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Asymptotic behavior of neutral delay differential equation of euler form with constant impulsive jumps
Applied Mathematics and Computation 219 (2013) 9906–9913
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Asymptotic behavior of neutral delay differential equation of euler form with constant impulsive jumps Fangfang Jiang, Jitao Sun ⇑ Department of Mathematics, Tongji University, Shanghai 200092, China
a r t i c l e
i n f o
Keywords: Asymptotic behavior Nonlinear neutral differential equation Unbounded delay Constant impulsive jumps Euler form
a b s t r a c t In this paper, we investigate the asymptotic behavior of solutions for a class of forced nonlinear neutral differential equation in first-order Euler form with constant impulsive jumps and unbounded delay. Several new sufficient conditions are given to guarantee that every non-oscillatory/oscillatory solution of system tends to zero as t ! 1. Our conclusions improve and generalize some related existing results in the literatures. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction It is well known that there are two basic methods in studying the asymptotic behavior of solutions of neutral delay differential equation with/without impulsive perturbations. One direction is by the construction of Lyapunov (functions) functionals, see [1–5]. In [1], Guan and Shen investigated the following impulsive neutral delay differential equation in Euler form
(
½xðtÞ CðtÞxðatÞ0 þ PðtÞ xðbtÞ ¼ 0; t P t 0 > 0; t – t k ; t Rt xðt þk Þ ¼ bk xðtk Þ þ ð1 bk Þ btkk Pðs=bÞ xðsÞds; k 2 Zþ s
ð1:1Þ
and obtained that every solution of system (1.1) tends to a constant as t ! 1. The other direction is by considering the asymptotic behavior of non-oscillatory and oscillatory solutions respectively, for example, see [6,7]. Such as in [6], the authors studied the asymptotic behavior of every non-oscillatory/ oscillatory solution for a forced nonlinear neutral differential equation with impulses and bounded delay
8 n X > < ½xðtÞ pxðt sÞ0 þ qi ðtÞf ðxðt ri ÞÞ ¼ hðtÞ; t – t k ; i¼1 > : þ k 2 Zþ : xðt k Þ xðt k Þ ¼ bk xðt k Þ;
ð1:2Þ
For the related stability results of impulsive functional differential equations, and the corresponding oscillatory theory on neutral differential equation in Euler form, the reader may want to consult [8–14] and the references therein. To the best of our knowledge, there is rarely result concerning the asymptotic behavior of non-oscillatory and/ or oscillatory solutions of impulsive neutral differential system. Especially, only paper [7] studies the asymptotic behavior of solutions for a nonlinear neutral delay differential equation with constant impulsive jumps. Moreover, there is hardly any result concerning the asymptotic behavior of solutions for a class of neutral differential equation in Euler form with constant
⇑ Corresponding author. E-mail address: [email protected] (J. Sun). 0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.04.022
F. Jiang, J. Sun / Applied Mathematics and Computation 219 (2013) 9906–9913
9907
impulsive jumps. Therefore, inspired by the ideas from [1,7], in present paper, we consider the asymptotic behavior of every non-oscillatory/ oscillatory solution for the following forced nonlinear neutral differential equation in first-order Euler form with constant impulsive jumps and unbounded delay
8 n X > Pi ðtÞ < ½xðtÞ CðtÞxðsðtÞÞ0 þ f ðxðbi tÞÞ ¼ hðtÞ; t P t0 > 0; t – t k ; t i¼1 > : þ k 2 Zþ : xðtk Þ xðtk Þ ¼ ak ;
