finite length. We study the case of Erlang distributed random variables defining the moments of the occurrence of the impulses. The p-moment exponential stability of the solution is studied using Lyapunov functions. 2. Random Noninstantaneous Impulses in Differential Equations Let T 0 0 be a given point and the increasing sequence of positive points {T k } and the sequence of nonnegative numbers {d i} be given such that lim k T k = T. Denote d 0 = 0. Consider the following condition: H1. The positive numbers {d k } are such that lim n n d k =.

3 P-MOMENT EXPONENTIAL STABILITY OF DIFFERENTIAL H2. The positive numbers {d k } are such that lim n n d k = B <. Consider the initial value problemivp for the system of noninstantaneous impulsive differential equations NIDE with fixed points of impulses x = ft,xt for t T k +d k,t k+1 ], k = 0,1,2,..., xt = I k t,xt k 0 for t T k,t k +d k ], k = 1,2,..., 1 xt 0 = x 0 where x,x 0 R n, f : [0, R n R n, I i : [T i,t i + d i ] R n R n, i = 1,2,3,... Denote the solution of NIDE 1 by xt;t 0,x 0,{T k }. We will assume the following conditions are satisfied H3. ft,0 = 0 and I k t,0 = 0 for t 0, k = 1,2,... H4. For any initial value T 0,x 0 the ODE x = ft,x with xt 0 = x 0 has an unique solution xt = xt;t 0,x 0 defined for t [T 0,P where P = if condition H1 is satisfied and P = T +B is condition H2 is satisfied. In Section 5 we will need the following result for the initial value problem for a scalar linear differential inequality with noninstantaneous fixed moments of impulses: u m k u for T k +d k t T k+1, k = 0,1,2,..., ut b k ut k 0, for T k < t T k +d k, k = 1,2,..., ut Proposition 1. Let m k > 0, k = 0,1,2... and b k > 0, k =,2,... be real constants. Then mt 0 for t T 0. Proof. Let t [T 0,T 1 ]. Then the function ut is continuous on [T 0,T 1 ] and ut ut 0 e m 0t T 0 0. Let t T 1,T 1 +d 1 ]. Then the function ut b 1 ut Let t [T 1 +d 1,T 2 ]. Then the function ut is continuous on [T 1 +d 1,T 2 ] and ut ut 1 +d 1 e m 1t T 1 d 1 0. Continue this process. Let the probability space Ω,F,P be given. Let {τ k } be a sequence of random variables defined on the sample space Ω. Assume τ k = with probability 1.

4 12 R. Agarwal, S. Hristova, D. O Regan, P. Kopanov Remark 1. Therandomvariables τ k will definethe timebetween two consecutive impulsive moments of the impulsive differential equation with random impulses. We will assume the following condition is satisfied H5. The random variables {τ k }, τ k Erlangα k,λ are independent with two parameters: a positive integer shape α k and a positive real rate λ. We will recall some properties of the Erlang distribution: i If X Erlangα 1,λ and Y Erlangα 2,λ are independent random variables, then X +Y Erlangα 1 +α 2,λ; ii The cumulative distribution function CDF of Erlangα, λ is α 1 Fx;α,λ = 1 e λx j=1 λx j j! = 1 α 1! λx 0 y α 1 e y dy, x 0 3 and the probability density function PDF is fx;α,λ = λ λxα 1 α 1! e λx, x > 0. Proposition 2. Let condition H5 be satisfied and the sequence of random variables {Ξ k } be such that Ξ n = n τ i, n = 1,2,... Then Ξ n Erlang n α i,λ. Define the increasing sequence of random variables {ξ k } k=0 by ξ k = T 0 + k k 1 τ i + d i, k = 0,1,2,... 4 where T 0 0 is a fixed point. Also, Ξ k = n τ i = ξ k T 0 k 1 d i, k = 1,2,... Remark 2. The random variable ξ n will be called the waiting time and it gives the arrival time of n-th impulses in the impulsive differential equation with random impulses.

