Oscillation of the bounded solutions of impulsive differential-difference equations of second order

Applied Mathematics and Computation 114 (2000) 61±68 www.elsevier.nl/locate/amc

Oscillation of the bounded solutions of impulsive di€erential-di€erence equations of second order D.D. Bainov

a,*

, M.B. Dimitrova b, A.B. Dishliev

c

a

c

Medical University of So®a, P.O. Box 45, 1504 So®a, Bulgaria b Technical University, 8800 Sliven, Bulgaria University of Chemical Technologies and Metallurgy, So®a, Bulgaria

Abstract The present paper is devoted to a class of impulsive di€erential-di€erence equations of second order. The moments of impulse e€ect are ®xed ones. Sucient conditions are found for unboundedness of the solutions and oscillation of the bounded solutions. Ó 2000 Elsevier Science Inc. All rights reserved. Keywords: Oscillation of the bounded solutions; Impulsive di€erential-di€erence equations of second order

1. Introduction The impulsive di€erential equations form a mathematical apparatus for modelling of processes which at certain moments of their development undergo rapid changes. The duration of these changes is neglible in comparison to the whole duration of the process and that is why, it can be supposed that the changes take place by jumps. In the last years were published various monographs, devoted to the impulsive di€erential equations. We refer to the monographs [1,2].

*

Corresponding author. E-mail address: [email protected] (D.D. Bainov).

0096-3003/00/$ - see front matter Ó 2000 Elsevier Science Inc. All rights reserved. PII: S 0 0 9 6 - 3 0 0 3 ( 9 9 ) 0 0 1 0 2 - 2

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In the present paper we obtain sucient conditions for unboundedness of the solutions and oscillation of the bounded solutions of impulsive di€erentialdi€erence equations of second order. 2. Preliminary notes We consider the initial problem ÿ
0 r…t†y 0 …t† ÿ p…t†y…t ÿ h† ˆ 0; 0

0

t 6ˆ sk ; k ˆ 1; 2; . . . ;

0

Dy …sk † ˆ y …sk ‡ 0† ÿ y …sk † ˆ bk y…sk †; y…t† ˆ u…t†; ÿh 6 t 6 0;

…1† …2† …3†

where y…sk ÿ 0† ˆ y…sk ‡ 0† ˆ y…sk †; y 0 …sk ÿ 0† ˆ y 0 …sk †; h > 0; r; p : R‡ ! R‡ ; bk 2 R;

R‡ ˆ ‰0; 1†;

R‡ ˆ …0; 1†;

k ˆ 1; 2; . . . ;

u: ‰ÿh; 0Š ! R; s1 ; s2 ; . . . are the moments of impulse e€ect. We introduce the following conditions: H1: H2:

r; p 2 C…R‡ ; R‡ †: Z 1 ds ˆ ‡1: r…s† 0

H3:

0 < s1 < s2 <; . . . ;

H4:

bk P 0;

H5:

u 2 C 2 …‰ÿh; 0Š; R†:

lim sk ˆ ‡1:

k!1

k ˆ 1; 2; . . . :

De®nition 1. The function y : ‰ÿh; 1† ! R is said to be continuable to in®nity solution of problem (1), (2), (3) if 1. For ÿh 6 t 6 0 the equality y…t† ˆ u…t† is satis®ed. 2. If 0 < t 6 s1 , then the solution y…t† coincides with the solution of the problem without impulses (1) and (3). 3. If sk < t 6 sk‡1 ; k ˆ 1; 2; . . ., the solution of problem (1), (2), (3) coincides with the solution of the integro-di€erential equation

