Dynamics and stability of impulsive hybrid setvalued integro-differential equations with delay

Nonlinear Analysis 65 (2006) 2082–2093 www.elsevier.com/locate/na

Dynamics and stability of impulsive hybrid setvalued integro-differential equations with delay Bashir Ahmad a , S. Sivasundaram b,∗ a Department of Mathematics, Faculty of Science, King Abdul Aziz University, P.O.Box 80203,

Jeddah 21589, Saudi Arabia b Department of Mathematics, Embry-Riddle Aeronautical University, Daytona Beach, FL 32114, USA

Abstract We study the dynamics and stability theory for impulsive hybrid set integro-differential equations with delay. Sufficient conditions for the stability of the null solution of impulsive hybrid set integro-differential equations with delay are presented. c 2006 Published by Elsevier Ltd  MSC: 34K45; 45J05; 34K20; 34D20 Keywords: Impulsive hybrid set integro-differential equations with delay; Existence; Stability

1. Introduction The study of set differential equations has been initiated as an independent subject and several results of interest can be found in [4,5,13–16]. The interesting feature of the set differential equations is that the results obtained in this new framework become the corresponding results of ordinary differential equations as the Hukuhara derivative and the integral used in formulating the set differential equations reduce to the ordinary vector derivative and integral when the set under consideration is a single valued mapping. Moreover, in the present setup, we have only semilinear complete metric space to work with, instead of complete normed linear space required in the study of the ordinary differential systems. Furthermore, set differential equations, that are generated by multivalued differential inclusions, when the multivalued functions involved do not ∗ Corresponding author.

E-mail address: [email protected] (S. Sivasundaram). c 2006 Published by Elsevier Ltd 0362-546X/$ - see front matter  doi:10.1016/j.na.2005.11.055

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possess convex values, can be used as a tool for studying multivalued differential inclusions [23]. Set differential equations can also be utilized to investigate fuzzy differential equations [14]. Impulsive hybrid systems are composed of some continuous variable dynamic systems along with certain reset maps that define impulsive switching among them. The switching performs resets to the modes and changes the continuous state of the system. There are three classes of impulsive hybrid systems, namely, impulsive differential systems, sample data control systems and impulsive switched systems. In recent years, a number of research papers have dealt with dynamical systems with impulse effect as a class of general hybrid systems. Examples include the pulse frequency modulation, optimization of drug distribution in the human body and control systems with changing reference signal. Impulsive dynamical systems are characterized by the occurrence of abrupt changes in the state of the system which occur at certain time instants over a period of negligible duration. The dynamical behavior of such systems is much more complex than the behavior of dynamical systems without impulse effects. The presence of impulse means that the state trajectory does not preserve the basic properties which are associated with non-impulsive dynamical systems. Thus, the theory of impulsive differential equations is quite interesting and has attracted the attention of many scientists, see for instance, [3,17,19, 22]. Moreover, in certain situations, the future state of the physical problems depends not only on the present state but also on its past history. Thus, introduction of the delay in the governing equations ensures a better modelling of the processes involved [7,20]. In this paper, we investigate the dynamics and stability theory for impulsive hybrid set integrodifferential equations with delay. The integral equation formulation of the physical problems is elegant. The auxiliary initial and boundary conditions are automatically satisfied in the integral equation formulation. The governing equations in the problems of biological sciences such as the spreading of disease by the dispersal of infectious individuals [2,8], the reaction–diffusion models in ecology to estimate the speed of invasion [6,11], etc. are integro-differential equations. In view of the extensive occurrence of the integro-differential equations in the mathematical modelling of physical problems, the theory and applications of integro-differential equations is an important area of investigation [12,18,21]. In Section 3, we study some basic results for set integro-differential equations with delay while the results on impulsive hybrid set integrodifferential equations with delay are developed in Section 4. 2. Terminology and preliminaries Let K c (R n ) denote the collection of nonempty, compact and convex subsets of R n . We define the Hausdorff metric as 
D[X, Y ] = max sup d(y, X), sup d(x, Y ) , y∈Y

(2.1)

x∈X

where d(y, X) = inf[d(y, x) : x ∈ X] and X, Y are bounded subsets of R n . Notice that K c (R n ) with the metric is a complete metric space. Moreover, K c (R n ) equipped with the natural algebraic operations of addition and nonnegative scalar multiplication becomes a semilinear metric space which can be embedded as a complete cone into a corresponding Banach space [1,23]. The Hausdorff metric (2.1) satisfies the following properties: ∀ X, Y, Z ∈ K c (R n ) and µ ∈ R+ , we have D[X + Z , Y + Z ] = D[X, Y ] D[µX, µY ] = µD[X, Y ], D[X, Y ] ≤ D[X, Z ] + D[Z , Y ].

and

D[X, Y ] = D[Y, X],

(2.2) (2.3) (2.4)