5 P-MOMENT EXPONENTIAL STABILITY OF DIFFERENTIAL Let the points t k be arbitrary values of the corresponding random variables τ k, k = 1,2,... Define the increasing sequence of points T k = T 0 + k k 1 t i + d i, k = 1,2,... 5 NoteT k arevaluesoftherandomvariablesξ k. Thesetofallsolutionsxt;T 0,x 0, {T k } of NIDE 1 for any values t k of the random variables τ k, k = 1,2,... generates a specific stochastic process with state space R n. We denote it by xt;t 0,x 0,{τ k } and we will say that it is a solution of the following initial value problem for differential equations with noninstantaneous random moments of impulses RNIDE x t = ft,xt for t T 0, ξ k +d k < t < ξ k+1, k = 0,1,..., xt = I k t,xξ k, for ξ k < t < ξ k +d k, k = 1,2,..., xt 0 = x 0. 6 Definition 1. The solution xt;t 0,x 0,{T k } of the IVP for the IDE with fixed points of impulses 1 is called a sample path solution of the IVP for the RIDE 6. Definition 2. A stochastic process xt;t 0,x 0,{τ k }is said tobeasolution of theivp for the system of RIDE 6 if for any values t k of the randomvariable τ k, k = 1,2,3,... andt k = T 0 + k t i,k = 1,2,... thecorrespondingfunction xt;t 0,x 0,{T k } is a sample path solution of the IVP for RIDE Preliminary Results for Erlang Distributed Moments of Impulses For any t T 0 consider the events S 0 t = {ω Ω : t T 0 < τ 1 ω}, S k t = {ω Ω : ξ k ω+d k < t < ξ k+1 ω}, k = 1,2,... and W k t = {ω Ω : ξ k ω < t < ξ k ω+d k }, k = 1,2,... where the random variables ξ k, k = 1,2,... are defined by 4.

6 14 R. Agarwal, S. Hristova, D. O Regan, P. Kopanov holds. holds. Proposition 3. For any t T 0 the equality PS 0 t =1 1 α 1 1! α 1 1 =e λt T 0 j=1 λt T0 0 λt T 0 j j! y α 1 1 e y dy Corollary 1. Upper bound of S 0 t. For any t T 0 the inequality PS 0 t e λt T 0 λt T0 α 1 1 α 1 1! Proof. We have the following α 1 1 j=1 λt T 0 j j! = eλt T 0 Γα 1,λt T 0 α 1 1! 1. Apply the inequality Γa,x xa e x x a 1 see [7] for the upper incomplete gamma function Γa,x = x ya 1 e y dy and obtain α 1 1 j=1 λt T 0 j j! eλt T 0 Γα 1,λt T 0 α 1 1! eλt T 0 1 λt T 0 α 1 e λt T 0 α 1 1! λt T 0 α λt T 0 α 1 1. α 1 1! λt T Lemma 1. Let conditions H1,H5 be satisfied and t T 0.

7 P-MOMENT EXPONENTIAL STABILITY OF DIFFERENTIAL Then the probability that there will be exactly k impulses until time t is 0, for j > k, e λt T 0 A j+1 1 j d λt T i 0 j d i m PS j t = m=a j λt T0 j d i y A j 1 = 0 A j 1! yaj+1 1 e y dy, for j k, A j+1 1! 8 where A j = j α i and T 0 + k d i t < T 0 + k+1 d i. Proof. From the definition of S j t we get j j j+1 j PS j t = PT 0 + τ i + d i < t < T 0 + τ i + d i = PΞ j < t T 0 j d i PΞ j+1 < t T 0 j d i. 9 Let j > k. From the definition of k it follows that j k+1 t T 0 d i t T 0 d i < 0 and therefore PΞ j < t T 0 j d i = 0, PΞ j+1 < t T 0 j d i = 0 and PS j t = 0. Now, let j k. Then t T 0 j d i t T 0 k d i 0. From Proposition 2, equality 9 and formula 3 we obtain PS j t = j+1 α i 1 m=1 j α i 1 m=1 λt T 0 j d i m λt T 0 j d i m e λt T 0 j d i e λt T 0 j d i. 10