D.D. Bainov et al. / Appl. Math. Comput. 114 (2000) 61±68

Z

r…t†y 0 …t† ˆ r…sk †y 0 …sk ‡ 0† ‡

t

sk

63

p…s†y…s ÿ h† ds

ÿ
ˆ r…sk † y 0 …sk † ‡ bk y…sk † ‡

Z

t sk

p…s†y…s ÿ h† ds

with initial condition (3). The solution of the problem under consideration is a continuous function with piecewise continuous derivative, which has points of discontinuity s1 ; s2 ; . . .. At the points of discontinuity the function y 0 …t† is continuous from the left. The requirement of the initial function u…t† to admit continuous second derivative (condition H5) is due to necessity of preserving the class which contains the solution of the considered problem. De®nition 2. The solution y…t† of problem (1), (2), (3) is said to be oscillatory if for each a > 0 we have ft: y…t† > 0; t > ag 6ˆ ;;

ft: y…t† < 0; t > ag 6ˆ ;:

Otherwise, the solution y…t† is called nonoscillatory. 3. Main results Theorem 1. Let the conditions H1±H5 be ful®lled. Then: 1. If u…t† P 0, ÿh 6 t 6 0, u0 …0† > 0, then limt!‡1 y…t† ˆ ‡1: 2. If u…t† 6 0, ÿh 6 t 6 0, u0 …0† < 0, then limt!‡1 y…t† ˆ ÿ1: Proof. We shall prove Assertion 1. The proof of Assertion 2 is analogous. We integrate (1) from 0 to t…t P 0† and obtain r…t†y 0 …t† ˆ r…0†y 0 …0† ‡

X

  r…sk † y 0 …sk ‡ 0† ÿ y 0 …sk † ‡

0
ˆ r…0†u0 …0† ‡

0
Z ‡

X

tÿh 0

Z bk r…sk †y…sk † ‡

0 ÿh

Z

t 0

p…s†y…s ÿ h† ds

p…s ‡ h†u…s† ds

p…s ‡ h†y…s† ds:

We divide the last inequality to r…t†, integrate and obtain

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D.D. Bainov et al. / Appl. Math. Comput. 114 (2000) 61±68

Z

0

y…t† ÿ u…0† ˆ r…0†u …0† Z ‡

t 0

ds r…s†

Z

0

ÿh

0

t

ds ‡ r…s†

Z

t 0

p…u ‡ h†u…u† du ‡

Z

1 r…s†

0

t

ds r…s†

X 0
Z

0

! bk r…sk †y…sk †

sÿh

ds

p…u ‡ h†y…u† du ds:

Having in mind conditions H1, H2, H4, as well as the assumption that u…t† P 0, ÿh 6 t 6 0 and u0 …0† > 0, we deduce from the above equality that limt!‡1 y…t† ˆ ‡1:  Theorem 2. Let the following conditions be ful®lled: 1. Conditions H1±H5 are met. Rs R1 2. 0 R…s†p…s† ds ˆ ‡1, where R…s† ˆ 0 du=r…u†: Then all bounded solutions of problem (1), (2), (3) either tend to zero (as t ! ‡1), or oscillate. Proof. Let y…t† be a bounded, nonoscillatory solution of problem (1), (2), (3). For the sake of de®niteness, let us suppose that y…t† > 0 for t P t1 P 0. It is clear that y…t ÿ h† > 0 for t P t2 ˆ t1 ‡ h: Let k0 be the number for which the inequalities sk0 ÿ1 6 t2 < sk0 , s0 ˆ 0 are valid. Let t2 6 t < s and let t and s belong to one and the same interval of the next ones ‰t2 ; sk0 †; …sk0 ; sk0 ‡1 †; …sk0 ‡1 ; sk0 ‡2 †; . . . : Then, integrating (1) from t to s we obtain Z s p…u†y…u ÿ h† du > 0: r…s†y 0 …s† ÿ r…t†y 0 …t† ˆ t

