Asymptotic theory for bi-dimensional linear impulsive differential systems, nonelliptic case

Nonlinear Analysis 35 (1999) 425–447

Asymptotic theory for bi-dimensionallinear impulsive di�erential systems,

nonelliptic case

Samuel Castillo∗, Manuel Pinto1

Departamento de Matem�aticas, Facultad de Ciencias, Universidad de Chile, Casilla 653,Santiago, Chile

Received 29 April 1996; accepted 12 November 1996

Keywords: Asymptotic theory; Bi-dimensional linear impulsive di�erential systems; Nonelliptic

1. Introduction

To study the asymptotic integration of the solutions of second order linear ordinarydi�erential equations in the nonelliptic case, Hartman and Wintner, see [1, p. 375],introduce the solutions of type Z of a binary system

v′(t)= �(t)z(t);

z′(t)= (t)v(t):(1)

System (1) on 0≤ t¡! (≤∞) is called of type Z at t=! if for every solution(v; z) of (1)

z(!)= limt→!

z(t) exists and z(!) 6=0 for some solution: (2)

This is equivalent to the existence of linearly independent solutions (vj; zj) (j=0; 1),

∗ Present address: Universidad de B��o B��o, Concepcion, Chile.1 Supported by D.T.I. E 3063-9222 Universidad de Chile and Fondecyt 0839-93 and Fundaci�on Andes for

the �rst author.

0362-546X/98/$19.00 ? 1998 Elsevier Science Ltd. All rights reserved.PII S0362-546X(97)00587 -7

426 S. Castillo, M. Pinto / Nonlinear Analysis 35 (1999) 425–447

such that

z0(!)= 0 and z1(!)= 1:

These asymptotic behaviors of zj imply corresponding behaviors of vj and hence ofsolutions of system (1).We wish to extend Hartman–Wintner’s study to di�erential systems with �xed mo-

ments of impulse e�ect. Let �=(tn)n∈N a sequence in [0; ![ such that tn¡tn+1 andtn→! as n→+∞. Consider the 2× 2 impulsive di�erential systems:

[I�] :

v′= �(t)z; t 6= tn;z′= (t)v; t 6= tn;�v(tn)= �(tn)z(tn); z(t−n )= z(tn);

�z(tn)= (tn)v(tn); v(t−n )= v(tn);

where ; � : [0; ![−�→C are continuous functions and �(tn); (tn)∈C. The basic theoryand the large �eld of applications of the impulsive di�erential systems can be seen in[2–6]. A solution (v; z) of [I�] is a piecewise continuous vector on [0; ![, continuous anddi�erentiable on (ti; ti+1], where the dynamic is determined by the di�erential equation,and for t= ti a change by jump of the solution (v; z) occurs in terms of the impulsiveequation.

De�nition 1. We will say that system [I�] is of type Z if: For any solution (v; z) of[I�]; z(!) := lim z(t) as t→!, exists and z(!) 6=0 for some of these solutions.

Let (v1; z1) and (v; z) be solutions of the impulsive system [I�]. Since v; z arecontinuous and di�erentiable functions in the intervals (ti; ti+1], we get the constantWronskian ci:

z1(t)v(t)− z(t)v1(t)= ci for t ∈ (ti; ti+1]; (3)

because the left term has derivative zero. We will prove that the constant ci satis�es:

ck = c0k∏i=1

(1− (ti)�(ti)):

We consider a condition, which insures that ci 6=0, for all i∈N.

Assumption 1. �(tn) (tn) 6=1 for any n∈N.

No condition about the convergence of cn as n→+∞ is needed.

Remark 1. Under Assumption 1, [I�] is of type Z if and only if there exist linearlyindependent solutions (vj; zj) (j=0; 1) of [I�] such that

z0(t)→ 0 and z1(t)→ 1 as t→!: (4)

S. Castillo, M. Pinto / Nonlinear Analysis 35 (1999) 425–447 427

We will establish impulsive integrability conditions which ensure that system [I�] beof type Z . These conditions are practically necessary for system [I�] be of type Z andallow to prove the existence of solutions (vj; zj) (j=0; 1) of [I�] satisfying

(a) v0∼ ckt and z0 = o(ckt =∫ t

�(s) ds+kt∑i=1

�(ti)); as t→!; and

(b) v1∼∫ t

�(s) ds+kt∑i=1

�(ti) and z1∼ 1; as t→!:

Moreover these integrability conditions guarantee the existence of principal andnonprincipal solutions of system [I�]. For that we have de�ned this concept for theimpulsive system [I�]. Finally knowing the principal and nonprincipal solution of anonperturbed impulsive system we obtain the asymptotic integration of a general per-turbed system. Several examples and applications are shown. The results are a faith-ful extension of the ordinary Hartman–Wintner results to the impulsive situation. Theproofs show a new “impulsive calculus”. To our knowledge there is no similar workin the literature [2–10].Let u : [0; !)→C be a continuous function on (ti; ti+1] such that u(t+i )= lims→t+i

u(s) (the right lateral limit) exists. The vector space of such functions u will be denotedby C+� ([0; ![). De�ne the linear operators:

�u(t)= u(t+)− u(t) (5)

and

Eu(t)= u(t+) (6)

We have the product rule:

�(uv)= (�u)v+ (Eu)�v (7)

and

�(uv

)=(�u)v− u�v

〈v〉2 ; (8)

where

〈u〉2 = uEu: (9)

These “di�erential” operators arise an interesting di�erential–integral calculus whoseplay will appear throughout the paper. As an example, we have:

∫ t

T

�(s)B2(s)

ds+k∑i=n

�(ti)〈B〉2(ti) =

1B(T )

− 1B(t)

;


where k = kt de�ned by tk¡t≤ tk+1; tn−1¡T ≤ tn¡tk¡t≤ tk+1 and

B(t)=kt∑i=1

�(ti) +∫ t

t0�(s) ds:

2. Preliminary facts

We will need some Lemmata. First a generalization of Gronwall inequality to im-pulsive case:

Lemma 1 ([2]). Let u; v be nonnegative functions on [0; ![ and continuous except fort= ti; satisfying

u(t)≤ �+∫ t

t0u(s)v(s) ds+

∑ti∈(t0 ; t)

u(ti)v(ti):

Then

u(t)≤ � exp∫ t

t0v(s) ds+

∑ti∈(t0 ; t)

v(ti)

: (10)

Now, we establish some integrability conditions insuring that a system [I�] is oftype Z :

Lemma 2. Assume that:

(∑)

+∞∑i=1

| (ti)|¡+∞; (∫)∫ !

0| (s)| ds¡+∞;

(∫ ∫)∫ !

0| (t)|

(∫ t

0|�(s)| ds

)dt¡+∞;

( ∫ ∑) ∫ !

0| (t)|

kt∑

j=1

|�(tj)| dt¡+∞;

(∑∫

)+∞∑i=1

| (ti)|∫ ti

0|�(s)| ds¡+∞;

(∑∑

)+∞∑i=1

| (ti)|i−1∑j=1

|�(tj)|¡∞; (11)


where kt ∈N is de�ned by tkt¡t≤ tkt+1; or more generally; let the conditions

�(s)= sups≤t¡!

∣∣∣∣∣+∞∑k=nt+1

(tk) +∫ !

t (�) d�

∣∣∣∣∣¡∞∫ !

|�(s)|�(s) ds¡+∞;+∞∑i=1

|�(ti)|�(ti)¡+∞;(12)

where nt ∈N is de�ned by tnt ≤ t¡tnt+1. Then [I�] is of type Z .

Remark. Interchanging the integration orders we have that condition (11) impliescondition (12).

Proof of Lemma 2. Let n= kt and i0 = kt0 + 1. From [I�], we have v(t)= v(ti0 ) +∑ni=i0 �v(ti) +

∫ tt0�z and then

v(t)= v(ti0 ) +n∑j=i0

�(tj)z(tj) +∫ t

t0�z;

z(t)= z(t0) +n∑i=1

(ti)v(ti) +∫ t

t0 v:

Therefore

z(t) = z(t0) + v(t0)n∑i=i0

(ti) + v(t0)∫ t

t0 +

n∑i=i0

(ti)i−1∑j=i0

�(tj)z(tj) +n∑i=i0

(ti)∫ ti

t0�z

+∫ t

t0 (s)

ks∑j=i0

�(tj)z(tj) +∫ t

t0 (�)

[ ∫ s

t0�z]ds: (13)

So, interchanging the integration limits in this expression, we get

z(t) = z(t0) + v(t0)

(nt∑i=i0

(ti) +∫ t

t0

)+

n∑j=i0

�(tj)z(tj)

n∑i=j+1

(ti) +∫ t

tj

+∫ t

t0�(s)z(s)

[n∑

i=ns+1

(ti) +∫ t

s

]ds: (14)

Now, for t≥ r and k ≥ i,∣∣∣∣∣n∑

nr+1

(tk) +∫ t

r

∣∣∣∣∣ ≤∣∣∣∣∣+∞∑nr+1

(tk) +∫ !

r

∣∣∣∣∣+∣∣∣∣∣+∞∑k=nt+1

(tk) +∫ !

t

∣∣∣∣∣ ≤ 2�(r)


Thus,

|z(t)| ≤ �+n∑j=i0

|�(tj)| |z(tj)|�(tj) +∫ t

t0|�(s)| |z(s)|�(s) ds; (15)

where �= |z(t0)|+ 2|v(t0)|�(t0).From Lemma 1 and (15) we get that z(t) is bounded. Applying this fact to (14), we

have that z(t) converges as t→!. In order to show that z(!) 6=0 for some solutionof [I�], choose the initial conditions v(t0)= 0; z(t0)= 1 in (14). Thus

z(t)− 1 =∫ t

t0�(s)

(n∑

i=ns+1

(ti) +∫ t

s

)ds+

n∑j=i0

�(tj)

(n∑j=i

(ti) +∫ t

ti

)

+∫ t

t0�(s)

n∑j=ns+1

(tj) +∫ t

s

(z(s)− 1) ds

+n∑i=i0

�(ti)

(n∑j=i

(tj) +∫ t

ti

)(z(ti)− 1):

Taking �= �(t0)=∫ !t0|�(s)|�(s) ds+∑∞

i=i0 |�(ti)|�(ti), from Lemma 1 and (12), z(w)is su�ciently near to 1 as t0→!. So, taking t0 su�ciently near to !, we have z(!) 6=0.

Lemma 3. Let ; �∈C+� ([0; ![) with constant sign and Sign =Sign ; Sign �=Sign �. Then (

∑); (∫); (∫ ∫); (∫ ∑

); (∑∫

); (∑∑

) are true if [I�] is of type Z .