8 16 R. Agarwal, S. Hristova, D. O Regan, P. Kopanov Also, 1 PS j t = A j 1! λt T0 j d i 0 1 A j+1 1! λt T0 j d i 0 y A j 1 e y dy y A j+1 1 e y dy. Remark 3. Let conditions H1,H5 be satisfied and α j+1 = 1, i.e. τ j+1 Expα. Then the formula 8 reduces to PS j t = e λt T 0 j d i λt T 0 j d i A j. A j! Remark 4. Note A j+1 1 m=a j with positive coefficients. x m is a polynomial R j x of power A j+1 1 Corollary 2. Let conditions H1,H5 be satisfied with α i = α, i = 1,2,... and t T 0 + k d i. Then the probability that there will be exactly k impulses until time t is PS k t = e λt T 0 kα+α 1 k d i j=kα λt T 0 k d i j. 11 j! Lemma 2. Let conditions H2,H5 be satisfied and t T 0.Then the probability that there will be exactly k impulses until time t is 0, for t < T 0 +B and j > k, e λt T 0 A j+1 1 j d λt T i 0 j d i m PS j t = m=a j λt T0 j d i y A j 1 = 0 A j 1! yaj+1 1 e y dy, A j+1 1! for t < T 0 +B and j k or t T 0 +B, where A j = j α i and T 0 + k d i t < T 0 + k+1 d i. 12

9 P-MOMENT EXPONENTIAL STABILITY OF DIFFERENTIAL Proof. The proof of the case t < T 0 +B is similar to the proof in Lemma 1. Let t T 0 + B. Then for all natural number j the inequality t T 0 j d i > t T 0 d i T T 0 +B 0 holds and the proof is similar to the one of Lemma 1. Lemma 3. Upper bound of S k t. Let condition H5 and one of H1 or H2 be satisfied. Then for any natural number j we have PS j t K j e λt T 0 j d i λt T 0 A j. 13 holds where K j = max{1,λt T 0 α j+1 1 α j+1 1}. Proof. From equality 12, inequality λt T 0 j d i m λt T 0 m λt T 0 A j+1 1 λt T 0 A j+1 1, and Remark 4 it follows that 13 is true. Lemma 4. Let conditions H1, H5 hold. Then the probability the time t is immediately after the k-th random impulse but not far away than d k from it is given by 0, for j > k, A j 1 e PW j t = λg λgi m λg j j 1 m e λd k = m=1 1 A j 1! λgj 1 λg j y A k 1 e y dy, for j k, 14 where A j = j α i, D j = j d i, g j = t T 0 D j and T 0 + k d i t < T 0 + k+1 d i.

10 18 R. Agarwal, S. Hristova, D. O Regan, P. Kopanov Proof. From the definition of W j t and the random variables Ξ j we get PW j t = Pξ j < t < ξ j +d k j j j 1 = Pt T 0 d i < τ i < t T 0 d i j 1 = PΞ j < t T 0 d i PΞ j < t T 0 j d i. 15 If j > k then similar to the proof in Lemma 1 we get PW j t = 0. Now, let j k. Thent T 0 j 1 d i t T 0 j d i t T 0 k d i 0. From Proposition 2 and equality 3 we obtain Also, PW j t = PW j t = = A j 1 m=1 λg j m 1 λgj 1 A j 1! 1 A j 1! 0 λgj 1 λg j A j 1 e λg j m=1 y A j 1 e y dy y A j 1 e y dy. λg j 1 m e λg j 1. 1 λgj y Aj 1 e y dy A j 1! 0 Lemma 5. Let conditions H2,H5 be satisfied and t T 0. Then the probability the time t is immediately after the k-th random impulse but not far away than d k from it is given by 0 for t < T 0 +B and j > k e λg A j j 1 λgj m λg j 1 m e λd k m=1 PW j t = 1 λgj 1 = A j 1! λg j y Ak 1 e y dy for t < T 0 +B and j k or t T 0 +B, 16 where A j = j α i, D j = j d i, g j = t T 0 D j and T 0 + k d i t < T 0 + k+1 d i.

