A Contribution to the Study of Functional Differential Equations with Impulses

Math. Nachr. 218 (2000), 49 – 60

A Contribution to the Study of Functional Differential Equa-

tions with Impulses

By Daniel Franco of Madrid, Eduardo Liz of Vigo, Juan J. Nieto of Santiago de Com-

postela and Yuri V. Rogovchenko of Gazimagusa

(Received February 19, 1998)

(Revised Version July 8, 1999)

Abstract. A periodic boundary value problem for a special type of functional differential equa-

tions with impulses at fixed moments is studied. A comparison result is presented that allows to

construct a sequence of approximate solutions and to give an existence result. Several particular

cases are considered.

1. Introduction

Differential equations with impulses are a basic tool to study evolution processes thatare subjected to abrupt changes in their state. For instance, many biological, physicaland engineering applications exhibit impulsive effects (see the monographs [13, 21] andthe references cited therein). In particular, in optimal control theory, we may observeimpulsive behavior of solutions of some problems [2, 3]. It should be noted that therecent progress in the development of the qualitative theory of impulsive differentialequations has been stimulated primarily by a number of interesting applied problems.We mention here the Kruger –Thiemer model for drug distribution [13], stabilizationof an inverted pendulum [16], the model for the heat conduction and vibration of astring with impulsive self – support [19], the model of population growth where theimpulses are used to prevent the extinction or unbounded growth of a species [1], thecompetition model with abrupt harvesting or epidemics [10], and this list is still to becompleted.The simpler case of impulsive differential equations is that in which impulse effects

1991 Mathematics Subject Classification. Primary: 34A37; Secondary: 34K10.Keywords and phrases. Functional differential equations, differential equations with impulses,

maximum principle, upper and lower solutions.

50 Math. Nachr. 218 (2000)

occur at fixed moments of the time, that is,

u′(t) = f(t, u(t)) , t ∈ [0, T ] , t �= tk , k = 1 , 2 , . . . p ,

u(t+k)

= Ik(u(tk)) , k = 1 , 2 , . . . , p ,

where 0 < t1 < · · · < tp < T .In general, the impulse functions Ik are used to regulate the process (for instance,

the control of the amount of drug ingested by a patient at certain moments in themodel for drug distribution [13] or the permanence of population density in the modelsof population dynamics [1]). It is of interest to consider the situation when at theinstant tk the function Ik depends not only on the value u(tk), but also on some otherparameters. For example, considering the model of drug distribution, one can takeinto account the maximum value of the drug concentration umax corresponding to thethreshold dose of drugs that can be ingested at one time to avoid overdose or fataldose. Another option is to deal with the minimum value of u corresponding to thelowest necessary dose that produces the minimal medical effect. Finally, one can beinterested in the mean value of u on the interval (tk−1, tk) corresponding to the averagedose ingested over past interval of time with a weighting factor which gives informationabout how much emphasis should be given to the amount of drugs ingested at earliertimes to determine the present effect of the medicine taken.On the other hand, functional differential equations have many applications in the

natural sciences (see, for instance, [7, 8, 11, 18]). It is well known that one of thedeficiencies of modeling population dynamics with the ordinary differential equationsis that the birth rate is considered to act simultaneously whereas it is reasonable toassume that there may be a delay due to the finite gestation period or because of alengthy maturation period. On the other hand, incorporation of delay into differentialequations may be also caused by a finite time required for the conversion of consumednutrient to viable biomass or dispersion of the population from one patch to its nearestneighbors.The importance of the investigation of periodic problems for the delay differential

equations can be easily seen also from the observation that there are many acutephysiological diseases where the initial symptoms are manifested by an alteration orirregularity in a control system which is normally periodic, or by the onset of an os-cillation in a hitherto non –oscillatory process. Such physiological diseases have beencalled dynamical diseases by Glass and Mackey [5, 6] who have made a partic-ular study of several important examples like Cheyne –Stokes respiration, a humanrespiratory aliment manifested by an alteration in the regular breathing pattern, orregulation of hematopoiesis, the formation of blood cell elements in the body. Wealso remark that clinical data show [18] that one of manifestations of leukemia is theperiodic oscillations observed in, for example, the white cell count.In consequence, the study of impulsive functional differential equations (IFDEs) is