Proof. Let (v; z) be a solution of [I�] such that z(!)= 1. So, with t0 = T and k = kT ,(13) implies

z(t) = z(T ) + v(T )n∑i=k

(ti) + v(T )∫ t

T +

n∑i=k

(ti)i−1∑j=k+1

�(tj)z(tj)

+n∑i=k

(ti)∫ ti

T�z +

∫ t

T (s)

[ ∫ s

T�z]ds+

∫ t

T (s)

ks∑i=k+1

�(ti)z(ti) ds: (16)

We can suppose, without loss of generality, that v(T )= 0 for some T . If this doesnot hold, we can add to (v; z) a suitable multiple of a solution (v0; z0) 6=0 of [I�] withz0(!)= 0. In fact, v0 = 0 cannot hold, for then z0≡ 0.By (16) with v(T )= 0 and T so large that 1

2 ≤ z(t)≤ 32 for t≥T and since any

double integral have the same sign, then (∑∑

); (∑∫

); (∫ ∑

) and (∫ ∫) hold. Using

that and taking T such that v(T ) 6=0, we get (∫ ) and (∑).Lemma 4. We have

ck = c0k∏i=1

(1− (ti)�(ti)): (17)


Proof. Using the piecewise continuity of v and z we get

z1(t+i )v(t+i )− v1(t+i )z(t+i )= ci;

z1(ti)v(ti)− v1(ti)z(ti)= ci−1:(18)

So, by (5) and [I�] we have

�(z1v)(ti) =�z1(ti)v(t+i ) + z1(ti)�v(ti)

= (ti)v1(ti)v(t+i ) + �(ti)z(ti)z1(ti):

Similarly, �(v1z)(ti)= (ti)v(ti)v1(t+i ) + �(ti)z1(ti)z1(ti). Then

ci − ci−1 =�(z1v)−�(v1z)= (ti)(v1(ti)v(t+i )− v(ti)v1(t+i ))= (ti)(v1(ti)v(t+i )− v1(ti)v(ti) + v1(ti)v(ti)− v(ti)v1(t+i ))= (ti)(v1(ti)�v(ti)− v(ti)�v1(ti))= (ti)�(ti)(v1(ti)z(ti)− v(ti)z1(ti))=− (ti)�(ti)ci−1:

So, ci=(1− (ti)�(ti))ci−1 and hence (17) follows.

Lemma 5. If the system [I�] satis�es the summability hypotheses (12), we have;

limt→!

B(t)�(t)= 0

where �(t) is as in Lemma 2 and B(t)=∑kt

j=i0 �(tj) +∫ tt0�(s) ds.

Proof. Let H (t)=B(t)�(t) and, for T ∈]t0; t[ �xed, h1(t)=B(T )�(t) and h2(T )=∑+∞j=kT+1 �(tj)�(tj) +

∫ !T �(s)�(s) ds. Then,

|H (t)| ≤ h1(t) + kT∑j=kt+1

�(tj) +∫ t

T�(s) ds

�(t)

So, |H (t)| ≤ h1(t) + h2(T ). From (12), �(t)→ 0 as t→!. Hence, limt→!

h1(t)= 0. So, limt→!|H (t)| ≤ h2(T ).From (12), we obtain h2(T )→ 0 as T→!. Since T is arbitrary, limt→!

H (t)= 0.

Lemma 6. We have

�(z(ti)z1(ti)

)=

(ti)〈z1(ti)〉2 · ci−1

and (19)

�(v(ti)v1(ti)

)=− �(ti)

〈v1(ti)〉2 · ci−1:


Moreover;

z(t)z1(t)

=z(t0)z1(t0)

+n∑i=1

ci−1

( (ti)

〈z1(ti)〉2 +∫ ti

ti−1

z21

)+ cn

∫ t

tn

z21

and (20)

v(t)v1(t)

=v(t0)v1(t0)

+n∑i=1

ci−1

(�(ti)

〈v1(ti)〉2 +∫ ti

ti−1

�v21

)+ cn

∫ t

tn

�v21:

Proof. By the quotient rule (8) and [I�],

�(z(ti)=z1(ti)) = [�z(ti)z1(ti)−�z1(ti)z(ti)]=〈z1(ti)〉2

= [ (ti)v(ti)z1(ti)− (ti)v1(ti)z(ti)]=〈z1(ti)〉2

= (ti)ci−1=〈z1(ti)〉2:Similarly,

�(v(ti)=v1(ti)) = ((�v(ti)v1(ti)− v(ti)�v1(ti))=〈v1(ti)〉2

= (�(ti)z(ti)v1(ti)− �(ti)z1(ti)v(ti))=〈v1(ti)〉2

=−�(ti)ci−1=〈v1(ti)〉2:In a similar way we have(

zz1

)′= ciz21; t ∈ (ti; ti+1]; (21)

from where

z(t)z1(t)

=z(t0)z1(t0)

+n∑i=1

�(z(ti)z1(ti)

)+

n∑i=1

∫ ti

ti−1

ci−1 z21

+ cn

∫ t

tn

z21: (22)

Similarly

v(t)v1(t)

=v(t0)v1(t0)

+n∑j=1

�(v(ti)v1(ti)

)+

n∑i=1

∫ ti

ti−1

�v21ci−1 + cn

∫ t

tn

�v21

(23)

where tn¡t≤ tn+1. Thus, (19) implies (20).