11 P-MOMENT EXPONENTIAL STABILITY OF DIFFERENTIAL Lemma 6. Upper bound of W k t. Let condition H5 and one of H1 or H2 be satisfied. Then for any natural number j we have j PW j t d j e λt T 0 j λt T 0 α i d i A j 1!t T Proof. Using the Integral mean value Theorem and j α i 1 we obtain λgj 1 λg j y A j 1 e y dy λd j λg j 1 j α i 1 e λg j λd j λt T j 0 α i 1 e λt T 0 j di. 4. Linear Equations with Random Noninstantaneous Impulses Consider the initial value problem for a scalar linear differential equation with random noninstantaneous moments of impulses: u = m k u for ξ k +d k < t < ξ k+1, k = 0,1,2,..., ut = b k uξ k, for ξ k < t < ξ k +d k, k = 1,2,..., ut 0 = u 0, 18 where u 0 R, m k > 0, k = 0,1,2... and b k 1, k =,2,... are real constants. Lemma 7. Let the following conditions be satisfied: 1. Condition H5 and one of the conditions H1 or H2 is fulfilled. 2. m i +λ m k for all natural i,k : i < k. Then the solution of the IVP for the scalar linear differential equation with random noninstantaneous moments of impulses 18 is u 0 e k 1 k i=0 m iτ i+1 b i for ξ k < t ut;t 0,u 0,{τ k } = ξ k +d k, k = 1,2,..., u 0 e k 1 k i=0 m iτ i+1 b 19 i e m kt ξ k d k for ξ k +d k < t < ξ k+1, k = 0,1,2,...

12 20 R. Agarwal, S. Hristova, D. O Regan, P. Kopanov and the expected value of the solution is { E ut;t 0,u 0,{τ k } = u 0 e m 0t T 0 PS 0 t + + k b i λ αi PW k t m i 1 +λ k b i λ } αi e m kg k PS k t. m i 1 m k +λ 20 Proof. The sample path solution of 18 is given by u 0 e k 1 i=0 m it i+1 T i d i k b i for T k < t T k +d k, k = 1,2,..., ut;t 0,u 0,{T k } = u 0 e k 1 i=0 m it i+1 T i d i k b i e m kt T k d k for T k +d k < t < T k+1, k = 0,1,2,... The above equality and Definition 2 establishes 19. From formula 19 and the independence of the random variables τ k we obtain E ut;t 0,u 0,{τ k } = u 0 e m 0t T 0 PS 0 t + + u 0 k b i E e k 1 i=0 m iτ i+1 PW k t k u 0 b i e m kg k E e k m k m i 1 τ i PS k t. 21 Using the definition of the density function of the Erlang distribution and substituting m i 1 +λx = s we get Ee m i 1τ i = e m i 1x λα i x αi 1 e λx dx 0 Γα i 1 = mi 1 +λ λ α i α i e s s αi 1 λ αi 22 ds = α i 1! m i 1 +λ 0

13 P-MOMENT EXPONENTIAL STABILITY OF DIFFERENTIAL and substituting m i 1 m k +λx = s we get Therefore, Ee m k m i 1 τ i = e m k m i 1 x λα i x αi 1 e λx dx 0 Γα i 1 = mi 1 m k +λ λ α i α i e s s αi 1 ds α i 1! 0 λ αi. = m i 1 m k +λ E e k m k m i 1 τ i = k λ αi. 23 m i 1 m k +λ Substitute 22 and 23 in 21 and obtain 20. Corollary 3. Let the conditions of Lemma 7 be satisfied with m k = m, k = 1,2,... Then for any t T 0 E ut;t 0,u 0,{τ k } = u 0 e mt T0 PS 0 t + u 0 k λ αi b i PW k t m+λ + u 0 e mt T 0 k b i e md i PS k t. 24 Lemma 8. Upper bound of the expected value. Let the conditions of Lemma 7 be satisfied. Then E ut;t 0,u 0,{τ k } = u 0 e λt T 0 { e m 0t T 0 λt T0 α 1 1 α 1 1! k + b i e λd i λ 2 t T 0 αi 25 m i 1 m k +λ K k e m kt T 0 + d } k. t T 0

14 22 R. Agarwal, S. Hristova, D. O Regan, P. Kopanov Proof. According to equality20, Corollary 1, Lemma 3 and Lemma 6 we obtain { E ut;t 0,u 0,{τ k } u 0 e m 0t T 0 e λt T λt T0 α α 1 1! + k b i λ αi d k e λt T 0 k d i m i 1 +λ λt T 0 k α i A k 1!t T 0 k + b i λ } αi e m k+λg k K k λt T 0 A k m i 1 m k +λ { u 0 e λt T 0 e m 0t T 0 λt T0 α 1 1 α 1 1! k + b i e λd i λ 2 t T 0 αi K k e m kt T 0 + d k. m i 1 m k +λ t T 0 Corollary 4. Upper bound of the expected value Let the conditions of Lemma 7 be satisfied and there exists positive constants M,M k,µ,µ k : 0 < µ k λ, k = 0,1,2,...,0 < µ λ such that for any t T 0 e m 0t T 0 λt T0 α 1 1 α 1 1! M 0 e µ 0t T 0 and K k e m kt T 0 + d k k b i e λd λ 2 t T i 0 αi M k e µ kt T 0 t T 0 m i 1 m k +λ with Then M k e µ kt T 0 Me µt T0. k=0 E ut;t 0,u 0,{τ k } M u 0 e λ µt T0.

