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Asymptotic behavior of solutions for nonlinear delay differential equation with impulses
Applied Mathematics and Computation 213 (2009) 449–454
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Asymptotic behavior of solutions for nonlinear delay differential equation with impulses q Jianhua Shen a,*, Yanjun Liu b a b
Department of Mathematics, Hangzhou Normal University, Hangzhou, Zhejiang 310036, China Department of Mathematics, Changsha University of Science and Technology Changsha, Hunan 410058, China
a r t i c l e
i n f o
a b s t r a c t This paper studies the asymptotic behavior of solutions of the nonlinear delay differential equation with impulses
Keywords: Asymptotic behavior Liapunov functional Differential equation Impulse
(
x0 ðtÞ þ pðtÞf ðxðt sÞÞ ¼ 0; t P t0 ; t – t k ; Rt xðt k Þ ¼ bk xðt k Þ þ ð1 bk Þ t ks pðs þ sÞf ðxðsÞÞds; k 2 Z þ : k
Sufficient conditions are obtained under which every solution of the equation tends to a constant or zero as t ! 1. Our results improve and generalize some known ones. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction and Preliminaries Qualitative theory of delay differential equations has taken the shape of a well-developed theory presented in monographs [1–3]. In particular, asymptotic behavior of solutions of delay differential equations has been studied by many authors (see [4–7] and the references cited therein). However, the asymptotic behavior of solutions of delay differential equations with impulses has been less developed due to numerous theoretical and technical difficulties caused by their peculiarities [8,9]. In this paper, we study the asymptotic behavior of solutions of a nonlinear delay differential equation with impulses
(
x0 ðtÞ þ pðtÞf ðxðt sÞÞ ¼ 0; t P t 0 ; t – tk ; R tk xðt k Þ ¼ bk xðtk Þ þ ð1 bk Þ tk s pðs þ sÞf ðxðsÞÞds; k 2 Z þ ;
ð1Þ
where Z þ denotes the set of all positive integers, s 2 ð0; 1Þ; f 2 CðR; RÞ; pðtÞ 2 Cð½t0 ; 1Þ; ½0; 1ÞÞ; 0 ¼ t 0 < t 1 < < tk < t kþ1 , with limk!1 tk ¼ 1 and bk ; k ¼ 1; 2; . . ., are constants, xðt k Þ denotes the left limit of xðtÞ at t ¼ t k . By a solution of (1) we mean a real valued function xðtÞ defined on ½t 0 s; 1Þ which is piecewise right continuous on ½t 0 s; 1Þ and satisfies (1). In system (1) the impulsive term is also delayed, that is, it contains an integral term. A more general form was considered in [12,13], in which the existence and uniqueness of solutions and the stability was studied for the following more general impulsive differential equation
x0 ðtÞ ¼ f ðt; xt Þ;
t P t0 ; t – t k
MxðtÞ ¼ Ik ðt; xt Þ; t ¼ t k ; k ¼ 1; 2; . . . q
Supported by the NNSF of China (Nos. 10571050, 10871062) and Hunan Province Department of Education. * Corresponding author. E-mail address: [email protected] (J. Shen).
0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.03.043
450
J. Shen, Y. Liu / Applied Mathematics and Computation 213 (2009) 449–454
We note that some special form of the above system have been studied by many authors (for example, see [8,13]), and they also concerned stability of solutions. We note also that though the impulse in (2) is a special form of the impulse term form, the method given in this paper will mark this impulse term form. That is, by using the method in our paper cannot establish similar results for the following equation
x0 ðtÞ þ pðtÞf ðxðt sÞÞ ¼ 0; t P t 0 ; t – tk ; xðt k Þ ¼ bk xðt k Þ;
t ¼ t k ; k ¼ 1; 2; . . . ;