of great interest both from the theoretical and practical point of view. The goal ofmany control systems is to maintain a quantity at a constant level. Therefore, in themodel of the formation of blood cells or that of drug distribution, one can use impulses(that, as we have already mentioned, may have rather complicated structure involvingfunctional dependence) to maintain a constant value of the total level of cells through

Franco/Liz/Nieto/Rogovchenko, Functional Differential Equations with Impulses 51

the change of the level of oxygen in the blood or the concentration of drugs in thebody of the patient.However, the development of the general theory of IFDEs is not very rapid due to

several theoretical and technical problems. It should be noted that even a mere notionof solution for a general IFDE is to be defined. One of the reasons for additionaldifficulties that arise in comparison to impulsive differential equations (IDEs) withoutfunctional arguments involves the state of an IFDE. Whereas the state at the fixed timeof a one – dimensional IDE is a single real number, the state of an IFDE at the fixedtime is given by a function and therefore the state of an IFDE is infinite –dimensional.It is also known that in certain FDEs the increase of the delay results in the transitionto chaos which may lead to serious difficulties with the study of the correspondingIFDE.In this paper, we deal with a periodic boundary value problem and we consider

different types of functional dependence not only for the impulse action, but alsofor the right – hand side of the differential equation that governs the process. Thiswork was intended as a partial contribution to the theory of IFDEs. We also remarkthat this work complements and improves recent results on differential equations withmaxima [9, 22].The paper is organized as follows. First, we prove a new maximum principle for a

general problem. Then we show that it is possible to construct monotone sequencesconverging to the extremal solutions of the periodic problem in a sector by using themethod of upper and lower solutions coupled with the monotone iterative technique(see [12, 20]).

2. Preliminaries

Let J = [0, T ], 0 = t0 < t1 < · · · < tp < tp+1 = T . We introduce the followingspaces of functions:

PC(J) ={u : J → IR : u is continuous for any t ∈ J \

{t1, . . . , tp

};

u(0+), u(T−), u(t+k ), u(t−k ) exist, and u

(t−k)= u

(tk), k = 1, . . . , p

},

and

PC1(J) = {u ∈ PC(J) : u is continuously differentiable for anyt ∈ J \

{t1, . . . , tp

}; u′(0+), u′(T−), u′(t+k ), u′(t−k ) exist, k = 1, . . . , p,

}.

PC(J) and PC1(J) are Banach spaces with the norms

‖u‖PC(J) = sup{|u(t)| : t ∈ J} and ‖u‖PC1(J) = ‖u‖PC(J) + ‖u′‖PC(J) .

For u ∈ PC(J), we consider the functions

uk : Jk −→ IR , k = 1 , 2 , . . . , p+ 1 ,

where Jk = [tk−1, tk] and uk(t) = u(t) if t ∈ (tk−1, tk], uk(tk−1) = u(t+k−1

). Then

‖u‖PC(J) = sup{‖uk‖C(Jk) : k = 1, 2, . . . , p+ 1

}, and, in this sense, PC(J) is equiv-

alent to∏p+1

k=1C(Jk), where C(Jk) is the Banach space of real continuous functionsdefined on Jk, endowed with the usual supremum norm.

52 Math. Nachr. 218 (2000)

If u, v ∈ PC(J) satisfy u(t) ≤ v(t), t ∈ J , then we write u ≤ v and we define thesector

[u, v] = {z ∈ PC(J) : u ≤ z ≤ v} .