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Definition 2.1. The set Z ∈ K c (R n ) satisfying X = Y + Z is known as the Hukuhara difference of the sets X and Y in K c (R n ) and is denoted as X − Y . Definition 2.2. For any interval I ∈ R, the mapping F : I → K c (R n ) has a Hukuhara derivative D H F(t0 ) at a point t0 ∈ I , if there exists an element D H F(t0 ) ∈ K c (R n ) such that the limits lim

h→0+

F(t0 + h) − F(t0 ) h

and

lim

h→0+

F(t0 ) − F(t0 − h) , h

(2.5)

exist in the topology of K c (R n ) and each one is equal to D H F(t0 ). Now we consider the set-integro differential equation  t D H U (t) = F(t, U (t)) + K (t, η, U (η))dη, U (t0 ) = U0 ∈ K (R n ), t0 > 0, (2.6) t0

(R n ), K

n n n where F ∈ C[R+ × K c c (R )], K ∈ C[R+ × R+ × K c (R ), K c (R )]. 1 n The mapping U ∈ C [ J, K c (R )], J = [t0 , t0 + T ], T > 0, is said to be a solution of (2.6) on J if it satisfies (2.6) on J . Since U (t) is continuously differentiable, we have  t U (t) = U0 + D H U (η)dη, t ∈ J, (2.7) t0

which can be put in the form [18]  t  U (t) = U0 + F(η, U (η)) +

η

t0

t

 K (σ, η, U (η))dσ dη,

t ∈ J,

(2.8)

where the integral is in the sense of the Hukuhara integral [9,10]. Thus, we can say that U (t) is a solution of (2.8) if and only if it satisfies (2.8) on J . Now, we state some properties which are useful in the proof of our results. If F : [t0 , T ] → K c (R n ) is integrable, then  t2  t1  t2 F(t)dt = F(t)dt + F(t)dt, t0 ≤ t1 ≤ t2 ≤ T, (2.9) 

t0 T

t0



µF(t)dt = µ

t1 T

F(t)dt,

t0

t0

µ ∈ R+ .

(2.10)

Moreover, if F, G : [t0 , T ] → K c (R n ) are integrable, then D[F(.), G(.)] : [t0 , T ] → R is integrable and  t   t  t D F(σ )dσ, G(σ )dσ ≤ D[F(σ ), G(σ )]dσ. t0

t0

t0

We observe that D[ A, θ ] = A = sup a,

(2.11)

a∈A

where A ∈ K c (R n ) and θ , the zero element of K (R n ), is regarded as a one point set. 3. Some basic results in set integro-differential equations with delay For any τ > 0, let C = C[[−τ, 0], K c (R n )]. For any Φ, Ψ ∈ C, we define the metric D0 [Φ, Ψ ] = max−τ ≤s≤0 D[Φ(s), Ψ (s)]. In other words, we can write φ0 = D0 [Φ, θ ]. For

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any t ∈ J0 = [t0 − τ, t0 + a], a > 0, U ∈ [ J0 , K c (R n )], let Ut denote a translation of the restriction of U to the interval [t − τ, t], that is, Ut ∈ C be defined by Ut (s) = U ((t + s)), −τ ≤ s ≤ 0. Consider the set integro-differential equation with delay  t K (t, η, Uη )dη, Ut0 = Φ0 ∈ C, t0 > 0, (3.1) D H U (t) = F(t, Ut ) + t0

where F ∈ C[ J × C, K c

(R n )], K

∈ C[ J × J × C, K c (R n )] and J = [t0 , t0 + a].

Theorem 3.1 (Comparison Result). Assume that F ∈ C[R+ × C, K c (R n )], K ∈ C[R+ × R+ × C, K c (R n )] and for t ∈ R+ , X, Y, ∈ C,    t  t K (t, η, X)dη, F(t, Y ) + K (t, η, Y )dη D F(t, X) + t0

t0



t

≤ q(t, D0 [X, Y ]) +

Q(t, η, D0 [X, Y ])dη,

t0

where q ∈ C[R+ × R+ , R+ ] and Q ∈ C[R+ × R+ × R+ , R+ ]. Moreover, we require that there exists the maximal solution r (t, t0 , w0 ) of the scalar integro-differential equation  t Q(t, η, w(η))dη, w(t0 ) = w0 ≥ 0, t ≥ t0 . w (t) = q(t, w(t)) + t0

Then, if U (t) = U (t0 , Φ0 )(t), V (t) = V (t0 , Ψ0 )(t) are any two solutions of (3.1), we have D[U (t), V (t)] ≤ r (t, t0 , w0 ), t ≥ t0 , provided that D0 [Φ0 , Ψ0 ] ≤ w0 . Proof. Since U (t), V (t) are solutions of (3.1), the differences U (t + h) − U (t), V (t + h) − V (t) exist for small h > 0. For t ∈ R+ , we set m(t) = D(U (t), V (t)). Using the properties (2.2)–(2.4) of Hausdorff metric, we have m(t + h) − m(t) = D[U (t + h), V (t + h)] − D[U (t), V (t)]      t ≤ D U (t + h), U (t) + h F(t, Ut ) + K (t, η, Uη )dη + D U (t) 