Lemma 7. Assume that conditions (12) holds. Let (vi; zi) (i=0; 1) be solutions of[Iv] satisfying (4) and suppose

v1(t) ∼kt∑i=i0

�(ti) +∫ t

t0�(s) ds: (24)


Then there is a constant M such that

|z0(t)| ≤Mckt�(t) (25)

Proof. Since (v0; z0) is a solution of [Iv] and z0(!)= 0, from (14) we have

z0(t)=−v0(t)�(t)−+∞∑

i0=nt+1

�(tj)�(tj)z(tj)−∫ !

t�(s)z(s)�(s) ds; (26)

where

�(t) :=+∞∑i=nt+1

(ti) +∫ !

t (s) ds:

From (3), we obtain

v0(t)= v1(t)z0(t)z1(t)

+cktz1(t)

:

Let

�(t)= 1 +v1(t)z1(t)

· �(t):

Replacing this in (26), we get

z0(t)�(t)=− cktz1(t)

�(t)−+∞∑j=kt+1

�(tj)�(tj)z0(tj)−∫ !

t�(s)z0(s)�(s)z0(s) ds:

Thus,

|z0(t)|�(t)≤ |ckt ||z1(t)| |�(t)|+

+∞∑j=kt+1

�(tj)|�(tj)||z0(tj)|+∫ !

t�(s)|�(s)||z0(s)| ds:

Since

|�(t)− 1| ≤ 1|z1(t)|

(kt∑i=i0

�(ti) +∫ t

t0�

)�(t);

by Lemma 5 and z1(!)= 1; �(t)→ 1 as t→!. Let M1 = supt≥t01

�(t)|z1(t)| and M2 =supt≥t0

1|�(t)| . Then

|z0(t)| ≤M1|ckt |�(t) +M2+∞∑j=kt+1

�(tj)|�(tj)||z0(tj)|+M2∫ !

t�(s)|�(s)||z0(s)| ds:

(27)


Now, since

| (tkt )| ≤∣∣∣∣∣(+∞∑i=kt

(ti) +∫ !

tkt

)−(

+∞∑i=kt+1

(ti) +∫ !

tkt

)∣∣∣∣∣≤ 2�(�)|ckt−1|;for all �∈ (tkt−1; tkt ], we have

|ckt |�(t) = |1− (tkt )�(tkt )||ckt−1|�(t) ≤ [�(t) + | (tkt )|(�(tkt )�(tkt ))]|ckt−1|≤ (1 + �kt )�(�)|ckt−1|;

where �k =2�(tk)�(tk). Then if T ≥ t,

|ckT |�(T )≤[

+∞∏i=kt+1

(1 + �i)

]|ckt |�(tkt )≤M3|ckt |�(t); (28)

where M3 = exp(∑+∞

i=i0 �i)¡∞, by (12). By applying (28) in (27), we obtain

|z0(T )| ≤M4|ckt |�(t) +M2+∞∑j=kt+1

�(tj)|�(tj)||z0(tj)|+M2∫ !

t�(s)|�(s)||z0(s)| ds;

where M4 =M1 ·M3. Let z0(t)= supT≥t |z0(T )|. Then

z0(t)≤M4|ekt |�(t) +M2 +∞∑j=kt+1

�(tj)|�(tj)|+∫ !

t�(s)|�(s)| ds

z0(t):

Thus,

z0(t)�1(t)≤M4|ckt |�(t);where

�1(t)= 1−M2 +∞∑j=kt+1

�(tj)�(tj) +∫ !

t�(s)�(s) ds

:

Let M =suptM4�1(t)

. Then,

|z0(T )| ≤ z0(t)≤M |ckt |�(t); for all T ≥ t:

3. Asymptotic integration and principal solutions

Lemma 8. Let �; ; �; as in Lemma 2. Assume also that �; �≥ 0 and∫ !

�=∞; or+∞∑i=1

�(ti)=+∞: (29)


Then [I�] has two solutions (vj; zj) (j=0; 1) satisfying

(a) v0∼ ckt and z0 = o(ckt =(∫ t

� +kt∑i=1

�(ti))); and

(b) v1∼∫ t

� +kt∑�(ti) and z1∼ 1:

Proof. By Lemma 2, system [I�] has a solution (v1; z1) such that z1(!)= 1. We have

v1(t)= v1(t0) +kt∑i=1

�(ti)z1(ti) +∫ t

t0�(s)z1(s) ds:

Let r and R be two numbers such that r¡1¡R. Let T¿t0 such that r¡z1(t)¡R,for t≥T . Let

�1 = v(t0) +N0∑i=1

�(ti)z(ti) +∫ T

t0�z;

�t =kt∑

i=N0+1

�(ti)z(ti) +∫ t

T�z;

where N0 = kT . Then v(t)= �1 + �t . Let

D1 = v1(t0) +N0∑i=1

�(ti) +∫ T

t0�;

Dt =kt∑

i=N0+1

�(ti) +∫ t

T�:

By (29), Dt→+∞ as t→+∞. Moreover�1 + rDtD1 + Dt

≤ �1 + �tD1 + Dt

≤ �1 + RDtD1 + Dt

:

So,

r≤ limt→!

v(t)D1 + Dt

≤R

R¿1 and r¡1 arbitrary imply limt→!v(t)D1+Dt

=1. Then v1(t)∼D1 + Dt as t→!,i.e. part (b) is proved.Next, we will prove part (a). From Lemma 7, we have

B(t)|ckt |

|z0(t)| ≤MB(t)�(t):


From Lemma 5, limn→+∞B(t)|ckt | |z0(t)|=0, i.e.

z0(t)= o(cktB(t)

):

Finally, part (b) implies

z0(t)v1(t)= o(ckt ):

Now, from (3), v0(t)= 1z1(t)(z0(t)v1(t) + ckt ) and since z1(∞)= 1, we have

v0(t)∼ ckt .