Next we introduce the problem we shall study,

u′(t) = f(t, u(t), [ψkuk](t)

), t ∈ int (Jk) , k = 1 , 2 , . . . , p+ 1 ,(2.1)

u(t+k)

= Ik([φkuk](tk)

), k = 1 , 2 , . . . , p ,(2.2)

u(0) = u(T ) ,(2.3)

where ψk : C(Jk) → C(Jk) is continuous for k = 1, 2, . . . , p+ 1, φk : C(Jk) → C(Jk)and Ik : IR → IR are continuous for k = 1, 2, . . . , p, and f : J × IR × IR → IR iscontinuous at each point (t, x, y) ∈

(J \{t1, t2, . . . , tp

})× IR× IR. We assume that for

k = 1, 2, . . . , p and for all x, y ∈ IR there exist the limits

limt→t−

k

f(t, x, y) = f(tk, x, y) and limt→t+

k

f(t, x, y) .

By a solution of (2.1) – (2.3) we mean a function u ∈ PC1(J) satisfying (2.1) –(2.3).

3. Maximum principle

In order to prove a maximum principle which will permit us to develop the monotonemethod for (2.1) – (2.3), we require the following assumptions on functions ψk, k =1, 2, . . . , p+ 1:

(H1) For each k = 1, 2, . . . , p+1, ψk satisfies a Lipschitz condition, i. e., there existsan R > 0 such that

‖ψkx− ψky‖ ≤ R ‖x− y‖ for all x, y ∈ C(Jk) .

(H2) There exists a constant L > 0 such that

[ψkx](t) ≤ L maxs∈[tk−1,t]

x(s) , for any x ∈ C(Jk) and for all t ∈ Jk ,

k = 1 , 2 , . . . , p+ 1 .

Lemma 3.1. Let ck ∈ IR, k = 1, 2, . . . , p be constants, b ∈ PC(J), and suppose that(H1) holds. Then the periodic boundary value problem

u′(t) = −Mu(t) −N [ψkuk](t) + b(t) , t ∈ int (Jk) , k = 1 , 2 , . . . , p+ 1 ,

u(t+k)

= ck , k = 1 , 2 , . . . , p ,

u(0) = u(T ) ,

(3.1)

where M > 0, N ≥ 0 are constants, has a unique solution u ∈ PC1(J).


Proof . We consider the following initial value problems for k = 2, 3, . . . , p+ 1:

u′(t) = −Mu(t)−N [ψkuk](t) + b(t) , t ∈ (tk−1, tk] ,

u(t+k−1

)= ck−1 .

(3.2)

It is easy to see that to find a solution of (3.2) is equivalent to get a fixed point forthe operator Tk : C(Jk) → C(Jk) defined by

[Tku](t) = ck−1 +∫ t

tk−1

{b(s) −Mu(s) −N [ψkuk](s)} ds , t ∈ Jk .

Now, consider the following norm in C(Jk):

‖y‖∗ = sup{e−A(t−tk−1) |y(t)| : t ∈ Jk

}, A = M +NR,

and note that it is equivalent to the usual supremum norm.Then, for x, y ∈ C(Jk) and t ∈ Jk, we have that

∣∣[Tkx− Tky](t)∣∣ =

∣∣∣∣∣∫ t

tk−1

{M(y(s) − x(s)) +N

([ψkyk](s)− [ψkxk](s)

)}ds

∣∣∣∣∣≤∫ t

tk−1

eA(s−tk−1) {M ‖x− y‖∗ +N ‖ψkxk − ψkyk‖∗}ds

≤ A ‖x− y‖∗∫ t

tk−1

eA(s−tk−1) ds

=(eA(t−tk−1) − 1

)‖x− y‖∗ .

Thus,e−A(t−tk−1)

∣∣(Tkx− Tky)(t)∣∣ ≤

(1− e−A(t−tk−1)

)‖x− y‖∗ ,

for any x, y ∈ C(Jk) and t ∈ Jk.This implies that

‖Tkx− Tky‖∗ ≤(1− e−A(tk−tk−1)

)‖x− y‖∗ .