+ h F(t, Ut ) + 





t



t t0





+ D V (t) + h F(t, Vt ) + + h D F(t, Ut ) +







t0 t t0

t t0

t



t t0

 K (t, η, Vη )dη

K (t, η, Vη )dη , V (t + h) − D[U (t), V (t)]

   ≤ D U (t + h), U (t) + h F(t, Ut ) +





K (t, η, Uη )dη , V (t) + h F(t, Vt ) +

t0

+ D V (t) + h F(t, Vt ) +



t0

 K (t, η, Uη )dη

  K (t, η, Vη )dη , V (t + h)

K (t, η, Uη )dη, F(t, V (t)) +



t

t0

 K (t, η, Vη )dη ,

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which implies that

   t U (t + h) − U (t) m(t + h) − m(t) ≤D , F(t, Ut ) + K (t, η, Uη )dη h h t0    t V (t + h) − V (t) K (t, η, Vη )dη, + D F(t, Vt ) + h t0    t  t K (t, η, Uη )dη, F(t, V (t)) + K (t, η, Vη )dη . + D F(t, Ut ) + t0

Taking limsup as h →

0+

t0

yields

1 D + m(t) = limsuph→0+ [m(t + h) − m(t)] h   t  ≤ D F(t, Ut ) + K (t, η, Uη )dη, F(t, Vt ) + t0



t

≤ q(t, D0 [Ut , Vt ]) +

t0

 ≤ q(t, D0 [U0 , V0 ]) +

t

t t0

 K (t, η, Vη )dη ,

Q(t, η, D0 [Uη , Vη ])dη, Q(t, η, D0 [U0 , V0 ])dη,

t0

which together with the fact that D0 [Φ0 , Ψ0 ] ≤ w0 and by the comparison theorem for ordinary delay integro-differential equations [18] gives D[U (t), V (t)] ≤ r (t, t0 , w0 ),

t ≥ t0 .

This completes the proof of the theorem.




Theorem 3.2. Assume that (A1 ) F ∈ C[R+ × C, K c (R n )], and for (t, Φ) ∈ R+ × C, D[F(t, Φ), θ ] ≤ q(t, D0 [Φ, θ ]), where q ∈ C[R+ × R+ , R+ ] and q(t, w) is nondecreasing in w for each t ∈ R+ . (A2 ) K ∈ C[R+ × R+ × C, K c (R n )] be such that D[K (t, η, Φ), θ ] ≤ Q(t, η, D0 [Φ, θ ]), where Q(t, η, w) is nondecreasing in w for each (t, η) ∈ R+ × R+ . (A3 ) The solution w(t, t0 , w0 ) of  t Q(t, η, w(η))dη, w(t0 ) ≥ w0 , w (t) = q(t, w(t)) + t0

exists for t ≥ t0 . If F and K are smooth enough to assure the local existence, then the largest interval of existence of any solution U (t0 , Φ0 )(t) of (3.1) is [t0 , ∞). Proof. For the sake of contradiction, we assume that U (t0 , Φ0 )(t) is a solution of (3.1) existing on some interval [t0 − τ, β] and β cannot be increased. For t ∈ [t0 − τ, β], let us set m(t) = D[U (t0 , Φ0 )(t), θ ], m t = D[Ut (t0 , Φ0 ), θ ], and |m t |0 = D0 [Ut (t0 , Φ0 ), θ ]. Employing the procedure used in the proof of the comparison theorem, we obtain the differential inequality  t + Q(t, η, |m t |0 )dη, t0 ≤ t < β. D m(t) ≤ q(t, |m t |0 ) + t0

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Choosing |m t |0 = D0 [Φ0 , θ ] ≤ w0 , we get D[U (t0 , Φ0 )(t), θ ] ≤ r (t, t0 , w0 ),

t0 ≤ t < β.

Now q(t, w) ≥ 0 and Q(t, η, w) ≥ 0 imply that r (t, t0 , w0 ) is nondecreasing in t and hence D[Ut (t0 , Φ0 ), θ ] ≤ r (t, t0 , w0 ),

t0 ≤ t < β.