Consider now the impulsive system

[A�] :

x′=p(t)y; t 6= tn;y′= q0(t)x; t 6= tn;�x= p(tn)y; t= tn;

�y= q0(tn)x; t= tn;

and a perturbed system:

[B�] :

u′=p(t)w; t 6= tn;w′=(q0(t) + q(t))u; t 6= tn;�u= p(tn)w; t= tn;

�!=(q0(tn) + q(tn))u; t= tn;

where p1; q0 : [0; ![−�→R; q : [0; ![−�→R are continuous functions and p(tn); q0(tn)∈R, q(tn)∈C.

De�nition. We will say that [A�] is nonoscillatory if there exists T ∈ [0; ![ such thatx(t)x(T )¿0 for all t≥T , for all solution nonzero (x; y) of [A�].

Lemma 9. Assume that system [A�] is nonoscillatory and satis�es en−1p(tn)¿0 andektp(t)¿0 where

eN =

[N∏n=1

(1− p(tn)q0(tn))]e0: (30)

Then; [A�] has a pair of solutions (xj; yj) (j=0; 1) such that

+∞∑n=1

en−1

(p(tn)

〈x0(tn)〉2 +∫ tn

tn−1

p(s)x0(s)2

ds

)=+∞ (31)

and+∞∑n=1

en−1

(p(tn)

〈x1(tn)〉2 +∫ tn

tn−1

p(s)x1(s)2

ds

)¡+∞ (32)


Proof. Let (x; y) be a solution of [A�]. Since [A�] is nonoscillatory 〈x(t)〉2 is positivefor t large enough. From en−1p(tn)¿0 and ektp(t)¿0, we have that

F(t) :=kt∑n=1

en−1

(p(tn)〈x(tn)〉2 +

∫ tn

tn−1

px2

)+ ekt

∫ t

tkt

px2

is an increasing function. Hence, F(!)= �¡∞ or F(!)=+∞.From Lemma 6 and en 6=0, there exists a solution (�x; �y) of [A�] such that (x; y) and

(�x; �y) are linearly independent,

�x(t)= x(t)(F0 + F(t)); F0 constant (33)

and

x(t)= �x(t)(G0 + G(t)); G0 constant; (34)

where

G(t)=kt∑n=1

en−1

(p(tn)〈 �x(tn)〉2 +

∫ t

tn−1

p

�x2

)+ ekt

∫ t

tkt

p

�x2:

If F(!)=+∞, we can take x= x0 and we obtain (31). From (33), x(t)�x(t) → 0 as t→!and, from (34), G(t)→−G0 as t→!. So, we can take, x1 = �x and we obtain (32).If F(!)= �, we can take x0(t)= �x(t)− (�+F0)x(t). Then, x0(t)�x(t) → 0 as t→!. Thus,

taking x= x0 and x= x1 in (33), then (31) and (32) are satis�ed.

Lemma 10. For a system [A�] nonoscillatory; conditions (31) and (32) are equivalentto the existence of a pair of linearly independent solutions (xj; yj); j=0; 1; of [A�]such that:

x0(t)x1(t)

→ 0 as t→!:

Functions x1 and x0 satisfying Lemma 9 are called, respectively, principal and non-principal solutions.

Corollary 1. Under conditions of Lemma 8, any system [I�] has principal and non-principal solutions.

We will establish the conditions for the existence of solutions (uj; !j) j=0; 1 of[B�] such that uj ∼ xj, j=0; 1, where (x0; y0) are principal and nonprincipal solutionsof [A�], respectively.First, we will see the relations between the solutions of [B�] and of [A�].


Lemma 11. Let (v; z) be a solution of the impulsive equation

[C�] :

z′=x2

en−1qv; t 6= tn;

v′= en−1px2z; t 6= tn;

�z=〈x〉2enqv; t= tn;

�v= en−1p〈x〉2 z; t= tn:

If n= kt and

u= xv

w= vy + en−1z=x

where (x; y) is a solution of [A�].Then (u; w) is a solution of the perturbed system [B�].

Proof. Consider u= xv and w= en−1 zx + vy. Then,

u′ = x′v+ xv′=pyv+ xen−1p=x2v

=p(yv+ en−1z=x)=pw

and

w′ = v′y + vy′ + en−1z′x − zx′=x2

= en−1pzy=x2 + vq0x + en−1z′x=x2 − en−1 py z=x2

= vq0x + vqx=(q0 + q)xv=(q0 + q)u:

For the impulsive part we have,

�u(tn) = x(t+n )�v(tn) + �x(tn) · v(tn)

= x(t+n )en−1p(tn)〈x(tn)〉2 · z(tn) + p(tn)y(tn)v(tn)

= en−1p(tn)z(tn)x(tn)

+ p(tn)y(tn)v(tn)

= p(tn)(en−1

z(tn)x(tn)

+ y(tn)v(tn))= p(tn)w(tn)


and

�w(tn) =�(v(tn)u(tn) + en−1

z(tn)x(tn)

)

= v(t+n )�y(tn) + (�v(tn)) · y(tn) + �(en−1

z(tn)x(tn)

)

= v(t+n )q0(tn)x(tn) + en−1p(tn)〈x(tn)〉2 z(tn)y(tn)

+�(en−1z(tn))x(tn)− en−1z(tn)�x(tn)