Thus Tk is a contractive operator and Banach’s Contraction Principle assures thatit has a unique fixed point. This gives us the solution u(t) for t ∈ (t1, T ] and itdetermines a value u(T ) that we use as the initial value for the problem

u′(t) = −Mu(t) −N [ψ1u1](t) + b(t) , t ∈ [0, t1] ,

u(0) = u(T ) .(3.3)

Now, we can guarantee, by the same reasoning, existence and uniqueness of solutionfor (3.3).Finally, we construct a solution u of (3.1) by taking the solution of (3.2) on each

interval (tk−1, tk], k = 2, . . . , p+ 1 and the solution of (3.3) on [0, t1].

54 Math. Nachr. 218 (2000)

The uniqueness of solution for (3.1) is derived from the uniqueness of solution foreach problem in (3.2) and (3.3). ✷

Lemma 3.2. Assume that (H2) holds and let u ∈ PC1(J) satisfy the followinginequalities:

u′(t) ≥ −Mu(t)−N [ψkuk](t) , t ∈ int (Jk) , k = 1 , 2 , . . . , p+ 1 ,

u(t+k)

≥ 0 , k = 1 , 2 , . . . , p ,

u(0) ≥ u(T )

(3.4)

with M > 0, N ≥ 0 and

LNσeMσ ≤ 1 , where σ = max{tk+1 − tk : k = 0, 1, . . . , p} .

Then u(t) ≥ 0 for every t ∈ J .

Proof . Let v(t) = u(t)eMt. Then (3.4) yields

v′(t) ≥ −N [ψ∗kvk](t) , t ∈ int (Jk) , k = 1 , 2 , . . . , p+ 1 ,

v(t+k)

≥ 0 , k = 1 , 2 , . . . , p ,(3.5)

with ψ∗k : C(Jk) → C(Jk) defined by

[ψ∗kv](t) = [ψkv](t)eMt where v(t) = e−Mtv(t) .

Note that u(t) and v(t) have the same sign for every t ∈ J . On the other hand,observe that

v(T ) ≥ 0 =⇒ v(0) ≥ 0 .(3.6)

When N = 0, it follows immediately that v ≥ 0 on J (see Lemma 3.1 in [17]).Then we assume that N > 0. We show first that v(T ) ≥ 0. Indeed, let v(T ) < 0.

Since v(t+p)≥ 0, for any z ∈

(tp, T

)we have that

v′(z) ≥ −N[ψ∗

p+1vp+1

](z)

≥ −NLeMz maxs∈[tp,z]

{vp+1(s)e−Ms

}≥ −NLeM(z−tp) max

s∈[tp,z]vp+1(s)

≥ −NLeMσ maxs∈[tp,z]

vp+1(s) .

(3.7)

Define λ = maxs∈[tp,T ] vp+1(s) ≥ 0. If λ = 0 then vp+1

(tp)= v

(t+p)= 0. Applying

the Mean Value Theorem on the interval [tp, T ], we obtain a point z ∈(tp, T

)such

thatv(T ) − v

(t+p)

= v′(z)(T − tp

).

From this and (3.7), we obtain the contradiction

0 > v(T ) − v(t+p)

= v′(z)(T − tp

)≥ 0 .


Thus, λ > 0 and we can apply the Mean Value Theorem on the interval [µ, T ], wherevp+1(µ) = λ, to obtain a point z ∈ (µ, T ) such that

v′(z) =v(T ) − v(µ)

T − µ<

−λT − µ

≤ −λσ

,

with v(µ) = v(t+p)if µ = tp.

This, together with (3.7), implies that

LNσeMσ > 1 ,

but the latter is impossible by hypothesis.In consequence, v(T ) ≥ 0. Then, by (3.6), v(0) ≥ 0. Furthermore, v

(t+k)≥ 0 for

k = 0, 1, . . . , p.If there exists r1∈ [0, T ], where v(r1)<0, and assume for certainty that r1∈

(tj−1, tj

].