(3.2)

Selecting t1 , t2 such that t0 < t1 < t2 < β and using the properties (2.2)–(2.4) of Hausdorff metric, we have D[U (t0 , Φ0 )(t1 ), U (t0 , Φ0 )(t2 )]   = D U (t0 , Φ0 )(t1 ), U (t0 , Φ0 )(t1 ) +   = D θ,  ≤  ≤

t2



t1 t2



t2



D[F(s, Us ), θ ] +



 F(s, Us ) +

t1



s



t0 s

F(s, Us ) +

t1

t2

t0

  K (s, η, Uη )dη ds

t0





K (s, η, Uη )dη ds

 D[K (s, η, Uη ), θ ]dη ds 

q(s, D0 [Us (t0 , φ0 ), θ ] +

s t0

t1

s

 Q(s, η, D0 [Uη (t0 , φ0 ), θ ]dη ds.

In view of the nondecreasing property of g and Q in w and by relation (3.2), it follows that D[U (t0 , Φ0 )(t1 ), U (t0 , Φ0 )(t2 )]  t2   ≤ q(s, r (s, t0 , w0 ) +

s

 Q(s, η, r (η, t0 , w0 ))]dη ds

t0

t1

= r (t2 , t0 , w0 ) − r (t1 , t0 , w0 ). If we allow t1 , t2 → β in the above relation, then limt →β − U (t0 , Φ0 )(t) exists by Cauchy’s criterion of convergence. Now we define U (t0 , Φ0 )(β) = limt →β − U (t0 , Φ0 )(t) and consider χ0 = Uβ (t0 , Φ0 ) as the new initial function at t = β. Then, by the assumption of local existence, there exists a solution U (β, χ0 )(t) of (3.1) on [β −τ, β +γ ], γ > 0. This implies that the solution U (t0 , Φ0 )(t) can be continued beyond β, which contradicts our assumption that β cannot be increased. This completes the proof of the theorem. 4. Main results for impulsive hybrid set integro-differential equations with delay Consider the impulsive hybrid set integro-differential equations with delay  t    D U (t) = F(t, U ) + K (t, η, Uη )dη, t = tk , H t  U + = Ik (Utk ),    tk Ut0 = Φ0 ∈ C,

t0

t = tk ,

(4.1)

where F ∈ PC[R+ × C, K c (R n )], K ∈ PC[R+ × R+ × C, K c (R n )], Ik : C → C and {tk } is a sequence of points such that 0 ≤ t0 < t1 < · · · tk < · · · with limk→∞ tk = ∞. Here we emphasize that F ∈ PC[R+ × C, K c (R n )] and K ∈ PC[R+ × R+ × C, K c (R n )] mean that F : (tk−1 , tk ] × C → K c (R n ) and K : (tk−1 , tk ] × (tk−1 , tk ] × C → K c (R n ) are continuous and

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lim

(t,Yt )→(tk+ ,Ut )

F(t, Yt ) = F(tk+ , Ut ),

lim

(t,η,Yt )→(tk+ ,η,Ut )

K (t, η, Yt ) = K (tk+ , η, Ut ),

exist for each Ut ∈ C. By a solution of (4.1), we mean a piecewise continuous function U (t0 , Φ0 )(t) on [t0 , ∞) which is left continuous on (tk , tk+1 ] and is defined by  Φ0 , t = tk ,    U (t , Φ )(t), t  0 0 0 0 ≤ t ≤ t1 ,    U (t , Φ )(t), t  1 1 1 1 < t ≤ t2 ,    . .    . . U (t0 , Φ0 )(t) = (4.2) . .     Uk (tk , Φk )(t), tk < t ≤ tk+1 ,     . .     . .   . . where U (tk , Φk )(t) is a solution of the Impulsive hybrid set integro-differential equation with delay  t K (t, η, Uη )dη, Ut + = Φk , k = 0, 1, 2 . . . . D H U (t) = F(t, Ut ) + t0

k

Now we are in a position to prove an existence theorem for Impulsive hybrid set integrodifferential equations with delay (4.1). Theorem 4.1. Assume that (B1 ) F ∈ PC[R+ × C, K c (R n )] be such that D[F(t, Φ), θ ] ≤ q(t, D0 [Φ, θ ]), t = tk where q ∈ C[R+ × R+ , R+ ] and q(t, w) is nondecreasing in w for each t ∈ R+ . (B2 ) K ∈ PC[R+ × R+ ×C, K (R n )] be such that D[K (t, η, Φ), θ ] ≤ Q(t, η, D0 [Φ, θ ]), t = tk and Q(t, η, w) is nondecreasing in w for each (t, η) ∈ R+ × R+ . (B3 ) (D0 [I (Utk ), θ ]) ≤ Jk (D0 [U (tk ), θ ]), t = tk , where Jk (w) is a nondecreasing function of w. (B4 ) r (t, t0 , w0 ) is the maximal solution of the Impulsive hybrid scalar integro-differential equation  t     Q(t, η, w(η))dη, t = tk , w (t) = q(t, w(t)) + t0 (4.3) w + = Jk (w(tk )), t = tk ,    tk w(t0 ) = w0 , which exists on [t0 , ∞) and F, K are smooth enough to assure the local existence. Then there exists a solution for (4.1) on [t0 , ∞). Proof. we define J0 = [t0 , t1 ] and restrict F to J0 × C and K to J0 × J0 × C so that F and K are continuous on their respective domains. Consider the set integro-differential equation with delay   t  K (t, η, Uη )dη, D H U (t) = F(t, Ut ) + t0  Ut0 = Φ0 ,