〈x(tn)〉2

= v(t+n )q0(tn)x(tn) +�(en−1z(tn))

〈x(tn)〉2 x(tn);

since �x= py. But, from (30)

�(en−1z(tn)) = en�z(tn) + z(tn)�en

= en�z(tn)− p(tn)q0(tn)en−1z(tn):

So,

�w(tn) = v(t+n )q0(tn)x(tn) +en�z(tn)− p(tn)q0(tn)en−1z(tn)

〈x(tn)〉2 x(tn)

= v(t+n )q0(tn)x(tn)+(en�z(tn)〈x(tn)〉2 − p(tn)q0(tn)en−1

z(tn)〈x(tn)〉2

)x(tn)

= v(t+n )q0(tn)x(tn)+en

〈x(tn)〉2en

q(tn)v(tn)x(tn)

〈x(tn)〉2

− p(tn)q0(tn)en−1z(tn)

〈x(tn)〉2 x(tn)

= v(t+n )q0(tn)x(tn) + q(tn)x(tn)v(tn)− p(tn)q0(tn)en−1z(tn)

〈x(tn)〉2 x(tn)

= (�v(tn))q0(tn)x(tn) + (q0(tn) + q(tn))x(tn)v(tn)


〈x(tn)〉2 x(tn)


= en−1p(tn)q0(tn)〈x(tn)〉2 z(tn)x(tn) + (q0(tn) + q(tn))x(tn)v(tn)


〈x(tn)〉2 x(tn)

= (q0(tn) + q(tn))x(tn)v(tn)= (q0(tn) + q(tn))u(tn):

To state the next theorem, take:

(t)= x20(t)q(t)=en−1; �(t)= en−1p(t)=x20(t); t 6= tn; (tn)= 〈x0(tn)〉2q(tn)=en; �(tn)= en−1p(tn)=〈x0(tn)〉2;

(35)

where en is de�ned in (30) and the system

[C�] :

z′= v; t 6= tn;v′= �z; t 6= tn;�z= v; t= tn;

�v= �z; t= tn:

Assume that ; �; and � satisfy conditions (12). Then system [C�] is of type Z .Remark that (

∫) and (

∑) are smallness conditions for q and q, respectively. If

system [A�] is nonoscillatory then en−1p(tn)¿0 and ektp(t)¿0 and from (31) weobtain

(III)+∞∑n=1

�(tn)+∫ !

�(s) ds=+∞:

Theorem 1. Assume that system [A�] is nonoscillatory; let (x0; y0) and (x1; y1) beits respective nonprincipal and principal solutions. Assume that ; �; and givenby (35), satisfy the summability conditions (12). If en−1p(tn)¿0 and ektp(t)¿0 andp(tn)(q0(tn)+ q(tn)) 6=1 for all n∈N; then there exist solutions (uj; !j) (j=0; 1) of[B�] such that

u0∼ ckt x0; u1∼ x1;u′jpuj

=x′jpxj

+ o(en−1|x0x1|

); (36)

�uj(tn)p(tn)uj(tn)

=�xj(tn)p(tn)xj(tn)

+ o(en−1|x0x1|

);

where en is given by (30) and

cN =N∏n=1

1− p(tn)(q0(tn)+ q(tn))1− p(tn)q0(tn)

:


Proof. By (12) and (III), [C�] satis�es the hypotheses of Lemma 8. Hence, thereexist solutions (zj; vj); j=0; 1 satisfying (a) and (b) of Lemma 8. Then, by Lemma 11u0 = x0v0; u1 = x0v1 are solutions of [B�]. So, then t→! implies v0∼ 1 and henceu0∼ ckt x0 and

v1∼kt∑i=1

ei−1

(p

〈x0〉2 (ti)+∫ ti

ti−1

px20

)+ ek

∫ t

tk

px20=:F(t) (37)

implies

u1∼ x0F(t)= x1:Moreover, u= x0v implies u′= x′0v+ x0v

′. Then u′=u= x′0=x0 + v′=v and from (35),

u′

pu=x′0px0

+zx20ven−1: (38)

So,

u′0=u0 = x′0=px0 + z0en−1=x

20v0:

By applying of Lemma 8, v0∼ ckt ; z0 = 0 (ckt =F(t)) and x1 = x0F(t). Then we have

u0=pu0 = x′0=px0 + (en−1=x20) · o(1=F(t))

= x′0=px0 + o(en−1=|x0x1|):Now, taking u= u1; z= z1 and v= v1, in (38) we obtain

u′1=pu1 = x′0=px0 + z1en−1=x

20v1

= x′0=px0 + (1+ o(1))en−1=x0x1

since v1∼F(t) and z1 = 1+o(1). On the other hand,x′1px1

=x′0px0

+en−1x0x1

:

So, u′1=pu1 = x′1=px1− en−1=x0x1 + (1+ o(1))en−1=x0x1, and hence

u′1pu1

=x′1px1

+ o(en−1|x0x1|

)

as t→!. Moreover,

�u(tn)= v(tn)�x0(tn)+ x0(t+n )�v(tn):

Hence

�u(tn)u(tn)

=�x0(tn)x0(tn)

+x0(t+n )x0(tn)

· �v(tn)v(tn)


i.e.