For z ∈ Jj, using the same argument, we have

v′(z) ≥ −NLeMσ maxs∈[tj−1,z]

vj(s) .(3.8)

Define λ=maxs∈[tj−1,r1 ]vj(s). As above, we may consider that λ>0. Let µ∈(tj−1, r1

)be such that vj(µ) = λ. Then, for some z ∈ (µ, r1), we obtain that

v′(z) =v(r1) − v(µ)

r1 − µ<

−λr1 − µ

≤ −λσ

.

From the latter inequality and (3.8), it follows that LNσeMσ > 1. As before, ifµ = tj−1, then v(µ) = v

(t+j−1

). ✷

4. Monotone iterative technique

In the sequel, we assume the following hypotheses.(H3) For x, y ∈ PC(J) with x ≤ y,

f(t, y(t), [ψkyk](t)

)−f(t, x(t), [ψkxk](t)

)≥ −M(y(t)−x(t))−N ([ψkyk](t)− [ψkxk](t)),

where M > 0, N ≥ 0, t ∈ Jk, k = 1, 2, . . . , p+ 1.(H4) The function ψk satisfies

ψku− ψkv ≤ ψk(u− v) for v, u ∈ C(Jk) , v ≤ u , k = 1 , 2 , . . . , p+ 1 .

(H5) Ik, φk, k = 1, 2, . . . , p are nondecreasing.We note that we consider the partial order in C(Jk) defined by v ≤ u if v(t) ≤ u(t)

for every t ∈ Jk.

Definition 4.1. A function α ∈ PC1(J) is said to be a lower solution of (2.1) –(2.3) if it satisfies

α′(t) ≤ f(t, α(t), [ψkαk](t)

), t ∈ int (Jk) , k = 1 , 2 , . . . , p+ 1 ,

α(t+k)

≤ Ik([φkαk](tk)

), k = 1 , 2 , . . . , p ,

α(0) ≤ α(T ) .

56 Math. Nachr. 218 (2000)

Analogously, β ∈ PC1(J) is an upper solution for (2.1) – (2.3) if

β′(t) ≥ f(t, β(t), [ψkβk](t)

), t ∈ int (Jk) , k = 1 , 2 , . . . , p+ 1 ,

β(t+k)

≥ Ik([φkβk](tk)

), k = 1 , 2 , . . . , p ,

β(0) ≥ β(T ) .

Theorem 4.2. Suppose that (H1) – (H5) hold. Assume also that the constants M ,N and L satisfy

NLσeMσ ≤ 1 ,and there exist α and β, lower and upper solutions of (2.1) – (2.3) respectively, suchthat α ≤ β on J . Then there exist monotone sequences {αn}, {βn} with α0(t) = α(t),and β0(t) = β(t), t ∈ J , such that

limn→∞αn(t) = α(t) , lim

n→∞βn(t) = β(t)

uniformly on J , where α, β are the minimal and maximal solutions of (2.1) – (2.3)in the sector [α, β], i. e., if u is a solution of (2.1) – (2.3) such that α ≤ u ≤ β on J ,then α ≤ u ≤ β on J .

Proof . Let η ∈ [α, β]. We consider the following problem

u′(t) = F (t, ηk(t))−Mu(t)−N [ψkuk](t) , t ∈ int (Jk) , k = 1, 2, . . . , p+ 1 ,

u(t+k)

= Ik([φkηk](tk)

), k = 1 , 2 , . . . , p ,

u(0) = u(T )

(4.1)

withF (t, ηk(t)) = f

(t, η(t), [ψkηk](t)

)+Mη(t) +N [ψkηk](t) .