It follows from the last inequality that the function r…t†y 0 …t† is increasing function on each of the intervals ‰t2 ; sk0 †; …sk0 ; sk0 ‡1 †; …sk0 ‡1 ; sk0 ‡2 †; . . .. It follows from (2) that r…sk ‡ 0†y 0 …sk ‡ 0† ÿ r…sk †y 0 …sk † ˆ r…sk †Dy 0 …sk † ˆ bk y…sk † P 0; k ˆ k0 ; k0 ‡ 1; . . . : Therefore, the function r…t†y 0 …t† is monotone increasing for t P t2 : The next cases are possible: Case 1. r…t2 †y 0 …t2 † > 0. Then, having in mind the fact that r…t†y 0 …t† is monotone increasing function for t P t2 , we deduce y 0 …t† P

r…t2 †y 0 …t2 † c ˆ ; r…t† r…t†

whence y…t† P y…t2 † ‡ c

Z t2

t

du : r…u†

t P t2 ;

c ˆ r…t2 †y 0 …t2 † > 0;

D.D. Bainov et al. / Appl. Math. Comput. 114 (2000) 61±68

65

The last inequality and condition H2 yield limt!‡1 y…t† ˆ ‡1, which contradicts the assumption that y…t† is a bounded solution. Case 2. r…t†y 0 …t† 6 0, t P t2 . Having in mind that r…t†y 0 …t† is monotone increasing function, we obtain limt!‡1 r…t†y 0 …t† 6 0. Let us suppose limt!‡1 r…t†y 0 …t† ˆ c < 0. Then Z t du ; y…t† 6 y…t2 † ‡ c r…u† t2 whence, in view of condition H2, it follows limt!‡1 y…t† ˆ ÿ1, which contradicts the assumption that y…t† is bounded and positive solution of problem (1), (2), (3). Therefore, we have in this case lim r…t†y 0 …t† ˆ 0:

…4†

t!‡1

We integrate (1) from t2 to t…t P t2 † and arrive to the equality Z t X 0 0 bk r…sk †y…sk † ‡ p…s†y…s ÿ h† ds: r…t†y …t† ˆ r…t2 †y …t2 † ‡ t2

t2 6 sk
…5†

Having in mind (4), it follows from (5) after passing to limit as t ! ‡1 Z 1 1 X p…s†y…s ÿ h† ds: …6† r…t2 †y 0 …t2 † ˆ ÿ bk r…sk †y…sk † ‡ t2

kˆk0

We divide both sides of (5) to r…t† and integrate from t2 to t. Thus, Z t Z s Z t ds 1 0 ‡ p…u†y…u ÿ h† du ds y…t† ˆ y…t2 † ‡ r…t2 †y …t2 † t2 r…s† t2 r…s† t2 Z t 1 X bk r…sk †y…sk † ds; ‡ t2 r…s† t2 6 sk
t

Z

t

kˆk0

Z

t2

1 X ‰ R…t† ÿ R…s†Šp…s†y…s ÿ h† ds ‡ bk r…sk †y…sk † ds t2 t2 r…s† t2 6 sk
‡

t2

t

t2

kˆk0

1 X ‰ R…t† ÿ R…s†Šp…s†y…s ÿ h† ds ‡ ‰ R…t† ÿ R…t2 †Š bk r…sk †y…sk †

ˆ y…t2 † ‡ R…t2 †

Z t2

t

Z p…s†y…s ÿ h† ds ÿ

t2

t

kˆk0

R…s†p…s†y…s ÿ h† ds:

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D.D. Bainov et al. / Appl. Math. Comput. 114 (2000) 61±68

By the aid of (5), we deduce from the above inequality " # X 0 0 bk r…sk †y…sk † y…t† 6 y…t2 † ‡ R…t2 † r…t†y …t† ÿ r…t2 †y …t2 † ÿ t2 6 sk
…7†

It follows from y…t† > 0 and y 0 …t† < 0 for t P t2 that there exists the limit limt!‡1 y…t† P 0. Let us suppose limt!‡1 y…t† ˆ c > 0. Then, we obtain from (7) as t ! ‡1 that limt!‡1 y…t† ˆ ÿ1 which contradicts the assumption that y…t† is positive and bounded solution of problem (1), (2), (3). Therefore, limt!‡1 y…t† ˆ 0. Case 3. There exists a point t3 P t2 such that r…t3 †y 0 …t3 † P 0. Since r…t†y 0 …t† is monotone increasing function, it is clear that r…t†y 0 …t† P 0 for t P t3 . If we suppose that there exists a point t4 P t3 such that r…t4 †y 0 …t4 † > 0, then we arrive at a contradiction analogously to the Case 1. Therefore, r…t†y 0 …t† ˆ 0;

t P t3 ;