�u(tn)p(tn)u(tn)

=�x0(tn)p(tn)x0(tn)

+z(tn)en−1

x0(tn)2 · v(tn) : (39)

In particular, taking in the last equality u= u0 and v= v0 we have

�u0(tn)p(tn)u0(tn)

=�x0(tn)p(tn)x0(tn)

+z0(tn)

p(tn)x0(tn)2en−1:

Since v0∼ 1, from x1 = x0F and z0 = o(1=F(t)) as t→!, we have

�u0pu0

=�x0px0

+ o(en−1|x0x1|

)

as t→!. Again by (39) we have

�u1pu1

=�x0px0

+z1x20v1

en−1:

Since z1∼ 1 and v1∼F(t),�u1pu1

=�x0px0

+1+ o(1)x0x1

en−1

as t→!. On the other hand, since x1 = x0F we have,

�x1px1

=�x0px0

+en−1x0x1

: (40)

In fact, �x1 = (�x0)F(tn)+ x0(t+n )p(tn)en−1 =〈x0(tn)〉2(tn)= (�x0(tn))F(tn)+ p(tn)en−1=x0(tn).So, dividing this relation by px1 = px0F; we obtain (40). Thus,

�u1pu1

=�x1px1

+ o(en−1x0x1

):

4. Examples and applications

Example 1. Consider for t ≥ t0 = 1:

z′= t−4v; t 6= ti;v′= tz; t 6= ti;�z(n)= n−�v(n); t= ti; �¿2;

�v(n)= z(n); t= ti:

(41)

It is easy to see that this system satis�es the integrability conditions of Lemma 8. ThenLemma 8 implies the existence of two solutions (vi; zi) (i=0; 1) such that for t→∞,

v0∼ 1 and z0 = o

(1∫ t

t0�(s) ds+

∑kti=1 �(ti)

)= o(

1t2=2+ kt

)


and, since cn=∏n(i− k−�)c0 converges,

v1∼∫ t

�(s) ds+kt∑i=1

�(ti)= t2=2+ kt and z1∼ 1:

The following is an example where cn does not converge.

Example 2. Consider the system

z(t)=−2(t+1)e−(t+1)2v(t); t 6= n;v(t)= z(t) t 6= n;�z(n)= (e−(n+1)

2 − e−n2 )v(n); t= n;

�v(n)= en2z(n); t= n:

This system satis�es the integrability conditions of Lemma 2. Here, cN+1 = (2−e−2(N+1))cN and cN is not convergent. So, Lemma 8 implies the existence of twosolutions (vi; zi)(i=0; 1) such that n→+∞

v0(t)∼kt∏i=1

(1− e−(2i+1)) and z0(t)= o

(∏kti=1 (1− e−(2i+1))t+∑kt

i=1 ei2

)

v1(t)∼ t+kt∑i=1

ei2

and z1(t)∼ 1:

Now, we will show some applications:

Theorem 2. Consider the equation:

(B′)

[u′=! �u(tn)=!(tn);

!′= q(t)u �!(tn)= q(tn)u(tn);(42)

where q : [t0;∞[−�→C is a continuous function and (q(tn))⊆C satisfy∫ +∞

t0t|q(t)| dt¡+∞ and

∑n≥1

mn|q(tn)|¡+∞;

with mn=max{tn; n} or more generally; the conditional summability given by

Q(r)=: supr≤s

∣∣∣∣∣+∞∑n=ks+1

q(tn)+∫ +∞

sq(t) dt

∣∣∣∣∣¡+∞ for all r ≥ t0;

(∫ ∑

) ′∫ +∞

t0Q(r) dr¡+∞ and (

∑∫) ′

+∞∑N=1

Q(N )¡+∞:


Then system (42) has two solutions; {u0; u1} such that as t→∞u0∼ 1; u′0 = o

(1

t+ kt

); �u0 = o

(1

t+ kt

)u1∼ t+ kt ; u′1 = 1+ o(1); �u1 = 1+ o(1):

(43)

Proof. In Theorem 1, we identify system [B�] with (B′) and system [A�] with

(A′)

[x′=y �x(tn)=y(tn);

y′=0 �y(tn)= 0:

We calculate the solutions of (A′). From x′=0 and �y(tn)= 0, we get y=y0; y0constant. From x′=y we have x(t)=y0t+dn, for t ∈ (tn; tn+1]. Then dn+1−dn=�x(tn)=y0. So, dn=y0kt +d′ and hence x(t)=y0(t+ kt)+d′. Thus a fundamental sys-tem of solutions is given by

x0(t)= 1 and x1(t)= t+ kt ; t ∈ (tk ; tk+1]:

Clearly x0 is principal and x1 nonprincipal. On the other hand, the sequence ofWronskian {ei}∞i=1 in (36) is en=(1− q)en−1 = en−1, from where we can take en=1.So the system identi�ed with [C�] is

(C′)

[z′= q(t)v �z(tn)= q(tn)v(tn);

v′= z �v(tn)= z(tn):

We have en−1p(tn)= 1= ektp(t)¿0 and q(tn) 6=0 for n large and the integrabilityconditions of Theorem 1 are satis�ed. So, from Theorem 1, (B′) has two solutions{u0; u1} such that

u0∼ ckt =[kt∏i=1

(1− q(ti))]c0∼ 1 and u1∼ t+ kt ;

u′juj=x′jxj+ o(

1|x0x1|

)= o(

1t+ kt

); j=0; 1; t =∈ �

and

�ujuj

=�xjxj+ o(

1t+ kt

)j=0; 1; t ∈ �: (44)

Since u0∼ 1 and x0 = 1, from (44) we have and

u′0 = o(

1t+ kt

)and �u0(tn)= o

(1

tn+ ktn

)= o(

1tn+ n− 1

):


Using (44) again and since u1∼ t+ kt as t→!, we get

u′1u1=

1t+ kt

+ o(

1t+ kt

)and

�u1(tn)u1(tn)

=1

tn+ n− 1 + o(

1tn+ n− 1

)

Hence u′1∼ 1 and �u1∼ 1, for n su�ciently large.