Lemma 3.1 permits us to assure that this problem has a unique solution u ∈ PC1(J).Then we can define the operator

B : [α, β] −→ PC(J)

by [Bη](t) = u(t), t ∈ J , where u is the unique solution of (4.1).This operator possesses the following two properties.(a) B([α, β]) ⊂ [α, β].(b) η ≥ ξ ⇒ B(η) ≥ B(ξ).To prove (a), we consider the function v(t) = u(t) − α(t), where u is the solution

of (4.1). By the definition of lower solution, we have that v(0) ≥ v(T ). Moreover, inview of (H3) and (H4),

v′(t) = u′(t) − α′(t)≥ f


)− f(t, α(t), [ψkαk](t)

)+M(η(t) − u(t)) +N

([ψkηk](t)− [ψkuk](t)

)≥ M(α(t)− η(t)) +N

([ψkαk](t)− [ψkηk](t)

)+M(η(t) − u(t)) +N

([ψkηk](t)− [ψkuk](t)

)= −M(u(t) − α(t)) −N

([ψkuk](t)− [ψkαk](t)

)≥ −Mv(t) −N [ψkvk](t) ,


for t ∈ int (Jk), k = 1, 2, . . . , p+ 1.Now, using (H5), we have that

v(t+k)

= u(t+k)−α(t+k)

≥ Ik([φkηk](tk)

)−Ik

([φkαk](tk)

)≥ 0 , k = 1 , 2 , . . . , p .

Thus, Lemma 3.2 implies that v = u− α ≥ 0. Analogously one can show that u ≤ β.Now, to prove (b), let us consider u = B(η), w = B(ξ) with η ≥ ξ and v = u − w.

Then

v′(t) = (u−w)′(t)= f


)− f(t, ξ(t), [ψkξk](t)

)+M(η(t) − u(t) +w(t)− ξ(t))+ N

([ψkηk](t)− [ψkuk](t) + [ψkwk](t)− [ψkξk](t)

)≥ −M(η(t) − ξ(t)) −N

([ψketak](t)− [ψkξk](t)

)+M(η(t) − u(t) +w(t)− ξ(t))+ N

([ψkηk](t)− [ψkuk](t) + [ψkwk](t)− [ψkξk](t)

)≥ −Mv(t) −N [ψkvk](t)

for t ∈ int (Jk), k = 1, 2, . . . , p+ 1 and

v(t+k)

= u(t+k)−w

(t+k)

= Ik([φkηk](t)

)− Ik

([φkξk](t)

)≥ 0 , k = 1 , 2 , . . . , p .

It follows from Lemma 3.2 that v(t) ≥ 0 for all t ∈ J , which proves that B ismonotone increasing. Now, starting at α0 = α and β0 = β, we can recursively definetwo sequences {αn} and {βn} by

αn = Bαn−1 ; βn = Bβn−1 , n ≥ 1 .

From the properties of B it follows that {αn} is increasing, {βn} is decreasing, andαn ≤ βn for all n ≥ 0.In view of the definition of (4.1), we have that {α′

n} and {β′n} are bounded in PC(J).

Hence, {αn} converges to α uniformly on J , and {βn} converges to β uniformly on J .Moreover, employing the integral representation of the solutions of (4.1), given in

the proof of Lemma 3.1, we conclude, using the uniform convergence, that α and βare solutions of the problem (2.1) – (2.3).Finally, to show that α and β are the extremal solutions of (2.1) – (2.3) in [α, β], let

u be any solution of (2.1) – (2.3) in [α, β]. It is obvious that α0 ≤ u ≤ β0. Then, byinduction and using property (b), one can easily see that

αn ≤ u ≤ βn , n ≥ 0 .

Thus, passing to the limit, we conclude thatα(t) ≤ u(t) ≤β(t), t ∈ J . This completesthe proof. ✷

Remark 4.3. Theorem 4.2 is also valid if we replace hypothesis (H5) by the follow-ing weaker condition:

(H5’) Ik ◦ φk : C(Jk) → IR are nondecreasing functions for k = 1, 2, . . . , p.