where the condition bk ¼ 1 not hold for all k ¼ 1; 2; . . .. Set PC ¼ PCð½s; 0; RÞ ¼ fu : ½s; 0 ! R; uðtÞ is continuous everywhere except a finite number of points s at which uðsþ Þ and uðs Þ exist and uðsþ Þ ¼ uðsÞg. For t P t0 ; xt 2 PC is defined by xt ðsÞ ¼ xðt þ sÞ for s 6 s 6 0. For a given r 2 ½tm1 ; tm Þ for some m 2 Z þ and u 2 PCð½s; 0; RÞ, the initial-value problem of (1) is
8 0 t P r; t – t k ; > < x ðtÞ þ pðtÞf ðxðt sÞÞ ¼ 0;R t xðt k Þ ¼ bk xðt k Þ þ ð1 bk Þ tkks pðs þ sÞf ðxðsÞÞds; k P m; > : xr ¼ u : The solution xðtÞ ¼ xðt; r; uÞ of the above initial-value problem is characterized by the following (i) For (ii) For
r s 6 t 6 r, the solution coincides with the function u. r 6 t < tm , the solution coincides with the solution of
x0 ðtÞ þ pðtÞf ðxðt sÞÞ ¼ 0; t P r; xr ¼ u :
(iii) For tm 6 t < t mþ1 , the solution coincides with the solution of
x0 ðtÞ þ pðtÞf ðxðt sÞÞ ¼ 0; t P t m ; xtm ¼ um ;
where
(
um ¼
xðtÞ; tm s 6 t < t m ; R tm
bm xðt m Þ þ ð1 bm Þ
tm s
pðs þ sÞf ðxðsÞÞds; t ¼ tm :
(iv) Suppose that the solution xðtÞ has already been determined in the interval ½r s; tmþk Þ. Then for tmþk 6 t < t mþkþ1 , the solution xðtÞ coincides with the solution of
(
x0 ðtÞ þ pðtÞf ðxðt sÞÞ ¼ 0; t P tmþk ; xtmþk ¼ umþk ;
where
(
umþk ¼
xðtÞ; t mþk s 6 t < t mþk ; bmþk xðt mþk Þ þ ð1 bmþk Þ
R tmþk
t mþk s
pðs þ sÞf ðxðsÞÞds; t ¼ t mþk :
Thus, in its interval of definition ½r; 1Þ the solution xðtÞ is a continuous function for t–t k ; k P m, and at t ¼ t k , it has discontinuities of the first kind and is continuous from the right. The above initial-value problem of the impulsive delay differential equation is an appropriate mathematical model for describing evolutionary systems which at certain instants in time are subjected to rapid changes [10,11]. As is customary, a solution of (1) is called oscillatory if it is neither eventually positive nor eventually negative. Otherwise, it is called nonoscillatory. In this paper, we obtain sufficient conditions for every solution of (1) to tend to zero as t ! 1. Our results improve and generalize some known results. See Corollaries 1, 3 and Remarks 1, 2 in Section 2. 2. Main results Theorem 1. Let the following conditions be fulfilled. P (i) 0 < bk 6 1 and 1 k¼1 ð1 bk Þ < 1; (ii) there is a constant M > 0 such that jf ðxÞj 6 Mjxj; x 2 R; xf ðxÞ > 0; x – 0; R tþs (iii) lim supt!1 ts pðs þ sÞds < M2 . Then every solution of (1) tends to a constant as t ! 1.
451
J. Shen, Y. Liu / Applied Mathematics and Computation 213 (2009) 449–454
Proof. Let xðtÞ be any solution of (1). We will prove that the limit limt!1 xðtÞ exists and is finite. For this purpose, we rewrite (1) in the form
8h i0 R < xðtÞ t pðs þ sÞf ðxðsÞÞds þ pðt þ sÞf ðxðtÞÞ ¼ 0; t P t0 ; t – t k ; ts : xðt Þ ¼ b xðt Þ þ ð1 b Þ R tk pðs þ sÞf ðxðsÞÞds; k 2 Zþ : k k k k t k s
ð2Þ
Now we introduce two functionals as
Z V 1 ðtÞ ¼ xðtÞ
2 pðs þ sÞf ðxðsÞÞds
t
ts
and
V 2 ðtÞ ¼
Z
t
pðs þ 2sÞ
Z
ts
t
pðu þ sÞf 2 ðxðuÞÞ du ds:
s
As t – tk , calculating dV 1 =dt and dV 2 =dt along the solution of (1), we have
Z t Z t dV 1 pðs þ sÞf ðxðsÞÞ ds pðt þ sÞf ðxðtÞÞ ¼ pðt þ sÞ 2xðtÞf ðxðtÞÞ pðs þ sÞð2f ðxðtÞÞf ðxðsÞÞÞ ds ¼ 2 xðtÞ dt ts ts Z t Z t 6 pðt þ sÞ 2xðtÞf ðxðtÞÞ f 2 ðxðtÞÞ pðs þ sÞ ds pðs þ sÞf 2 ðxðsÞÞ ds ts
ts
and
dV 2 ¼ pðt þ sÞ dt
Z
t
pðs þ sÞf 2 ðxðsÞÞds þ pðt þ sÞf 2 ðxðtÞÞ
Z
ts
t
pðs þ 2sÞds:
ts
Set VðtÞ ¼ V 1 ðtÞ þ V 2 ðtÞ. We shall prove that
Z tþs dV 2 pðs þ sÞds ; 6 pðt þ sÞf 2 ðxðtÞÞ dt M ts
t – tk :
ð3Þ
Indeed, (3) is obvious for xðtÞ ¼ 0. For xðtÞ–0, from the above two inequalities, we obtain
Z t Z t dV pðs þ sÞds f 2 ðxðtÞÞ pðs þ 2sÞds 6 pðt þ sÞ 2xðtÞf ðxðtÞÞ f 2 ðxðtÞÞ dt ts ts Z t Z t 2xðtÞ 2 ¼ pðt þ sÞf ðxðtÞÞ pðs þ sÞds pðs þ 2sÞds f ðxðtÞÞ ts ts Z t Z tþs Z t 2 2 pðs þ sÞds pðs þ 2sÞds ¼ pðt þ sÞf 2 ðxðtÞÞ pðs þ sÞds ; 6 pðt þ sÞf 2 ðxðtÞÞ M M ts ts ts