ð1:3Þ
It should be noted that in previous references, such as [1,6,7], the condition limt!1 jCðtÞj ¼ C < 1 is satisfied, while in our work, we do not require limt!1 jCðtÞj exists. In fact, we obtain our results just require lim supt!1 jCðtÞj < 1. So the obtained conclusions further improve and generalize some related existing results. In addition, it is worth mentioning that in system (1.3), we cannot simply and directly apply the above two methods to derive sufficient conditions such that every non-oscillatory/ oscillatory solution of system (1.3) tends to zero as t ! 1. Therefore, for the purpose of desirable results, we introduce the function HðtÞ of the form
(R1 Rt1
HðtÞ ¼
tk
hðsÞds;
t 2 ðt k ; tkþ1 ;
ð1:4Þ
hðsÞds þ aþk1 ; t ¼ t k ; k 2 Zþ ;
S where aþ f0g, and a0 ¼ 0. k ¼ maxfak ; 0g; k 2 Zþ In system (1.3), sðtÞ is monotone increasing for t > t0 and sðt0 Þ 6 t0 , 0 < bi < 1 satisfies b1 < b2 < < bn ; i 2 K; CðtÞ; Pi ðtÞ; hðtÞ 2 PCðR; RÞ; f 2 CðR; RÞ, where K , f1; 2; ; ng; R denotes the family of all real numbers, PCðR; RÞ denotes the set of all functions u : R ! R such that u is continuous everywhere except at some points þ t k ; k 2 Zþ , and the limits uðt þ k Þ ¼ limt!t uðtÞ; uðt k Þ ¼ limt!t k uðtÞ exist with uðt k Þ ¼ uðt k Þ; the sequence ft k g; k 2 Zþ is impulk
sive time which satisfying 0 < t 0 < t k < tkþ1 " 1 as k ! 1; fak g; k 2 Zþ is a constant impulsive perturbations sequence, where Zþ denotes the set of all positive integers. The rest of the paper is organized as follows. In the next section, we present some preliminaries. In Section 3, we give and prove our main results by a technique of construction. Finally, in Section 4, as an application of our results, we present an example to illustrate the usefulness of the obtained results. 2. Preliminaries In connection with the nonlinear function f, the forced term hðtÞ, and the impulsive perturbations sequence fak g in system (1.3), we assume that the following assumptions hold. (H1) there exists a constant M > 0 such that
jf ðxÞj 6 Mjxj;
x 2 R;
xf ðxÞ > 0 for x – 0;
(H2) for all t P t0 > 0, the integration
GðtÞ ¼
Z
1
hðsÞds is convergent;
t þ (H3) sðt k Þ is not an impulsive point for all k 2 Zþ , and limk!1 aþ k ¼ 0, where ak ¼ maxfak ; 0g.
With system (1.3), one associates with an initial value condition
xt0 ¼ uðdÞ;
n
d 2 ½q; 1;
ð2:1Þ
o where q ¼ min t0 ; b1 ; xt0 ¼ xðdt 0 Þ for q 6 d 6 1 and u 2 PCð½q; 1; RÞ , fu : ½q; 1 ! R : u is continuous everywhere exþ þ cept at some points tk ; k 2 Zþ , and uðt k Þ ¼ limt!t k uðtÞ, uðt k Þ ¼ limt!t uðtÞ exist with uðt k Þ ¼ uðt k Þg. sðt0 Þ
k
One can easily show the global existence and uniqueness of the solution of the initial value problem (1.3) and (2.1). In the following, we give two relevant definitions. Definition 2.1. A function xðtÞ is said to be a solution of system (1.3) satisfying the initial value condition (2.1), if (1) xðtÞ ¼ uðt=t 0 Þ for qt0 6 t 6 t0 , and xðtÞ is continuous for t P t0 ; t – tk ; k 2 Zþ ; (2) xðtÞ CðtÞxðsðtÞÞ is continuously differentiable for t > t0 ; t – tk =bi ; i 2 K; k 2 Zþ and satisfies system (1.3); (3) xðtþ k Þ and xðt k Þ exist with xðt k Þ ¼ xðt k Þ for all k 2 Zþ , and satisfies system (1.3).
Definition 2.2. A solution xðtÞ of system (1.3) is said to be eventually positive (negative) if it is positive (negative) for all sufficiently large t. It is said to be oscillatory if it is neither eventually positive nor eventually negative. Otherwise, it is said to be non-oscillatory.