…8†

whence we deduce y 0 …t† ˆ 0 and therefore y…t† ˆ c > 0, t P t3 . That is why, if we denote t4 ˆ t3 ‡ h, integration of (1) from t4 to t…t P t4 † yields r…t†y 0 …t† ˆ r…t4 †y 0 …t4 † ‡ "Z Pc

t t4

Z

t t4

p…s† ds ‡

cp…s† ds ‡

X t4 6 sk
X t4 6 sk
The last inequality contradicts (8).

cbk r…sk †

# bk r…sk † > 0;

t P t4 :



Theorem 3. Let the following conditions be ful®lled: 1. Conditions H1±H5 R t are met. 1 …s ‡ h ÿ t†p…s† ds > 1: 2. lim supt!‡1 r…t† tÿh 3. r 2 C 1 …R‡ ; R‡ † and r0 …t† P 0; t 2 R‡ : Then all bounded solutions of problem (1), (2), (3) are oscillatory.

D.D. Bainov et al. / Appl. Math. Comput. 114 (2000) 61±68

67

Proof. Let y…t† be a bounded and nonoscillatory solution of problem (1), (2), (3) for t P 0. For the sake of de®niteness, we assume y…t† > 0 for t P t1 P 0. Then y…t ÿ h† > 0 for t P t2 ˆ t1 ‡ h: As in the proof of Theorem 2, we conclude that the function r…t†y 0 …t† is monotone increasing for t P t2 , and that the inequality r…t†y 0 …t† > 0 is impossible for some t P t2 (cf. Cases 1 and 3 in the proof of Theorem 2). Therefore, r…t†y 0 …t† 6 0, t P t2 : We integrate (1) from s to t…t2 6 s < t† and obtain 0

0

0 P r…t†y …t† ˆ r…s†y …s† ‡

X s 6 sk
Z bk r…sk †y…sk † ‡

t s

p…u†y…u ÿ h† du:

Integrating the above inequality from t ÿ h to t; t P t3 ˆ t2 ‡ h, we arrive at Z t X Z t 0 r…s†y …s† ds ‡ bk r…sk †y…sk † ds 0P tÿh

tÿh s 6 sk
Z

‡ Z ‡

t tÿh t tÿh

Z

…u ÿ t ‡ h†p…u†y…u ÿ h† du P

t

tÿh

r…s†y 0 …s† ds

…u ÿ t ‡ h†p…u†y…u ÿ h† du:

…9†

From the fact that r…t† is nondecreasing function in R‡ and y…t† is nonincreasing function for t P t2 , it follows from (9) the inequality Z t …u ÿ t ‡ h†p…u† du; 0 P r…t ÿ h†‰ y…t† ÿ y…t ÿ h†Š ‡ y…t ÿ h† tÿh

whence

Z

y…t ÿ h†

t tÿh

…u ÿ t ‡ h†p…u† du 6 r…t ÿ h†y…t ÿ h† 6 r…t†y…t ÿ h†;

i.e., 1 r…t†

Z

t tÿh

…u ÿ t ‡ h†p…u† du 6 1;

t P t3 :

The last inequality contradicts condition 2 of Theorem 3.



Acknowledgements The present investigation was supported by the Bulgarian Ministry of Education and Science under Grant MM±702.

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D.D. Bainov et al. / Appl. Math. Comput. 114 (2000) 61±68

References [1] D.D. Bainov, P.S. Simeonov, Systems with Impulse E€ect: Stability, Theory and Applications, Ellis Horwood, Chichester, 1989. [2] D.D. Bainov, V. Lakshmikantham, P.S. Simeonov, Theory of Impulsive Di€erential Equations, World Scienti®c, Singapore, 1989.