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on J0 . Clearly the hypothesis of Theorem 3.2 is satisfied and there exists a solution U (t0 , Φ0 )(t), t ∈ J0 for the set integro-differential equation with delay on J0 . At t = t1 , U0 (t1 ) = U (t0 , Φ0 )(t1 ) and U0,t + = I1 (U0,t1 ). Now we set Φ1 = U0,t + , J1 = 1 1 (t1 , t2 ] and consider  t   D U (t) = F(t, U ) + K (t, η, Uη )dη, t ∈ J1 , H t (4.4) t0 U + = Φ . 1

t1

Now, restricting the domains of F and K to J1 × C and J1 × J1 × C respectively and employing the Impulsive hybrid condition (B3 ), it follows, by Theorem 3.2, that there exists a solution U1 (t1 , Φ1 )(t), t ∈ J1 of (4.4). Thus, we have U1 (t2 ) = U1 (t1 , Φ1 )(t2 ) and U1,t + = I2 (U1,t2 ). 2 Setting Φ2 = U1,t + , J2 = (t2 , t3 ] and employing the earlier arguments, it can be shown that 2 there exists a solution U2 (t2 , Φ2 )(t), t ∈ J2 of the set integro-differential equation with delay on J2 . The repetition of the above process proves the existence of a solution of the Impulsive hybrid set integro-differential equation with delay on [t0 , ∞).
Next we present a comparison theorem for impulsive hybrid set integro-differential equations with delay. Theorem 4.2. Assume that (C1 ) F ∈ PC[R+ × C, K c (R n )] be such that D[F(t, Φ), F(t, Ψ )] ≤ q(t, D0 [Φ, Ψ ]), where t ∈ R+ , t = tk , Φ, Ψ ∈ C and q ∈ PC[R+ × R+ , R+ ]. (C2 ) K ∈ PC[R+ × R+ × C, K (R n )] be such that D[K (t, η, Φ), K (t, η, Ψ )] ≤ Q(t, η, D0 [Φ, Ψ ]), where Q(t, η, w) ∈ PC[R+ × R+ × R+ , R+ ]. (C3 ) (D0 [Ik (Utk ), Ik (Vtk )]) ≤ Jk (D0 [Utk , Vtk ]), t = tk , where Jk (w) is a nondecreasing function of w. (C4 ) r (t, t0 , w0 ) is the maximal solution of the Impulsive hybrid scalar integro-differential equation (4.3) which exists on [t0 , ∞). Then, if U (t) = U (t0 , Φ0 )(t), V (t) = V (t0 , Ψ0 )(t) are any two solutions of (4.1) on [t0 , ∞), we have D[U (t), V (t)] ≤ r (t, t0 , w0 ),

t ≥ t0 ,

provided that D0 [Φ0 , Ψ0 ] ≤ w0 . Proof. Define J0 = [t0 , t1 ] and restrict F to J0 × C and K to J0 × J0 × C so that F and K are continuous on their respective domains and the hypothesis of Theorem 3.1 is satisfied. Hence we conclude that D[U (t), V (t)] ≤ r (t, t0 , w0 ),

t ∈ J0 .

Obviously D[U (t1 ), V (t1 )] ≤ r (t1 , t0 , w0 ). Using the assumption (C3 ) for t = t1+ , we have D0 [Ut + , Vt + ] = D0 [I1 (Ut1 ), I1 (Vt1 )] ≤ J1 (D0 [Ut1 , Vt1 ]) ≤ J1 (r (t1 )) ≡ r (t1+ ). 1

1

(4.5)

Now, we let J1 = (t1 , t2 ] and restrict the domains of F and K to J1 × C and J1 × J1 × C respectively. Using the impulsive condition (C2 ) and (4.5) together with Theorem 3.1, we obtain D[U (t), V (t)] ≤ r (t, t0 , w0 ),

t ∈ J1 .

By repeating the above procedure, the conclusion of the theorem is achieved.




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Next, we present a comparison theorem which provides a basis to investigate the stability criteria of impulsive hybrid set integro-differential equation with delay (4.1) in terms of Lyapunov-like functions. Theorem 4.3. Assume that (E1 ) V : R+ ×K c (R n )×C → R+ be such that V (t, U, Φ) is continuous in (tk−1 , tk ]×K c (R n )× C and for each U ∈ K c (R n ), Φ ∈ C, the following limit exists lim

(t,W,Φ )→(tk+,U,Φ )

V (t, W, Φ) = V (tk+ , U, Φ),

k = 1, 2, . . . .