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F. Jiang, J. Sun / Applied Mathematics and Computation 219 (2013) 9906–9913
3. Main results Theorem 3.1. Let ðH1 Þ-ðH3 Þ hold. Assume that for any n > 0, there exists a constant g > 0 such that
jf ðxÞj P g for jxj P n:
ð3:1Þ
Suppose
lim sup jCðtÞj ¼ C < 1
ð3:2Þ
t!1
and n X Pi ðt=bi Þ
t
i¼1
P 0;
Z
1
t0
n X Pi ðt=bi Þ
t
i¼1
dt ¼ 1
ð3:3Þ
and for sufficiently large t, there exists a constant k > 0 such that
XZ bi
rt
bi t
þ X Z bi t Pi ðs=bi Þ Pi ðs=bi Þ 1C ds þ ds 6 k < ; s s M b >r rt
ð3:4Þ
i
þ n o n o iÞ iÞ iÞ iÞ where r 2 ð0; bn , and Pi ðs=b ¼ max Pi ðs=b ; 0 , Pi ðs=b ¼ max Pi ðs=b ; 0 . Then every non-oscillatory solution of system s s s s (1.3) tends to zero as t ! 1. Proof. Choose a positive integer N sufficiently large such that there exists an integer m large enough which satisfying
sðtm Þ > tN and (3.4) holds for t P tN , where N is the largest subscript of satisfying sðtm Þ > tN . Let xðtÞ be a non-oscillatory solution of system (1.3). Without loss of generality, we will assume that xðtÞ is eventually positive solution, the case where xðtÞ is eventually negative is similar and we here omit it. Let xðtÞ > 0 for t P tN . For all t P t N , define
aðtÞ ¼
aþNt ; t > tNþ1 ; 0;
ð3:5Þ
t 2 ½t N ; tNþ1 ;
where N t corresponds to the largest subscript of impulsive points in the interval ðtN ; tÞ. Set
yðtÞ ¼ xðtÞ CðtÞxðsðtÞÞ
n Z X i¼1
rt
bi t
Pi ðs=bi Þ f ðxðsÞÞds þ HðtÞ aðtÞ; s
ð3:6Þ
where HðtÞ is as in (1.4). When t – tk , we choose Dt sufficiently small such that there is no impulsive point in the interval ðt; t þ DtÞ, then
a0 ðtÞ ¼ lim
Dt!0
aðt þ DtÞ aðtÞ Dt
¼ 0;
t – tk :
From this and (3.6), ðH2 Þ-ðH3 Þ, we see that for t – t k ; t – t k =bi ; i 2 K; k 2 Zþ ,
y0 ðtÞ ¼ ½xðtÞ CðtÞxðsðtÞÞ0
n X Pi ðrt=bi Þ
t
i¼1
f ðxðrtÞÞ þ
n X Pi ðtÞ i¼1
n X Pi ðrt=bi Þ f ðxðbi tÞÞ hðtÞ ¼ f ðxðrtÞÞ t t i¼1
ð3:7Þ
and for t ¼ tk ; k ¼ N þ 1; N þ 2; ,
Hðtþk Þ Hðtk Þ ¼ aþk1 :
ð3:8Þ
Moreover, when k ¼ N þ 1,
yðtþNþ1 Þ yðtNþ1 Þ ¼ aNþ1 aðt þN Þ aðt þNþ1 Þ 6 0; when k ¼ N þ 2; N þ 3; . . ., it follows from (H3), (3.5) and (3.8) that
yðtþk Þ yðt k Þ ¼ xðtþk Þ xðtk Þ þ Hðtþk Þ Hðt k Þ aðtþk Þ þ aðtk Þ ¼ ak aþk1 aþk þ aþk1 6 0: Which together with (3.7) imply that yðtÞ is nonincreasing on trN ; 1 . Let L ¼ limt!1 yðtÞ, we claim that L 2 R. Otherwise, L ¼ 1, then xðtÞ is unbounded. In fact, if xðtÞ is bounded, it follows from (3.6) and (H1) that for some constant G > 0,
yðtÞ P xðtÞ CxðsðtÞÞ G
" XZ bi
rt bi t
# þ X Z bi t Pi ðs=bi Þ Pi ðs=bi Þ ds þ ds þ HðtÞ aðtÞ: s s b >r rt i
Furthermore, it follows from ðH2 Þ-ðH3 Þ and (3.4) that L > 1. This is a contradiction and so xðtÞ is unbounded.