58 Math. Nachr. 218 (2000)

5. Particular cases

In this section, we present some examples of functions ψk and φk which satisfy thehypotheses required for our results.Case 1: If the functions ψk, k = 1, 2, . . . , p+1 and φk, k = 1, 2, . . . , p are the identity,

then problem (2.1) – (2.3) is the periodic boundary value problem for the first orderordinary differential equation with impulses at fixed moments:

u′(t) = f(t, u(t)) , t ∈ J \{t1, t2, . . . , tp

},(5.1)

u(t+k)

= Ik(u(tk)

), k = 1 , 2 , . . . , p ,(5.2)

u(0) = u(T ) ,(5.3)

and Theorem 4.2 gives us the result of Theorem 3.1 in [17].Case 2: If the functions ψk, k = 1, 2, . . . , p+ 1 are the identity, we have a problem

with functional dependence only in the impulse functions. Now, if we suppose thatIk, k = 1, 2, . . . , p, are nondecreasing, then Theorem 4.2 is valid for the problem

u′(t) = f(t, u(t)) , t ∈ J \{t1, t2, . . . , tp

},(5.4)

u(t+k)

= Ik

(sup

s∈(tk−1,tk]

u(s)

), k = 1 , 2 , . . . , p ,(5.5)

u(0) = u(T ) .(5.6)

Note that (5.5) is generated by the functions

[φku] (t) = maxs∈[tk−1,t]

u(s) , t ∈ Jk , k = 1 , 2 , . . . , p .

Instead of (5.5), we may consider also

u(t+k)

= Ik

(inf

s∈(tk−1,tk]u(s)

), k = 1 , 2 , . . . , p ,

u(t+k)

= Ik

(∫ tk

tk−1

u(s)ds

), k = 1 , 2 , . . . , p ,

that are generated by the relations

[φku] (t) = mins∈[tk−1,t]

u(s) , t ∈ Jk , k = 1 , 2 , . . . , p ,

[φku] (t) =∫ t

tk−1

u(s) ds , t ∈ Jk , k = 1 , 2 , . . . , p ,

respectively.Note that in the previous situations, the impulse action at the instant tk depends

respectively on the maximum, minimum and the mean value on the interval (tk−1, tk].


Finally, we can also consider functional dependence in the nonlinearity f as, forinstance, in the following two cases:Case 3: Let us define the function ψk by

[ψku](t) = maxs∈[tk−1,t]

u(s) .

It is not difficult to see that this function satisfies conditions (H1), (H2), and (H4)with R = L = 1. Thus, we improve results in [9, 22].Case 4: Now, let us consider the function

[ψku](t) =∫ t

tk−1

u(s) ds .

In this situation, the conditions (H1) and (H2) are satisfied with

R = L = max{|tk − tk−1| : k = 1, 2, . . . , p+ 1} .

In addition, (H4) is trivially verified since ψk is a linear function.In summary, we have shown the validity of the monotone iterative technique to

approximate the extremal solutions for a variety of impulsive problems where theimpulses involve a functional dependence on the history of the solution. Our resultsapply to a special type of impulsive differential equations with maximum, as well asto some particular impulsive integro – differential equations.

Acknowledgements

Research partially supported by D. G.E. S. I. C., project PB97 – 0552 (Spain), and by IN-TAS, project 96 – 0915.

Yuri V. Rogovchenko is on leave from Institute of Mathematics, National Academy ofSciences, 252601 Kyiv, Ukraine.

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Departamento de Matematica Aplicada IE.T. S. I. IndustrialesUniversidad Nacional de Educacion a DistanciaMadridSpain

Departamento de Matematica AplicadaE.T.S. I. TelecomunicacionUniversidad de VigoSpain

Departamento de Analisis MatematicoFacultad de MatematicasUniversidad de Santiago de CompostelaSpain

Department of MathematicsEastern Mediterranean UniversityGazimagusa, TRNCMersin 10Turkey

Documents

A Contribution to the Study of Functional Differential Equations with Impulses