(E2 ) V satisfies |V (t, U, Φ) − V (t, W, Φ)| ≤ L D[U, W ], L > 0. (E3 ) For q ∈ PC[R+ × R+ , R+ ] and Q ∈ PC[R+ × R+ × R+ , R+ ],   1 D + V (t, U, Φ) = limsuph→0+ V t + h, U + h F (t, Ut ) h  

 t + K (t, η, Uη )dη , Ut +h − V (t, U, Ut ) t0



t

≤ q(t, V (t, U, Φ)) +

Q(t, η, V (t, U, Φ))dη,

t = tk ,

t0

(E4 ) V (tk+ , U (t0 , Φ0 )(tk+ ), Ut + (t0 , Φ0 )) ≤ Jk [V (tk , U (t0 , Φ0 )(tk ), Utk (t0 , Φ0 ))], t = tk , and k Jk (w) is a nondecreasing function of w. Then, if U (t0 , Φ0 )(t) is any solution of (4.1) existing on [t0 , ∞), we have V (t, U (t0 , Φ0 )(t), Ut (t0 , Φ0 )) ≤ r (t, t0 , w0 ),

t ∈ [t0 , ∞),

where r (t, t0 , w0 ) is the maximal solution of the Impulsive hybrid scalar integro-differential equation (4.3) which exists on [t0 , ∞). Proof. We define m(t) = V (t, U (t0 , Φ0 )(t), Ut (t0 , Φ0 )) so that m(t0 ) = V (t, U (t0 , Φ0 )(t0 ), Φ0 ) and assume that m(t0 ) ≤ w0 . For small h > 0 and t ∈ (tk−1 , tk ], k = 1, 2, . . ., we consider m(t + h) − m(t) = V (t + h, U (t0 , Φ0 )(t + h), Ut +h (t0 , Φ0 )) − V (t, U (t0 , Φ0 )(t), Ut (t0 , Φ0 )) = V (t + h, U (t0 , Φ0 )(t + h), Ut +h (t0 , Φ0 ))  

 t K (t, η, Uη )dη , Ut +h (t0 , Φ0 ) − V t + h, U (t0 , Φ0 )(t) + h F(t, Ut ) +  + V (t + h, U (t0 , Φ0 )(t) + h F(t, Ut ) +



t0 t

t0

 K (t, η, Uη )dη , Ut +h (t0 , Φ0 ))

− V (t, U (t0 , Φ0 )(t), Ut (t0 , Φ0 ))    ≤ L D U (t0 , Φ0 )(t + h), U (t0 , Φ0 )(t) + h F(t, Ut ) +   + V t + h, U (t0 , Φ0 )(t) + h F(t, Ut ) +

t0 t

t0

− V (t, U (t0 , Φ0 )(t), Ut (t0 , Φ0 )),

t

 K (t, η, Uη )dη 

K (t, η, Uη )dη , Ut +h (t0 , Φ0 )

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where we have used the Lipschitz condition described in (E 2 ). Thus, 1 D + m(t) = limsuph→0+ [m(t + h) − m(t)] ≤ D + V (t, U (t0 , Φ0 )(t), Ut (t0 , Φ0 )) h   1 + Llimsuph→0+ D U (t0 , Φ0 )(t + h), U (t0 , Φ0 )(t) h    t + h F(t, Ut ) + K (t, η, Uη )dη . t0

Letting U (t0 , Φ0 )(t +h) = U (t0 , Φ0 )(t)+ Z (t0 , Φ0 )(t), we say that Z (t0 , Φ0 )(t) is the Hukuhara difference of U (t0 , Φ0 )(t + h) and U (t0 , Φ0 )(t) for small h > 0 and is assumed to exist. Hence, employing the properties of D[·, ·], it follows that     t K (t, η, Uη )dη D U (t0 , Φ0 )(t + h), U (t0 , Φ0 )(t) + h F(t, Ut ) + t

0    = D U (t0 , Φ0 )(t) + Z (t0 , Φ0 )(t), U (t0 , Φ0 )(t) + h F(t, Ut ) +

   = D U (t0 , Φ0 )(t + h) − U (t0 , Φ0 )(t), h F(t, Ut ) +

t t0

t

t0

 K (t, η, Uη )dη 

K (t, η, Uη )dη

.

