IMPULSIVE FUNCTIONAL-DIFFERENTIAL EQUATIONS WITH NONLOCAL...

IJMMS 29:5 (2002) 251–256PII. S0161171202012887

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IMPULSIVE FUNCTIONAL-DIFFERENTIAL EQUATIONSWITH NONLOCAL CONDITIONS

HAYDAR AKÇA, ABDELKADER BOUCHERIF, and VALÉRY COVACHEV

Received 22 April 2001 and in revised form 7 August 2001

The existence, uniqueness, and continuous dependence of a mild solution of an impulsivefunctional-differential evolution nonlocal Cauchy problem in general Banach spaces arestudied. Methods of fixed point theorems, of a C0 semigroup of operators and the Banachcontraction theorem are applied.

2000 Mathematics Subject Classification: 34A37, 34G20, 34K30, 34K99.

1. Introduction. In this paper, we study the existence, uniqueness, and continuous

dependence of a mild solution of a nonlocal Cauchy problem for impulsive functional-

differential evolution equation. Such problems arise in some physical applications as a

natural generalization of the classical initial value problems. The results for semilinear

functional-differential evolution nonlocal problem [2] are extended for the case of

impulse effect. We consider the nonlocal Cauchy problem in the form

u(t)=Au(t)+f (t,ut), t ∈ (0,a], t ≠ τk,u(τk+0

)=Qku(τk)≡u(τk)+Iku(τk), k= 1,2, . . . ,κ,

u(t)+(g(ut1 , . . . ,utp))(t)=φ(t), t ∈ [−r ,0],(1.1)

where 0< t1 < ···< tp ≤ a, p ∈N, A and Ik (k= 1,2, . . . ,κ) are linear operators acting

in a Banach space E; f , g, and φ are given functions satisfying some assumptions,

ut(s) := u(t+ s) for t ∈ [0,a], s ∈ [−r ,0], Iku(τk) = u(τk+0)−u(τk−0) and the

impulsive moments τk are such that 0< τ1 < τ2 < ···< τk < ···< τκ < a, κ ∈N.

Theorems about the existence, uniqueness, and stability of solutions of differen-

tial and functional-differential abstract evolution Cauchy problems were studied in

[1, 2, 3]. The results presented in this paper are a generalization and a continua-

tion of some results reported in [1, 2, 3]. We consider classical impulsive functional-

differential equation in the case of nonlocal condition, reduced to the classical initial

functional value problem.

As usual, in the theory of impulsive differential equations [4, 5] at the points of

discontinuity τi of the solution t�u(t) we assume that u(τi)≡u(τi−0). It is clear

that, in general, the derivatives u(τi) do not exist. On the other hand, according to the

first equality of (1.1) there exist the limits u(τi∓0). According to the above convention,

we assume u(τi)≡ u(τi−0).

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252 HAYDAR AKÇA ET AL.

Throughout, we assume that E is a Banach space with norm ‖·‖, A is the infinites-

imal generator of a C0 semigroup {T(t)}t≥0 on E, D(A) is the domain of A, and

M := supt∈[0,a]

{∥∥T(t)∥∥BL(E,E)}. (1.2)

Let f : [0,a]×C([−r ,0],E)→ E. Introduce the following assumptions:

(H1) for every w ∈ C([−r ,a],E) and t ∈ [0,a], f (·,wt)∈ C([0,a],E);(H2) there exists a constant L > 0 such that

∥∥f (t,wt)−f (t,wt

)∥∥E

≤ L1

∥∥w−w∥∥C([−r ,t],E) for w,w ∈ C([−r ,a],E), t ∈ [0,a],∥∥Ikv∥∥E ≤ L2‖v‖E for v ∈ E, k= 1,2, . . . ,κ,

L=max{L1,L2

}.

(1.3)

Let g : [C([−r ,0],E)]p → C([−r ,0],E). Then we have the following assumptions:

(H3) there exists a constant K > 0 such that

∥∥(g(wt1 , . . . ,wtp))(t)−(g(wt1 , . . . ,wtp

))(t)∥∥≤K∥∥w−w∥∥C([−r ,a],E) (1.4)

for w,w ∈ C([−r ,a],E), t ∈ [−r ,0];(H4) assume that φ∈ C([−r ,0],E).A function u∈ C([−r ,a],E) satisfying the following conditions:

u(t)= T(t)φ(0)−T(t)[(g(ut1 , . . . ,utp))(0)]

+∫ t

0T(t−s)f (s,us)ds+ ∑

0<τk<tT(t−τk

)Iku

(τk), t ∈ [0,a],

u(t)+(g(ut1 , . . . ,utp))(t)=φ(t), t ∈ [−r ,0),

(1.5)

is said to be a mild solution of the nonlocal Cauchy problem (1.1).

2. Existence and uniqueness of a mild solution

Theorem 2.1. Suppose that assumptions (H1)–(H4) are satisfied and

M[K+L(a+1)

]< 1. (2.1)

Then the impulsive nonlocal Cauchy problem (1.1) has a unique mild solution.