15 P-MOMENT EXPONENTIAL STABILITY OF DIFFERENTIAL Corollary 5. Let the conditions of Lemma 7 with m k = m. Then { E ut;t 0,u 0,{τ k } u 0 e m+λt T λt T 0 0 α 1 1 α 1 1! + K k + d ke mt T 0 k b i e λd i λt T 0 α i. t T 0 26 If additionally there exist positive constants D,µ : µ < λ+m such that K k + d ke mt T 0 k b i e λd i λt T 0 α i De µt T0, 27 t T 0 then E ut;t 0,u 0,{τ k } M u 0 e νt T 0 where ν = min{m,m+λ µ} and M = 1+D. Proof. From26 using α1 1 λt T 0 α 1 1! e λt T 0 we obtain E ut;t 0,u 0,{τ k } u 0 e m+λt T 0 e λt T 0 +De µt T 0. Remark 5. Note inequality 27 is satisfied for α k H, d k 0,d], and b i : b k e λd ks α k A k 1! A k!, k = 1,2,..., where s 0,1, H,d are positive

16 24 R. Agarwal, S. Hristova, D. O Regan, P. Kopanov constants. Indeed, k b i e λd i λt T 0 α i K k + d ke mt T 0 t T 0 k s α A i i 1! λt T 0 α i 1+λt T 0 αk+1 1 α k+1 1 A i! sλ +d k e mt T 0 t T 0 sλ sλt T 0 A k A k! +de mt T 0 t T 0 +λα k+1 1t T 0 α k+1 sλt T 0 A k Le mt T0 +de mt T 0 A k! sλl+de mt T sλt T 0 0 A k sλl+de m+sλt T0. A k! 5. p-moment Exponential Stability for RNIDE The main goal of the paper is to define the exponential stability of the zero solution of RNIDE 6 with x 0 = 0 and to obtain some sufficient conditions for it. Definition 3. Let p > 0. Then the trivial solution x 0 = 0 of the RNIDE 6 is said to be p-moment exponentially stable if for any initial point x 0 R n thereexistconstantsα,µ > 0suchthattheinequalityE[ xt;t 0,x 0,{τ k } p ] < α x 0 p e µt T 0 holds for all t T 0, where xt;t 0,x 0,{τ k } is the solution of the IVP for the RNIDE 6. Remark 6. We note that the two-moment exponentially stability for stochastic equations is known as exponentially stability in mean square. Definition 4. Let J R + be a given interval and R n, 0 be a given set. We will say that the function Vt,x : J R +, Vt,0 0

17 P-MOMENT EXPONENTIAL STABILITY OF DIFFERENTIAL belongs to the class ΛJ, if it is continuous on J and locally Lipschitzian with respect to its second argument. For functions Vt,x ΛJ, we will use Dini derivatives defined by: { } 1 D + Vt,x = limsup Vt,x Vt h,x hft,x, t J,x, h 0 + h where there exists h 1 > 0 such that t h J, x hft,x for 0 < h h 1. Theorem 1. Let the following conditions be satisfied: 1. Conditions H3, H4, H5 and one of the conditions H1 or H2 hold. 2. The function V Λ[T 0,,R n and there exist positive constants a,b such that i a x p Vt,x b x p for t T 0 x R n ; ii there exists a constant m : 0 < m λ such that the inequality D + Vt,x mvt,x, for t > T 0, x R n holds; iii for any k = 1,2,... there exist functions w k CR +,R + such that Vt,I k x w k tvt,x for t T 0, x R n Thereexist positive constants D,µ : µ < λ+m and C k < 1, k = 1,2,... such that w k t C k for t T 0 and k C i e λd i λt T 0 α i K k + d ke mt T 0 De µt T0, t T t T 0 Then the trivial solution of the RNIDE 6 is p-moment exponentially stable. Proof. Let x 0 R n be an arbitrary initial point and the stochastic process x τ t = xt;t 0,x 0,{τ k } be a solution of the initial value problem for the RNIDE 6. Now consider the IVP for the scalar linear RNIDE 18 with m k = m, b k = C k for k = 1,2,..., and x 0 = VT 0,x 0. According to Lemma 7 the solution ut;t 0,VT 0,x 0,{τ k } of RNIDE 18 is given by 19. Let t k be arbitrary values of the random variables τ k, k = 1,2,... and T k = T 0 + k t i+ k d i,k = 1,2,... are values of the random variables ξ k.