Consequently, we find that      t 1 K (t, η, Uη )dη D U (t0 , Φ0 )(t + h), U (t0 , Φ0 )(t) + h F(t, Ut ) + h t0    t U (t0 , Φ0 )(t + h) − U (t0 , Φ0 )(t) =D , F(t, Ut ) + K (t, η, Uη )dη , h t0 which, in view of the fact that U (t0 , Φ0 )(t) is a solution of (3.1), yields      t 1 lim sup K (t, η, Uη )dη D U (t0 , Φ0 )(t + h), U (t0 , Φ0 )(t) + h F(t, Ut ) + t0 h→0+ h    t U (t0 , Φ0 )(t + h) − U (t0 , Φ0 )(t) , F(t, Ut ) + = lim sup D K (t, η, Uη )dη = 0. h t0 h→0+ Hence, by the assumption (E 3 ), we get the scalar integro-differential inequality D + m(t) ≤ D + V (t, U (t0 , Φ0 )(t), Ut (t0 , Φ0 ))  t ≤ q(t, V (t, U (t0 , Φ0 ))(t), Ut (t0 , Φ0 )) + Q(t, η, V (η, U (t0 , Φ0 )(η), Uη (t0 , Φ0 ))dη t0



t

= q(t, m(t)) +

Q(t, η, m(η))dη.

t0

For t = tk , by virtue of (E 4 ), we have m(tk+ ) = V (tk+ , U (t0 , Φ0 )(tk+ ), Ut + (t0 , Φ0 )) k

≤ Jk [V (tk , U (t0 , Φ0 )(tk ), Utk (t0 , Φ0 ))] = Jk [m(tk )].

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Following the procedure elaborated in [18], we obtain the desired estimate m(t) ≤ r (t, t0 , w0 ),

t ∈ [t0 , ∞).

This proves the assertion of the theorem.




We are now in a position to formulate the stability criteria for the null solution of an Impulsive hybrid set integro-differential equation with delay (4.1). Let us first define the stability of the trivial solution and set our notations. Definition 4.1. Let U (t0 , Φ0 )(t) be any solution of (4.1). Then the trivial solution U (t) ≡ θ is said to be stable if for each > 0 and t0 ∈ R+ , there exists a δ = δ(t0 , ) > 0 such that D0 [Φ0 , θ ] < δ implies that D[U (t), θ ] < , t ∈ [t0 , ∞). We assume that F(t, θ ) ≡ θ, K (t, η, θ ) ≡ θ, Ik (θ ) ≡ θ for all k and set S(ρ) = [U ∈ K c (R n ) : D[U, θ ] < ρ],

S1 (ρ) = [Φ ∈ C : D0 [Φ, θ ] < ρ],

K = [ν ∈ C[R+ , R+ ] : ν(0) = 0 and ν(U ) is strictly increasing]. Theorem 4.4. Assume that (F1 ) V : R+ × S(ρ) × S1 (ρ) → R+ and (E 1 ), (E 2 ) of Theorem 4.3 hold. Moreover,  t Q(t, η, V (t, U, Φ))dη, t = tk , D + V (t, U, Φ) ≤ q(t, V (t, U, Φ)) + t0

where q : R+ × R+ → R+ and Q : R+ × R+ × R+ → R+ are such that q(t, 0) ≡ 0, Q(t, η, 0) ≡ 0 and satisfy the assumptions stated in Theorem 4.3. (F2 ) There exists some ρ0 > 0 such that Utk ∈ S1 (ρ0 ) implies that I K (Utk ) ∈ S1 (ρ) for all k and V (tk+ , U (t0 , Φ0 )(tk+ ), Ut + (t0 , Φ0 )) ≤ Jk [V (tk , U (t0 , Φ0 )(tk ), Utk (t0 , Φ0 ))], k

t = tk , Utk ∈ S1 (ρ0 ) and Jk : R+ → R+ is nondecreasing and Jk (0) ≡ 0 for all k. (F3 ) γ (D[U, θ ]) ≤ V (t, U, Φ) ≤ ν(D0 [Φ, θ ]), where ν, γ ∈ K. Then the stability properties of the null solution of (4.3) imply the corresponding stability properties of the null solution of (4.1). Proof. We choose such that 0 < < min(ρ, ρ0 ) and suppose that the null solution of (4.3) is stable. Then, for γ ( ) > 0, t0 ∈ R+ , we can find a δ1 = δ(t0 , ) > 0 such that 0 ≤ w0 < δ1 implies that w(t, t0 , w0 ) < γ ( ), t ≥ t0 , where w(t, t0 , w0 ) is any solution of (4.3). Letting w0 = ν(D0 [Φ0 , θ ]), we select a δ = δ(t0 , ) such that ν(δ) < δ1 and claim that D0 [Φ0 , θ ] < δ implies D[U (t0 , Φ0 )(t), θ ] < , t ≥ t0 for any solution U (t0 , Φ0 )(t) of (4.1). Suppose that our claim is false and there exists a solution U (t0 , Φ0 )(t) of (4.1) with D0 [Φ0 , θ ] < δ and a t1 > t0 satisfying tk < t1 ≤ tk+1 for some k such that D[U (t0 , Φ0 )(t1 ), θ ] ≥ and D[U (t0 , Φ0 )(t), θ ] < , t0 ≤ t ≤ tk . In view of 0 < < ρ0 , it follows from (F2 ) that D[U (t0 , Φ0 )(tk ), θ ] < and D0 [Ut + (t0 , Φ0 ), θ ] = D0 [Ik (Utk (t0 , Φ0 )), θ ] < ρ. k