Proof. The mild solution of the impulsive system (1.1) with nonlocal condition

can be written in the form

u(t;φ)= (Fu)(t), (2.2)

IMPULSIVE FUNCTIONAL-DIFFERENTIAL EQUATIONS . . . 253

where

(Fw)(t) :=

φ(t)−(g(wt1 , . . . ,wtp))(t), t ∈ [−r ,0),

T(t)φ(0)−T(t)[(g(wt1 , . . . ,wtp))(0)]

+∫ t

0T(t−s)f (s,ws

)ds+

∑0<τk<t

T(t−τk

)Ikw

(τk), t ∈ [0,a],

(2.3)

such that w ∈ C([−r ,a],E) and F : C([−r ,a],E)→ C([−r ,a],E). Now, we show that

F is a contraction mapping on C([−r ,a],E). Therefore,

(Fw)(t)−(Fw)(t) :=

(g(wt1 , . . . ,wtp

))(t)−(g(wt1 , . . . ,wtp

))(t)

for w,w ∈ C([−r ,a],E), t ∈ [−r ,0),T(t)

[(g(wt1 , . . . ,wtp

))(0)−(g(wt1 , . . . ,wtp

))(0)]

+∫ t

0T(t−s)[f (s,ws

)−f (s,ws)]ds

+∑

0<τk<tT(t−τk

)[Ikw

(τk)−Ikw(τk)]

for w,w ∈ C([−r ,a],E), t ∈ [0,a].

(2.4)

From (2.4), we have

∥∥(Fw)(t)−(Fw)(t)∥∥≤ ∥∥T(t)∥∥·∥∥(g(wt1 , . . . ,wtp))(0)−(g(wt1 , . . . ,wtp

))(0)∥∥

+∫ t

0

∥∥T(t−s)∥∥·∥∥f (s,ws)−f (s,ws

)∥∥ds+

∑0<τk<t

∥∥T(t−τk)∥∥·∥∥Ikw(τk)−Ikw(τk)∥∥(2.5)

for w,w ∈ C([−r ,a],E), t ∈ [0,a]. Because of (2.5), in view of (1.2), and applying

assumptions (H1)–(H4) we obtain

∥∥(Fw)(t)−(Fw)(t)∥∥≤MK∥∥w−w∥∥C([−r ,a],E)+ML1

∫ t0

∥∥w−w∥∥C([−r ,a],E)ds+ML2

∥∥w(τk)−w(τk)∥∥E≤ (MK+MaL1+ML2

)∥∥w−w∥∥C([−r ,a],E)≤M[K+L(a+1)

]·∥∥w−w∥∥C([−r ,a],E)

(2.6)

for w,w ∈ C([−r ,a],E), t ∈ [0,a], which implies that

∥∥Fw−Fw∥∥C([−r ,a],E) ≤ β∥∥w−w∥∥C([−r ,a],E), w,w ∈ C([−r ,a],E), (2.7)

where β :=M[K+L(a+1)]. The operator F satisfies all the assumptions of the Banach

contraction theorem, and therefore, in the space C([−r ,a],E) there is only one fixed

point of F and this is the mild solution of the nonlocal Cauchy problem with impulse

effect. This completes the proof of the theorem.


3. Continuous dependence of a mild solution

Theorem 3.1. Suppose that the functions f , g, and Ik(u), k= 1, 2, . . . ,κ, satisfy the

assumptions (H1)–(H4) and M[K+L(a+1)] < 1. Then, for each φ1,φ2 ∈ C([−r ,a],E),and for the corresponding mild solutions u1, u2 of the problems,

u(t)=Au(t)+f (t,ut), t ∈ (0,a], t ≠ τk,u(τk+0

)=Qku(τk)≡u(τk)+Iku(τk), k= 1,2, . . . ,κ,

u(t)+(g(ut1 , . . . ,utp))(t)=φi(t) (i= 1,2), t ∈ [−r ,0],(3.1)

the following inequality holds:

∥∥u1−u2

∥∥C([−r ,a],E) ≤MeaML(1+ML)κ

{∥∥φ1−φ2

∥∥C([−r ,0],E)+K

∥∥u1−u2

∥∥C([−r ,a],E)

}.

(3.2)

Additionally, if

K <e−aML(1+ML)−κ

M, (3.3)

then

∥∥u1−u2

∥∥C([−r ,a],E) ≤

MeaML(1+ML)κ1−KMeaML(1+ML)κ

∥∥φ1−φ2

∥∥C([−r ,0],E). (3.4)

Proof. Assume that φi ∈ C([−r ,0],E) (i= 1,2) are arbitrary functions and let ui(i= 1,2) be the mild solutions of problem (3.1). Then

u1(t)−u2(t)= T(t)[φ1(0)−φ2(0)

]−T(t){[g((u1

)t1 , . . . ,

(u1)tp

)](0)−[g((u2

)t1 , . . . ,

(u2)tp

)](0)}

+∫ t

0T(t−s)[f (s,(u1

)s)−f (s,(u2

)s)]ds

+∑

0<τk<tT(t−τk

)[Iku1

(τk)−Iku2

(τk)]