t – tk :
As t ¼ tk , we have
" Vðt k Þ ¼ xðtk Þ
Z
#2
tk
pðs þ sÞf ðxðsÞÞds
þ
t k s
¼
þ ð1 bk Þ
¼
xðtk Þ
" 6
xðt k Þ
pðs þ 2sÞ
Z
Z
tk
pðs þ sÞf ðxðsÞÞds
tk
tk
#2 pðs þ sÞf ðxðsÞÞds
þ
t k s
Z
tk
pðu þ sÞf 2 ðxðuÞÞduds #2
pðs þ sÞf ðxðsÞÞds
Z
tk
þ
Z
tk
pðs þ 2sÞ
t k s
pðs þ 2sÞ
þ
Z
tk
pðs þ 2sÞ
t k s
t k s
#2 pðs þ sÞf ðxðsÞÞds
tk
t k s
t k s
Z
Z s
t k s
" 2 bk
tk
t k s
" bk xðtk Þ
Z
Z
tk
Z
tk
pðu þ sÞf 2 ðxðuÞÞ du ds
s
pðu þ sÞf 2 ðxðuÞÞdu ds
s
Z s
tk
pðu þ sÞf 2 ðxðuÞÞdu ds ¼ Vðt k Þ:
i.e.,
Vðt k Þ 6 Vðt k Þ: (4), together with (iii) and (3), imply
pðt þ sÞf 2 ðxðtÞÞ 2 L1 ðt 0 ; 1Þ;
ð4Þ
452
J. Shen, Y. Liu / Applied Mathematics and Computation 213 (2009) 449–454
and, hence, for any q P 0 we have
lim
t!1
Z
t
pðs þ sÞf 2 ðxðsÞÞds ¼ 0:
tq
Moreover,
V 2 ðtÞ ¼
Z
Z t Z t Z t pðs þ 2sÞ pðu þ sÞf 2 ðxðuÞÞdu ds 6 pðs þ 2sÞds pðu þ sÞf 2 ðxðuÞÞdu ts s ts ts Z t 2 pðu þ sÞf 2 ðxðuÞÞdu ! 0 as t ! 1: 6 M ts t
On the other hand, by (iii), (3) and (4) we see that VðtÞ is eventually decreasing. Hence, the limit limt!1 VðtÞ ¼ c exists and is finite. So limt!1 V 1 ðtÞ ¼ c, that is,
Z lim xðtÞ
t!1
2 pðs þ sÞf ðxðsÞÞds ¼ c:
t
ð5Þ
ts
Let yðtÞ ¼ xðtÞ
Rt
ts
pðs þ sÞf ðxðsÞÞds, then
yðtk Þ ¼ xðt k Þ
Z
tk
t k s
"
¼ bk xðtk Þ
pðs þ sÞf ðxðsÞÞds ¼ bk xðtk Þ þ ð1 bk Þ
Z
#
tk
Z
tk
pðs þ sÞf ðxðsÞÞds
t k s
Z
tk
pðs þ sÞf ðxðsÞÞds
t k s
pðs þ sÞf ðxðsÞÞds ¼ bk yðtk Þ:
t k s
Moreover, in view of (2) and (5), we have
y0 ðtÞ þ pðt þ sÞf ðxðtÞÞ ¼ 0; t P t0 ; t – t k ;
ð6Þ
k 2 Zþ ;
yðt k Þ ¼ bk yðt k Þ; and
lim y2 ðtÞ ¼ c:
ð7Þ
t!1
If c ¼ 0, then limt!1 yðtÞ ¼ 0. If c > 0, then there exists a large enough T such that yðtÞ–0 for any t > T. Therefore, for any tk > T; t 2 ½tk ; t kþ1 Þ we have yðtÞ > 0 or yðtÞ < 0 because yðtÞ is continuous on ½tk ; tkþ1 Þ. Without loss of generality, we assume that yðtÞ > 0 on ½t k ; t kþ1 Þ. It follows that yðt kþ1 Þ ¼ bk yðt kþ1 Þ > 0, thus yðtÞ > 0 on ½t kþ1 ; t kþ2 Þ. By induction, we conclude that yðtÞ > 0 on ½t k ; 1Þ, from which, in view of (7) we have that limt!1 yðtÞ exists and is finite. Let
Z lim yðtÞ ¼ lim xðtÞ
t!1
t!1
pðs þ sÞf ðxðsÞÞds ¼ k:
t
ð8Þ
ts
In view of (6), we have
Z
t
pðs þ sÞf ðxðsÞÞds ¼ yðt sÞ yðtÞ þ
ts
lim
X yðt k Þ yðtk Þ ¼ yðt sÞ yðtÞ ð1 bk Þyðt k Þ:
ts
We let t ! 1 and note that
t!1
X
Z
P1
k¼1 ð1
ts
bk Þ < 1, then we have
t
pðs þ sÞf ðxðsÞÞds ¼ 0:
ð9Þ
ts
From (8), we have limt!1 xðtÞ ¼ k: The proof is complete.
h
By Theorem 1, we have the following result immediately. Theorem 2. Assume that the conditions in Theorem 1 hold, then every oscillatory solution of (1) tends to zero as t ! 1. We note that subject to the impulsive differential Eq. (1), the differential equation
x0 ðtÞ þ pðtÞf ðxðt sÞÞ ¼ 0 is called the corresponding continuous equation or the corresponding differential equation without impulse effect, of (1). This may be obtained by putting bk 1. Thus, by Theorem 2 and by taking f ðxÞ x, we get the following Corollary 1. Assume that
lim sup t!1
Z
tþs
ts
pðs þ sÞds < 2:
ð10Þ
J. Shen, Y. Liu / Applied Mathematics and Computation 213 (2009) 449–454
453
Then every oscillatory solution of
x0 ðtÞ þ pðtÞxðt sÞ ¼ 0
ð11Þ
tends to zero as t ! 1. One also might say that the assertion in Corollary 1 can be proved following the same procedure as the procedure used in the proof of Theorem 1 (i.e., taking bk ¼ 1; k ¼ 1; 2; . . .). Remark 1. Corollary 1 improves Theorem 2 in [4] by relaxing the following condition in [4]
lim sup t!1
Z
t
pðsÞds < 1:
ð12Þ
ts
Theorem 3. The conditions in Theorem 1 together with (i) for any a > 0 there exists b > 0 such that
jf ðxÞj P b;
for jxj P a;
and (ii)
Z
1
pðtÞdt ¼ 1;
t0
imply that every solution of (1) tends to zero as t ! 1.