18 26 R. Agarwal, S. Hristova, D. O Regan, P. Kopanov Define vt = Vt,xt;T 0,x 0,{T k }, t T 0, t T k and vt k = VT k,xt k 0;T 0,x 0,{T k }, k = 1,2,... Let t T k,t k + d k+1 ], k = 0,1,2,... Using the continuity and monotonicity of the function Vt,x and condition 2iii we obtain for t T k,t k + d k+1 ], k = 0,1,2,... vt =Vt,I k xt k 0 w k tvt,xt k 0 w k tvt k,xt k 0 =w k tvt k 0 C k vt k Now, consider any interval T k + d k+1,t k+1 ]. Then using vt k + d k+1 = VT k +d k+1,xt k +d k+1 ;T 0,x 0,{T k } we obtain v t =D + vt = D + Vt,xt;T 0,x 0,{T k } mvt,xt;t 0,x 0,{T k } = mvt,t T k +d k+1,t k+1 ]. 31 Therefore, from 30 and 31 it follows the function vt satisfies the linear impulsive differential inequalities with fixed points of noninstantaneous impulses v t m vt for T k +d k+1 < t < T k+1, k = 1,2,..., vt k + C k vt k, for T k < t T k +d k+1, k = 1,2,..., vt 0 = VT 0,x Consider the function mt = vt ut;t 0,VT 0,x 0,{T k }, t T 0 which is piecewise continuous function and according to Proposition 1 the function mt is nonpositive on [T 0, i.e. vt ut;t 0,x 0,{T k } for t T Note inequality 33 is satisfied for any arbitrary given sequence of fixed points of impulses {T k }. Therefore, the generated by vt stochastic process v τ t satisfies the inequality v τ t ut;t 0,x 0,{τ k }. From Corollary 5 and inequality 27 with m i = m and b i = C i and condi-

19 P-MOMENT EXPONENTIAL STABILITY OF DIFFERENTIAL tion 2i of Theorem 1 we obtain the inequalities E x τ t p = 1 a Ea x τt p 1 a EVt,x τt 1 a Ev τt 1 a Eut;T 0,x 0,{τ k } M a VT 0,x 0 e ν t T 0 Mb a x 0 p e ν t T 0, t T 0, where ν = min{m,m+λ µ} and M = 1+D. Inequality 34 proves the p-moment exponential stability. 34 Acknowledgments Research was partially supported by the Fund Scientific Research No. MU15FM IIT008, Plovdiv University. References [1] R. Agarwal, D. O Regan, S. Hristova, Stability of Caputo fractional differential equations with non-instantaneous impulses, Commun. Appl. Anal., , [2] R. Agarwal, D. O Regan, S. Hristova, Stability by Lyapunov like functions of nonlinear differential equations with non-instantaneous impulses, J. Appl. Math. Comput., accepted. [3] R. Agarwal, D. O Regan, S. Hristova, Stability of Caputo fractional differential equations by Lyapunov functions,appl. Math., , [4] A. Anguraj, A. Vinodkumar, Existence, uniqueness and stability results of random impulsive semilinear differential systems, Nonlinear Anal. Hybrid Syst., , [5] D.D. Bainov, P.S. Simeonov, Systems with Impulsive Effect. Stability, Theory and Applications, Ellis Horwood Series: Mathematics and its Applications, Ellis Horwood Ltd., Chichester Halster Press, John Wiley &Sons, Inc., New York, [6] D. Bainov, P. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, Pitman Monographs and Surveys in Pure and Applied Mathematics, 66, Longman Sciencific & Technical, Harlow; Co-published in USA, John Wiley & Sons, Inc., New York, 1993.

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