(3.5)

for t ∈ [0,a] and

u1(t)−u2(t)=φ1(t)−φ2(t)−{[g((u2)t1 , . . . ,

(u2)tp

)](t)−[g((u1

)t1 , . . . ,

(u1)tp

)](t)}

(3.6)

for t ∈ [−r ,0). From (3.5), (1.2), and using (H2) we get

∥∥u1(ξ)−u2(ξ)∥∥≤M∥∥φ1−φ2

∥∥C([−r ,0],E)+MK

∥∥u1−u2

∥∥C([−r ,a],E)

+ML1

∫ ξ0

∥∥u1−u2

∥∥C([−r ,s],E)ds+ML2

∑0<τk<ξ

∥∥u1(τk)−u2

(τk)∥∥E

≤M∥∥φ1−φ2

∥∥C([−r ,0],E)+MK

∥∥u1−u2

∥∥C([−r ,a],E)

+ML1

∫ t0

∥∥u1−u2


∑0<τk<t

∥∥u1(τk)−u2

(τk)∥∥E

(3.7)

IMPULSIVE FUNCTIONAL-DIFFERENTIAL EQUATIONS . . . 255

for 0≤ ξ ≤ t ≤ a. With this result, by virtue of (H3) it follows that

supξ∈[0,t]

∥∥u1(ξ)−u2(ξ)∥∥

≤M∥∥φ1−φ2

∥∥C([−r ,0],E)+MK

∥∥u1−u2

∥∥C([−r ,a],E)

+ML1

∫ t0

∥∥u1−u2


∑0<τk<t

∥∥u1(τk)−u2

(τk)∥∥E

(3.8)

for t ∈ [0,a]. At the same time, by (3.6) and (H3) we have

∥∥u1(t)−u2(t)∥∥≤M∥∥φ1−φ2

∥∥C([−r ,0],E)+MK

∥∥u1−u2

∥∥C([−r ,a],E) (3.9)

for t ∈ [−r ,0). Formulas (3.8) and (3.9) imply that∥∥u1(t)−u2(t)

∥∥≤M∥∥φ1−φ2

∥∥C([−r ,0],E)+MK

∥∥u1−u2

∥∥C([−r ,a],E)

+ML{∫ t

0

∥∥u1−u2

∥∥C([−r ,s],E)ds+

∑0<τk<t

∥∥u1(τk)−u2

(τk)∥∥E

}.

(3.10)

Applying Gronwall’s inequality for discontinuous functions (see [5]), from (3.10) it

follows that

∥∥u1(t)−u2(t)∥∥C([−r ,a],E) ≤

{M∥∥φ1−φ2

∥∥C([−r ,0],E)

+MK∥∥u1−u2

∥∥C([−r ,a],E)

}eaML(1+ML)κ

(3.11)

and therefore, (3.2) holds. Inequality (3.4) is a consequence of (3.2). This completes

the proof of the theorem.

Remark 3.2. If K = κ = 0, then (3.2) is reduced to the classical inequality

∥∥u1(t)−u2(t)∥∥C([−r ,a],E) ≤MeaML

∥∥φ1−φ2

∥∥C([−r ,0],E), (3.12)

which is characteristic for the continuous dependence of the semilinear functional-

differential evolution Cauchy problem with the classical initial condition.

Acknowledgement. The authors would like to thank King Fahd University of

Petroleum and Minerals, Department of Mathematical Sciences for providing excellent

research facilities. The present research was accomplished during the stay of the third

author at Fatih University, Istanbul, Turkey.

References

[1] H. Akça and V. Covachev, Periodic solutions of impulsive systems with delay, Funct. Differ.Equ. 5 (1998), no. 3-4, 275–286.

[2] L. Byszewski and H. Akça, On a mild solution of a semilinear functional-differential evolu-tion nonlocal problem, J. Appl. Math. Stochastic Anal. 10 (1997), no. 3, 265–271.

[3] , Existence of solutions of a semilinear functional-differential evolution nonlocal prob-lem, Nonlinear Anal. 34 (1998), no. 1, 65–72.

[4] V. Lakshmikantham, D. D. Baınov, and P. S. Simeonov, Theory of Impulsive DifferentialEquations, Modern Applied Mathematics, vol. 6, World Scientific Publishing, NewJersey, 1989.


[5] A. M. Samoılenko and N. A. Perestyuk, Impulsive Differential Equations, World ScientificSeries on Nonlinear Science. Series A: Monographs and Treatises, vol. 14, WorldScientific Publishing, New Jersey, 1995.

Haydar Akça: Department of Mathematical Sciences, King Fahd University ofPetroleum and Minerals, Dhahran 31261, Saudi Arabia

E-mail address: [email protected]

Abdelkader Boucherif: Department of Mathematical Sciences, King Fahd Universityof Petroleum and Minerals, Dhahran 31261, Saudi Arabia


Valéry Covachev: Institute of Mathematics, Bulgarian Academy of Sciences, Sofia1113, Bulgaria


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