Proof. By Theorem 2, all we have to prove is that every nonoscillatory solution of (1) tends to zero as t ! 1. Let xðtÞ be an eventually positive solution of (1), we shall prove limt!1 xðtÞ ¼ 0. As in the proof of Theorem 1, we can rewrite (1) in the form (6). Integrating from t 0 to t on both sides of (6) produces
Z
t
pðs þ sÞf ðxðsÞÞds ¼ yðt 0 Þ yðtÞ
t0
ð1 bk Þyðt k Þ:
t0
By using (8) and
Z
X
P1
k¼1 ð1
bk Þ < 1, we have
1
pðs þ sÞf ðxðsÞÞds < 1;
t0
which, together with (ii) yields lim inf t!1 f ðxðtÞÞ ¼ 0. We claim that
lim inf xðtÞ ¼ 0:
ð13Þ
t!1
Let fsm g be such that sm ! 1 as m ! 1 and limm!1 f ðxðsm ÞÞ ¼ 0. We must have lim inf m!1 xðsm Þ ¼ c ¼ 0. In fact, if c > 0, then there is a subsequence fsmk g such that xðsmk Þ P c=2 for k sufficiently large. By (i) we have f ðxðsmk ÞÞ P l for some l > 0 and sufficiently large k, which yields a contradiction because of limk!1 f ðxðsmk ÞÞ ¼ 0. Therefore, (13) holds. On the other hand, by Theorem 1, we have that limt!1 xðtÞ exists. Therefore, limt!1 xðtÞ ¼ 0. The proof is complete. h In Theorem 3, taking f ðxÞ x, we obtain Corollary 2. Assume that the following conditions hold. P (i) 0 < bk 6 1 and 1 ð1 bk Þ < 1; R tþs k¼1 (ii) lim supt!1 ts pðs þ sÞds < 2; R1 (iii) t0 pðtÞdt ¼ 1. Then every solution of
(
x0 ðtÞ þ pðtÞxðt sÞ ¼ 0; xðt k Þ ¼
bk xðtk Þ
þ ð1 bk Þ
R tk
t k s
t P t0 ; t – tk ; pðs þ sÞxðsÞds; k 2 Z þ
tends to zero as t ! 1 Similar to Corollary 1, by Theorem 3 and by taking bk 1, we have Corollary 3. Assume that (10) and (ii) in Theorem 3 hold. Then every solution of (11) tends to zero as t ! 1.
ð14Þ
454
J. Shen, Y. Liu / Applied Mathematics and Computation 213 (2009) 449–454
Remark 2. Corollary 3 improves Theorem 3 in [4] by relaxing condition (12) and lim inf t!1 pðtÞ > 0. 3. Example Consider the following impulsive differential equation
8 0 1þsin t xðt pÞ ¼ 0; < x ðtÞ þ 5ð1þx 2 ðtpÞ Þ 2 : xðt Þ ¼ k 1 xðt Þ þ 1 R tk 1þsinðsþpÞ k k 5 k2 k2 t p k
t P 0; t – t k ; xðsÞ ds; 1þx2 ðsÞ
ð15Þ k ¼ 2; 3; . . . ;
t x where pðtÞ ¼ 1þsin ; f ðxÞ ¼ 1þx 2 ; t k < t kþ1 with t k ! 1 as k ! 1. Since 5
1 1 X X 1 ð1 bk Þ ¼ < 1; 2 k k¼1 k¼1 x x2 1 þ x2 6 jxj; 1 þ x2 > 0 ðx – 0Þ and Z tþp 1 þ sinðs þ pÞ 2p lim sup ds ¼ < 2: 5 5 t!1 tp
By Theorem 1, it follows that the every solution of (15) tends to a constant as t ! 1. Acknowledgement The authors is very grateful to the referee for his/her valuable suggestions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
G.S. Ladde, V. Lakshimikantham, B.G. Zhang, Oscillation Theory of Differential Equations with Deviating Arguments, Marcel Dekker, New York, 1987. I. Gyori, G. Ladas, Oscillation Theory of Delay Differential Equations with Applications, Clarendon Press, Oxford, 1991. J.K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977. G. Ladas et al, Asymptotic behavior of solutions of retarded differential equations, Proc. Am. Math. Soc. 88 (1983) 247–253. J.H. Shen, J.S. Yu, Asymptotic behavior of solutions of neutral differential equations with positive and negative coefficients, J. Math. Anal. Appl. 195 (1995) 517–526. G. Ladas, Y.G. Sficas, Asymptotic behavior of oscillatory solutions, Hiroshima Math. J. 18 (1988) 351–359. X.P. Wang, L.S. Liao, Asymptotic behavior of solutions of nonlinear delay differential equations, Ann. Diff. Eq. 19 (2003) 552–558. A.M. Zhao, J.R. Yan, Asymptotic behavior of solutions of impulsive delay differential