Thus we can find a t2 such that tk < t2 ≤ t1 satisfying ≤ D[U (t0 , Φ0 )(t2 ), θ ] < ρ. As before, we set m(t) = V (t, U (t0 , Φ0 )(t), Ut (t0 , Φ0 )), t1 ≤ t ≤ t2 and use (F1 ), (F2 ) together with

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Theorem 4.3 to obtain the estimate m(t) ≤ r (t, t0 , ν(D0 [Φ0 , θ ])), t0 ≤ t ≤ t2 , where r (t, t0 , w0 ) is the maximal solution of (4.3). Now, using the assumption (F3 ), we arrive at the contradiction γ ( ) ≤ γ (D[U (t0 , Φ0 )(t2 ), θ ]) ≤ V (t2 , U (t0 , Φ0 )(t2 ), Ut2 (t0 , Φ0 )) ≤ r (t2 , t0 , ν(D0 [Φ0 , θ ])) < r (t2 , t0 , ν(δ)) < r (t2 , t0 , δ1 ) < γ ( ). This shows that our claim is not true and hence the null solution of (4.1) is stable. References [1] A.J. Brandao Lopes Pinto, F.S. De Blasi, F. Iervolino, Uniqueness and existence theorems for differential equations with compact convex valued solutions, Boll. Unione. Mat. Italy 3 (1970) 47–54. [2] C.M. Brasier, Rapid evolution of introduced plant pathogens via interspecific hybridization, Bioscience 2 (2001) 123–133. [3] V. Doddaballapur, P.W. Eloe, Y. Zhang, Quadratic convergence of approximate solutions of two-point boundary value problems with impulse, Electron. J. Differ. Eq., Conf. 01 (1997) 81–95. [4] T. Gnana Bhaskar, V. Lakshmikantham, Set differential equations and flow invariance, Appl. Anal. 82 (2003) 357–368. [5] T. Gnana Bhaskar, V. Lakshmikantham, Lyapunov stability for set differential equations, Dynam. Systems Appl. 13 (2004) 1–10. [6] K.P. Hadeler, Reaction transport systems in biological modeling, in: V. Capasso, O. Diekmann (Eds.), Mathematics Inspired by Biology, Springer, New York, 1999, pp. 95–150. [7] J. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977. [8] Z. Hubalek, J. Halouzka, West nile fever-a reemerging mosquito-born viral disease in Europe, Emerg. Infec. Dis. 5 (1999) 643–650. [9] M. Hukuhara, Sur l’aplication semicontinue dont la valeur est un compact convexe. Funkcial. Ekvak. (1967) 43–68. [10] M. Hukuhara, Integration des applications measurable dont la valeur est un compactconvex, Funkcial. Ekvak. (1967) 205–229. [11] M. Kot, W.M. Schaffer, Discrete-time growth-dispersal models, Math. Biosci. 1 (1986) 109–136. [12] M. Kot, M.A. Lewis, P.V. Driessche, Dispersal data and the spread of invading organisms, Ecology 7 (1996) 2027–2042. [13] V. Lakshmikantham, S. Leela, A.S. Vatsala, Setvalued hybrid differential equations and stability in terms of two measures, J. Hybrid Systems 2 (2002) 169–187. [14] V. Lakshmikantham, S. Leela, A.S. Vatsala, Interconnection between set and fuzzy differential equations, Nonlinear Anal. 54 (2003) 351–360. [15] V. Lakshmikantham, A. Tolstonogov, Existence and interrelation between set and fuzzy differential equations, Nonlinear Anal. 55 (2003) 255–268. [16] V. Lakshmikantham, S. Leela, A.S. Vatsala, Stability theory for differential equations, Dyn. Contin. Discrete impuls. syst. 11 (2004) 181–189. [17] V. Lakshmikantham, D.D. Bainov, P.S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989. [18] V. Lakshmikantham, M.R.M. Rao, Theory of Integro-Differential Equations, Gordon and Breach, London, 1995. [19] S. Leela, F.A. Mcrae, S. Sivasundaram, Controllability of impulsive differential equations, J. Math. Anal. Appl. 177 (1) (1993) 24–30. [20] F.A. Mcrae, J.V. Devi, Impulsive set differential equations with delay, Applicable Anal. 84 (2005) 329–341. [21] J.D. Murray, Mathematical Biology, second ed., Springer-Verlag, New York, 1993. [22] M.R.M. Rao, S. Sivasundaram, Stability of Volterra system with impulse effect, J. Appl. Math. Stoch. Anal. 4 (1) (1991) 83–93. [23] A. Tolstonogov, Differential Inclusions in a Banach Space, Kluwer Academic Publishers, Dordrecht, 2000.