Theorem 3. Let �∈ (0; 1) and assume∫ +∞

t0|q(t)| dt¡+∞;

∑n≥1

|q(tn)|¡+∞

or more generally; the conditional summability:

Q(s) := supt≥s

∣∣∣∣∣+∞∑k=nt+1

w2k−1e−2�tk q(tk)+∫ ∞

tw2k�e−2��q(�) d�

∣∣∣∣∣¡+∞;+∞∑n=1

w−2(n−1)e2�tnQ(tn)¡+∞;∫ ∞

t0w−2kte2�tQ(t) dt¡+∞;

(45)

where w= 1− �1 + � . Then;

(B′)

[u′=! �u(tn)=!(tn)

!′=(�2 + q(t))u �!(tn)= (�2 + q(tn))u(tn)

has two solutions: {u0; u1} such thatu0∏kt

i=1 (1−!−3q(ti))∼ u′0

−� ∼ (1− �)kte−�t ∼ �u0

−�

u1∼ u′1�∼ (1+ �)kte�t ∼ �u1

�:

Proof. In Theorem 1 we identify (B′) with [B�] and [A�] with

(A′) :

[x′=y �x(tn)=y(tn);

y′= �2x �y(tn)= �2x(tn):

We are going to compute the solutions of (A′). They are in the form

x(t)= ane�t + bne−�t ; tn¡t ≤ tn+1and

y(t)= an�e�t − bn�e−�t :


Since �x(tn)= (�an)e�tn +(�bn)e−�tn and �x(tn)=y(tn), we obtain

(�an)e�tn +(�bn)e−�tn = an�e�tn − bn�e−�tn : (46)

From �y(tn)= (�an)�e�tn − (�bn)�e−�tn and �y(tn)= �2x(tn) we get(�an)e�tn − (�bn)e−�tn = �ane�tn + �bne−�tn : (47)

Adding (46) and (47), we have (�an)= �an or an= a0(1+ �)n and substracting (46)to (47), we have (�bn)= �bn or bn=(1− �)nb0. So

x(t)= a0(1+ �)kte�t + b0(1− �)kte−�t :Then (A′) has to solutions, respectively: x0(t)= (1− �)kte−�t and x1(t)= (1+ �)kte�t .Since en=(1− �2)en−1¿0 the system identi�ed with [C�] is:

(C′′) :

z′=w2kte−2�tq(t)v;

v′=w−2kte2�tz;

�z(tn)=w2n−1e−2�tn q(tn)v(tn);

�v(tn)=w−2(n−1)e2�tn z(tn):

The integrability conditions of Lemma 8 are the conditions (45). Since (1− �2)= en−1p(tn)¿0 and ektp(t)= (1− �2)kt¿0, we have that by Theorem 1 there exist twosolutions {u0; u1} such that

u0∼[kt∏i=1

(1−!−3q(ti))

](1− �)kte−�t and u1∼ (1+ �)kte�t ;

(ii) u′0u0=−�+ o( 1

(1− �2)kt ) or u′0∼−�u0 + o( 1

(1− �2)kt )(1− �)kte−�t . So,

u′0∼−�u as t→!:

Analogously, �u0(tn)∼−�u0(tn) as n→+∞ and(iii) u′= �u1 + o( 1

(1− �2)kt )(1+ �)kte�t or

u′1∼ �u1Analogously, �u1(tn)∼ �u1(tn) as n→+∞.

References

[1] P. Hartman, Ordinary Di�erential Equations, Wiley, Chichester, 1964.[2] D. Bainov, P. Simeonov, Systems with Impulse E�ect, Ellis Horwood, 1989.[3] D. Bainov, P. Simeonov, Asymptotic properties of the solutions of di�erential equations with impulse

e�ect, Atti. Sem. Mat. Fis. Univ. Modena, XXXVIII (1990) 19–27.[4] M. Pinto, Ghizzetti’s theorem for piecewise continuous solutions, Internat. J. Math. and Math. Sciences

17(2) (1994) 283–286.[5] M. Pinto, Impulsive integral inequalities, Libertas Math. 12 (1992) 505–516.


[6] M. Pinto, Conditional stability for piecewise continuous solutions of di�erential equations, J. Di�. Eqs.and Dyn. Sys. 2(3) (1994) 217–230.

[7] P. Gonz�alez, M. Pinto, Asymptotic behavior of solutions of impulsive di�erential systems, RockyMountain J. Math. 26(3) (1996) 165–173.

[8] R. Medina and M. Pinto, Uniform asymptotic stability of impulsive di�erential systems, Dyn. Syst.Appl. 5(1) (1996) 97–108.

[9] R. Naulin, M. Pinto, Quasi-diagonalization of linear impulsive systems and applications, J. Math. Anal.Appl. 208 (1997) 281–297.

[10] M. Pinto, Dichotomies for di�erential systems with impulse e�ect, in: V. Lakshmikantham (Ed.),Proceeding of First World Congress of Nonlinear Analyst, Tampa, 1992, Walter de Gruyter, 1995.

Documents

Asymptotic theory for bi-dimensional linear impulsive differential systems, nonelliptic case