equations, J. Math. Anal. Appl. 201 (1996) 943–954. J.H. Shen, Z.C. Wang, Oscillatory and asymptotic behavior of solutions of delay differential equations with impulses, Ann. Diff. Eq. 10 (1994) 61–67. V. Lakshmikantham, D.D. Bainov, P.S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989. D.D. Bainov, P.S. Simenov, Systems with Impulse Effect: Stability Theory and Applications, Ellis Horwood, Chichester, 1989. Xinzhi Liu, G. Ballinger, Existence and continuability of solutions for differential equations with delays and state-dependent impulses, Nonlinear Anal. 51 (2002) 633–647. Jianhua Shen, Zhiguo Luo, Xinzhi Liu, Impulsive stabilization for functional differential equations via Liapunov functionals, J. Math. Anal. Appl. 240 (1999) 1–15.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Asymptotic behavior of solutions for nonlinear delay differential equation with impulses q Jianhua Shen a,*, Yanjun Liu b a b
Department of Mathematics, Hangzhou Normal University, Hangzhou, Zhejiang 310036, China Department of Mathematics, Changsha University of Science and Technology Changsha, Hunan 410058, China
a r t i c l e
i n f o
a b s t r a c t This paper studies the asymptotic behavior of solutions of the nonlinear delay differential equation with impulses
Keywords: Asymptotic behavior Liapunov functional Differential equation Impulse
(
x0 ðtÞ þ pðtÞf ðxðt sÞÞ ¼ 0; t P t0 ; t – t k ; Rt xðt k Þ ¼ bk xðt k Þ þ ð1 bk Þ t ks pðs þ sÞf ðxðsÞÞds; k 2 Z þ : k
Sufficient conditions are obtained under which every solution of the equation tends to a constant or zero as t ! 1. Our results improve and generalize some known ones. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction and Preliminaries Qualitative theory of delay differential equations has taken the shape of a well-developed theory presented in monographs [1–3]. In particular, asymptotic behavior of solutions of delay differential equations has been studied by many authors (see [4–7] and the references cited therein). However, the asymptotic behavior of solutions of delay differential equations with impulses has been less developed due to numerous theoretical and technical difficulties caused by their peculiarities [8,9]. In this paper, we study the asymptotic behavior of solutions of a nonlinear delay differential equation with impulses
(
x0 ðtÞ þ pðtÞf ðxðt sÞÞ ¼ 0; t P t 0 ; t – tk ; R tk xðt k Þ ¼ bk xðtk Þ þ ð1 bk Þ tk s pðs þ sÞf ðxðsÞÞds; k 2 Z þ ;
ð1Þ
where Z þ denotes the set of all positive integers, s 2 ð0; 1Þ; f 2 CðR; RÞ; pðtÞ 2 Cð½t0 ; 1Þ; ½0; 1ÞÞ; 0 ¼ t 0 < t 1 < < tk < t kþ1 , with limk!1 tk ¼ 1 and bk ; k ¼ 1; 2; . . ., are constants, xðt k Þ denotes the left limit of xðtÞ at t ¼ t k . By a solution of (1) we mean a real valued function xðtÞ defined on ½t 0 s; 1Þ which is piecewise right continuous on ½t 0 s; 1Þ and satisfies (1). In system (1) the impulsive term is also delayed, that is, it contains an integral term. A more general form was considered in [12,13], in which the existence and uniqueness of solutions and the stability was studied for the following more general impulsive differential equation
x0 ðtÞ ¼ f ðt; xt Þ;
t P t0 ; t – t k
MxðtÞ ¼ Ik ðt; xt Þ; t ¼ t k ; k ¼ 1; 2; . . . q
Supported by the NNSF of China (Nos. 10571050, 10871062) and Hunan Province Department of Education. * Corresponding author. E-mail address: [email protected] (J. Shen).
0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.03.043
450
J. Shen, Y. Liu / Applied Mathematics and Computation 213 (2009) 449–454
We note that some special form of the above system have been studied by many authors (for example, see [8,13]), and they also concerned stability of solutions. We note also that though the impulse in (2) is a special form of the impulse term form, the method given in this paper will mark this impulse term form. That is, by using the method in our paper cannot establish similar results for the following equation
x0 ðtÞ þ pðtÞf ðxðt sÞÞ ¼ 0; t P t 0 ; t – tk ; xðt k Þ ¼ bk xðt k Þ;
t ¼ t k ; k ¼ 1; 2; . . . ;
where the condition bk ¼ 1 not hold for all k ¼ 1; 2; . . .. Set PC ¼ PCð½s; 0; RÞ ¼ fu : ½s; 0 ! R; uðtÞ is continuous everywhere except a finite number of points s at which uðsþ Þ and uðs Þ exist and uðsþ Þ ¼ uðsÞg. For t P t0 ; xt 2 PC is defined by xt ðsÞ ¼ xðt þ sÞ for s 6 s 6 0. For a given r 2 ½tm1 ; tm Þ for some m 2 Z þ and u 2 PCð½s; 0; RÞ, the initial-value problem of (1) is
8 0 t P r; t – t k ; > < x ðtÞ þ pðtÞf ðxðt sÞÞ ¼ 0;R t xðt k Þ ¼ bk xðt k Þ þ ð1 bk Þ tkks pðs þ sÞf ðxðsÞÞds; k P m; > : xr ¼ u : The solution xðtÞ ¼ xðt; r; uÞ of the above initial-value problem is characterized by the following (i) For (ii) For
r s 6 t 6 r, the solution coincides with the function u. r 6 t < tm , the solution coincides with the solution of
x0 ðtÞ þ pðtÞf ðxðt sÞÞ ¼ 0; t P r; xr ¼ u :
(iii) For tm 6 t < t mþ1 , the solution coincides with the solution of
x0 ðtÞ þ pðtÞf ðxðt sÞÞ ¼ 0; t P t m ; xtm ¼ um ;
where
(
um ¼
xðtÞ; tm s 6 t < t m ; R tm
bm xðt m Þ þ ð1 bm Þ
tm s
pðs þ sÞf ðxðsÞÞds; t ¼ tm :
(iv) Suppose that the solution xðtÞ has already been determined in the interval ½r s; tmþk Þ. Then for tmþk 6 t < t mþkþ1 , the solution xðtÞ coincides with the solution of
(
x0 ðtÞ þ pðtÞf ðxðt sÞÞ ¼ 0; t P tmþk ; xtmþk ¼ umþk ;
where
(
umþk ¼
xðtÞ; t mþk s 6 t < t mþk ; bmþk xðt mþk Þ þ ð1 bmþk Þ
R tmþk
t mþk s
pðs þ sÞf ðxðsÞÞds; t ¼ t mþk :
Thus, in its interval of definition ½r; 1Þ the solution xðtÞ is a continuous function for t–t k ; k P m, and at t ¼ t k , it has discontinuities of the first kind and is continuous from the right. The above initial-value problem of the impulsive delay differential equation is an appropriate mathematical model for describing evolutionary systems which at certain instants in time are subjected to rapid changes [10,11]. As is customary, a solution of (1) is called oscillatory if it is neither eventually positive nor eventually negative. Otherwise, it is called nonoscillatory. In this paper, we obtain sufficient conditions for every solution of (1) to tend to zero as t ! 1. Our results improve and generalize some known results. See Corollaries 1, 3 and Remarks 1, 2 in Section 2. 2. Main results Theorem 1. Let the following conditions be fulfilled. P (i) 0 < bk 6 1 and 1 k¼1 ð1 bk Þ < 1; (ii) there is a constant M > 0 such that jf ðxÞj 6 Mjxj; x 2 R; xf ðxÞ > 0; x – 0; R tþs (iii) lim supt!1 ts pðs þ sÞds < M2 . Then every solution of (1) tends to a constant as t ! 1.
451
J. Shen, Y. Liu / Applied Mathematics and Computation 213 (2009) 449–454
Proof. Let xðtÞ be any solution of (1). We will prove that the limit limt!1 xðtÞ exists and is finite. For this purpose, we rewrite (1) in the form
8h i0 R < xðtÞ t pðs þ sÞf ðxðsÞÞds þ pðt þ sÞf ðxðtÞÞ ¼ 0; t P t0 ; t – t k ; ts : xðt Þ ¼ b xðt Þ þ ð1 b Þ R tk pðs þ sÞf ðxðsÞÞds; k 2 Zþ : k k k k t k s
ð2Þ
Now we introduce two functionals as
Z V 1 ðtÞ ¼ xðtÞ
2 pðs þ sÞf ðxðsÞÞds
t
ts
and
V 2 ðtÞ ¼
Z
t
pðs þ 2sÞ
Z
ts
t
pðu þ sÞf 2 ðxðuÞÞ du ds:
s
As t – tk , calculating dV 1 =dt and dV 2 =dt along the solution of (1), we have
Z t Z t dV 1 pðs þ sÞf ðxðsÞÞ ds pðt þ sÞf ðxðtÞÞ ¼ pðt þ sÞ 2xðtÞf ðxðtÞÞ pðs þ sÞð2f ðxðtÞÞf ðxðsÞÞÞ ds ¼ 2 xðtÞ dt ts ts Z t Z t 6 pðt þ sÞ 2xðtÞf ðxðtÞÞ f 2 ðxðtÞÞ pðs þ sÞ ds pðs þ sÞf 2 ðxðsÞÞ ds ts
ts
and
dV 2 ¼ pðt þ sÞ dt
Z
t
pðs þ sÞf 2 ðxðsÞÞds þ pðt þ sÞf 2 ðxðtÞÞ
Z
ts
t
pðs þ 2sÞds:
ts
Set VðtÞ ¼ V 1 ðtÞ þ V 2 ðtÞ. We shall prove that
Z tþs dV 2 pðs þ sÞds ; 6 pðt þ sÞf 2 ðxðtÞÞ dt M ts
t – tk :
ð3Þ
Indeed, (3) is obvious for xðtÞ ¼ 0. For xðtÞ–0, from the above two inequalities, we obtain
Z t Z t dV pðs þ sÞds f 2 ðxðtÞÞ pðs þ 2sÞds 6 pðt þ sÞ 2xðtÞf ðxðtÞÞ f 2 ðxðtÞÞ dt ts ts Z t Z t 2xðtÞ 2 ¼ pðt þ sÞf ðxðtÞÞ pðs þ sÞds pðs þ 2sÞds f ðxðtÞÞ ts ts Z t Z tþs Z t 2 2 pðs þ sÞds pðs þ 2sÞds ¼ pðt þ sÞf 2 ðxðtÞÞ pðs þ sÞds ; 6 pðt þ sÞf 2 ðxðtÞÞ M M ts ts ts
t – tk :
As t ¼ tk , we have
" Vðt k Þ ¼ xðtk Þ
Z
#2
tk
pðs þ sÞf ðxðsÞÞds
þ
t k s
¼
þ ð1 bk Þ
¼
xðtk Þ
" 6
xðt k Þ
pðs þ 2sÞ
Z
Z
tk
pðs þ sÞf ðxðsÞÞds
tk
tk
#2 pðs þ sÞf ðxðsÞÞds
þ
t k s
Z
tk
pðu þ sÞf 2 ðxðuÞÞduds #2
pðs þ sÞf ðxðsÞÞds
Z
tk
þ
Z
tk
pðs þ 2sÞ
t k s
pðs þ 2sÞ
þ
Z
tk
pðs þ 2sÞ
t k s
t k s
#2 pðs þ sÞf ðxðsÞÞds
tk
t k s
t k s
Z
Z s
t k s
" 2 bk
tk
t k s
" bk xðtk Þ
Z
Z
tk
Z
tk
pðu þ sÞf 2 ðxðuÞÞ du ds
s
pðu þ sÞf 2 ðxðuÞÞdu ds
s
Z s
tk
pðu þ sÞf 2 ðxðuÞÞdu ds ¼ Vðt k Þ:
i.e.,
Vðt k Þ 6 Vðt k Þ: (4), together with (iii) and (3), imply
pðt þ sÞf 2 ðxðtÞÞ 2 L1 ðt 0 ; 1Þ;
ð4Þ
452
J. Shen, Y. Liu / Applied Mathematics and Computation 213 (2009) 449–454
and, hence, for any q P 0 we have
lim
t!1
Z
t
pðs þ sÞf 2 ðxðsÞÞds ¼ 0:
tq
Moreover,
V 2 ðtÞ ¼
Z
Z t Z t Z t pðs þ 2sÞ pðu þ sÞf 2 ðxðuÞÞdu ds 6 pðs þ 2sÞds pðu þ sÞf 2 ðxðuÞÞdu ts s ts ts Z t 2 pðu þ sÞf 2 ðxðuÞÞdu ! 0 as t ! 1: 6 M ts t
On the other hand, by (iii), (3) and (4) we see that VðtÞ is eventually decreasing. Hence, the limit limt!1 VðtÞ ¼ c exists and is finite. So limt!1 V 1 ðtÞ ¼ c, that is,
Z lim xðtÞ
t!1
2 pðs þ sÞf ðxðsÞÞds ¼ c:
t
ð5Þ
ts
Let yðtÞ ¼ xðtÞ
Rt
ts
pðs þ sÞf ðxðsÞÞds, then
yðtk Þ ¼ xðt k Þ
Z
tk
t k s
"
¼ bk xðtk Þ
pðs þ sÞf ðxðsÞÞds ¼ bk xðtk Þ þ ð1 bk Þ
Z
#
tk
Z
tk
pðs þ sÞf ðxðsÞÞds
t k s
Z
tk
pðs þ sÞf ðxðsÞÞds
t k s
pðs þ sÞf ðxðsÞÞds ¼ bk yðtk Þ:
t k s
Moreover, in view of (2) and (5), we have
y0 ðtÞ þ pðt þ sÞf ðxðtÞÞ ¼ 0; t P t0 ; t – t k ;
ð6Þ
k 2 Zþ ;
yðt k Þ ¼ bk yðt k Þ; and
lim y2 ðtÞ ¼ c:
ð7Þ
t!1
If c ¼ 0, then limt!1 yðtÞ ¼ 0. If c > 0, then there exists a large enough T such that yðtÞ–0 for any t > T. Therefore, for any tk > T; t 2 ½tk ; t kþ1 Þ we have yðtÞ > 0 or yðtÞ < 0 because yðtÞ is continuous on ½tk ; tkþ1 Þ. Without loss of generality, we assume that yðtÞ > 0 on ½t k ; t kþ1 Þ. It follows that yðt kþ1 Þ ¼ bk yðt kþ1 Þ > 0, thus yðtÞ > 0 on ½t kþ1 ; t kþ2 Þ. By induction, we conclude that yðtÞ > 0 on ½t k ; 1Þ, from which, in view of (7) we have that limt!1 yðtÞ exists and is finite. Let
Z lim yðtÞ ¼ lim xðtÞ
t!1
t!1
pðs þ sÞf ðxðsÞÞds ¼ k:
t
ð8Þ
ts
In view of (6), we have
Z
t
pðs þ sÞf ðxðsÞÞds ¼ yðt sÞ yðtÞ þ
ts
lim
X yðt k Þ yðtk Þ ¼ yðt sÞ yðtÞ ð1 bk Þyðt k Þ:
ts
We let t ! 1 and note that
t!1
X
Z
P1
k¼1 ð1
ts
bk Þ < 1, then we have
t
pðs þ sÞf ðxðsÞÞds ¼ 0:
ð9Þ
ts
From (8), we have limt!1 xðtÞ ¼ k: The proof is complete.
h
By Theorem 1, we have the following result immediately. Theorem 2. Assume that the conditions in Theorem 1 hold, then every oscillatory solution of (1) tends to zero as t ! 1. We note that subject to the impulsive differential Eq. (1), the differential equation
x0 ðtÞ þ pðtÞf ðxðt sÞÞ ¼ 0 is called the corresponding continuous equation or the corresponding differential equation without impulse effect, of (1). This may be obtained by putting bk 1. Thus, by Theorem 2 and by taking f ðxÞ x, we get the following Corollary 1. Assume that
lim sup t!1
Z
tþs
ts
pðs þ sÞds < 2:
ð10Þ
J. Shen, Y. Liu / Applied Mathematics and Computation 213 (2009) 449–454
453
Then every oscillatory solution of
x0 ðtÞ þ pðtÞxðt sÞ ¼ 0
ð11Þ
tends to zero as t ! 1. One also might say that the assertion in Corollary 1 can be proved following the same procedure as the procedure used in the proof of Theorem 1 (i.e., taking bk ¼ 1; k ¼ 1; 2; . . .). Remark 1. Corollary 1 improves Theorem 2 in [4] by relaxing the following condition in [4]
lim sup t!1
Z
t
pðsÞds < 1:
ð12Þ
ts
Theorem 3. The conditions in Theorem 1 together with (i) for any a > 0 there exists b > 0 such that
jf ðxÞj P b;
for jxj P a;
and (ii)
Z
1
pðtÞdt ¼ 1;
t0
imply that every solution of (1) tends to zero as t ! 1.
Proof. By Theorem 2, all we have to prove is that every nonoscillatory solution of (1) tends to zero as t ! 1. Let xðtÞ be an eventually positive solution of (1), we shall prove limt!1 xðtÞ ¼ 0. As in the proof of Theorem 1, we can rewrite (1) in the form (6). Integrating from t 0 to t on both sides of (6) produces
Z
t
pðs þ sÞf ðxðsÞÞds ¼ yðt 0 Þ yðtÞ
t0
ð1 bk Þyðt k Þ:
t0
By using (8) and
Z
X
P1
k¼1 ð1
bk Þ < 1, we have
1
pðs þ sÞf ðxðsÞÞds < 1;
t0
which, together with (ii) yields lim inf t!1 f ðxðtÞÞ ¼ 0. We claim that
lim inf xðtÞ ¼ 0:
ð13Þ
t!1
Let fsm g be such that sm ! 1 as m ! 1 and limm!1 f ðxðsm ÞÞ ¼ 0. We must have lim inf m!1 xðsm Þ ¼ c ¼ 0. In fact, if c > 0, then there is a subsequence fsmk g such that xðsmk Þ P c=2 for k sufficiently large. By (i) we have f ðxðsmk ÞÞ P l for some l > 0 and sufficiently large k, which yields a contradiction because of limk!1 f ðxðsmk ÞÞ ¼ 0. Therefore, (13) holds. On the other hand, by Theorem 1, we have that limt!1 xðtÞ exists. Therefore, limt!1 xðtÞ ¼ 0. The proof is complete. h In Theorem 3, taking f ðxÞ x, we obtain Corollary 2. Assume that the following conditions hold. P (i) 0 < bk 6 1 and 1 ð1 bk Þ < 1; R tþs k¼1 (ii) lim supt!1 ts pðs þ sÞds < 2; R1 (iii) t0 pðtÞdt ¼ 1. Then every solution of
(
x0 ðtÞ þ pðtÞxðt sÞ ¼ 0; xðt k Þ ¼
bk xðtk Þ
þ ð1 bk Þ
R tk
t k s
t P t0 ; t – tk ; pðs þ sÞxðsÞds; k 2 Z þ
tends to zero as t ! 1 Similar to Corollary 1, by Theorem 3 and by taking bk 1, we have Corollary 3. Assume that (10) and (ii) in Theorem 3 hold. Then every solution of (11) tends to zero as t ! 1.
ð14Þ
454
J. Shen, Y. Liu / Applied Mathematics and Computation 213 (2009) 449–454
Remark 2. Corollary 3 improves Theorem 3 in [4] by relaxing condition (12) and lim inf t!1 pðtÞ > 0. 3. Example Consider the following impulsive differential equation
8 0 1þsin t xðt pÞ ¼ 0; < x ðtÞ þ 5ð1þx 2 ðtpÞ Þ 2 : xðt Þ ¼ k 1 xðt Þ þ 1 R tk 1þsinðsþpÞ k k 5 k2 k2 t p k
t P 0; t – t k ; xðsÞ ds; 1þx2 ðsÞ
ð15Þ k ¼ 2; 3; . . . ;
t x where pðtÞ ¼ 1þsin ; f ðxÞ ¼ 1þx 2 ; t k < t kþ1 with t k ! 1 as k ! 1. Since 5
1 1 X X 1 ð1 bk Þ ¼ < 1; 2 k k¼1 k¼1 x x2 1 þ x2 6 jxj; 1 þ x2 > 0 ðx – 0Þ and Z tþp 1 þ sinðs þ pÞ 2p lim sup ds ¼ < 2: 5 5 t!1 tp
By Theorem 1, it follows that the every solution of (15) tends to a constant as t ! 1. Acknowledgement The authors is very grateful to the referee for his/her valuable suggestions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
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