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F. Jiang, J. Sun / Applied Mathematics and Computation 219 (2013) 9906–9913
On the other hand, since xðtÞ is unbounded and limt!1 yðtÞ ¼ 1, we can choose a t P maxftN =b1 ; sðt m Þg (sufficiently large if necessary) such that yðt Þ Hðt Þ þ aðt Þ < 0 and xðt Þ ¼ maxfxðtÞ : maxftm ; t N g 6 t 6 t g. Thus, it follows from (3.4) that
0 > yðt Þ Hðt Þ þ aðt Þ
þ X Z bi t P i ðs=bi Þ Pi ðs=bi Þ xðsÞds M xðsÞds þ Hðt Þ aðt Þ Hðt Þ þ aðt Þ s s bir rt ( " #) X Z rt P i ðs=bi Þþ X Z bi t Pi ðs=bi Þ P xðt Þ 1 C M ds þ ds P xðt Þ½1 C Mk > 0: s s b r rt
P xðt Þ Cxðsðt ÞÞ M
XZ
rt
i
i
Which is a contradiction and so L 2 R. By integrating both sides of (3.7) from tN =b1 to t, we obtain that
Z
n X Pi ðrs=bi Þ
t
s
t N =b1 i¼1
f ðxðsrÞÞds ¼
Z
t
y0 ðsÞds ¼ yðt N =b1 Þ yðtÞ þ tN =b1
X
½yðt þk Þ yðtk Þ < yðtN =b1 Þ L:
ð3:9Þ
t N =b1
(3.9) together with (3.3) imply that
f ðxðtÞÞ 2 L1 ð½t N =b1 ; 1Þ; RÞ; and then lim inf t!1 f ðxðtÞÞ ¼ 0. We claim that
lim inf xðtÞ ¼ 0:
ð3:10Þ
t!1
Let fSm g satisfies 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 exists a subsequence fSmk g of fSm g such that xðSmk Þ P 2c for k sufficiently large. But due to (3.1), we have f ðxðSmk ÞÞ P gc for some gc > 0 and sufficiently large k. Which yields a contradiction because of limk!1 f ðxðSmk ÞÞ ¼ 0. Therefore, (3.10) holds. From (3.9), we obtain
Z
1
t0
n X Pi ðrs=bi Þ i¼1
s
f ðxðsrÞÞds < 1:
ð3:11Þ
Set
zðtÞ ¼ yðtÞ þ
n Z X
rt
bi t
i¼1
Pi ðs=bi Þ f ðxðsÞÞds HðtÞ þ aðtÞ: s
By the preceding proofs, one can see that xðtÞ is bounded. From ðH2 Þ-ðH3 Þ and (3.11), we have
limzðtÞ ¼ l exists:
t!1
Which together with (3.6) field that
lim½xðtÞ CðtÞxðsðtÞÞ ¼ l:
ð3:12Þ
t!1
If 0 < lim inf t!1 jCðtÞj < lim supt!1 jCðtÞj ¼ C < 1, one can see that CðtÞ is eventually positive or eventually negative. Otherwise, there is a sequence fsk g with sk ! 1 as k ! 1 such that Cðsk Þ ¼ 0, and so limk!1 Cðsk Þ ¼ 0. Which is a contradiction. Thus, one can find a sufficiently large T such that
0 < jCðtÞj < 1;
for all t > T:
Let
g ¼ lim sup xðtÞ: t!1
Which together with lim inf t!1 xðtÞ ¼ 0 imply that there exist two sequences fun g and fv n g with un ! 1; v n ! 1 as n ! 1, such that
limxðun Þ ¼ 0;
t!1
limxðv n Þ ¼ g:
t!1
On the other hand, it follows that there exists an integer n0 sufficiently large such that sðun Þ P tN ; sðv n Þ P tN for all n P n0 . We from the following two possible cases to discuss. Case 1. 1 < CðtÞ < 0 for t > maxfT; sðun0 Þ; sðv n0 Þg, where n0 sufficiently large, then we have
l ¼ n!1 lim ½xðun Þ Cðun Þxðsðun ÞÞ 6 lim supxðun Þ þ lim sup½Cðun Þxðsðun ÞÞ 6 C g n!1
and
n!1
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F. Jiang, J. Sun / Applied Mathematics and Computation 219 (2013) 9906–9913
l ¼ n!1 lim ½xðv n Þ Cðv n Þxðsðv n ÞÞ P lim inf xðv n Þ þ lim inf ½Cðv n Þxðsðv n ÞÞ P g; n!1 n!1 which mean that C g P g, it follows from g P 0 and 0 < C < 1 that g ¼ 0. This shows limt!1 xðtÞ ¼ 0. Case 2. 0 < CðtÞ < 1 for t > maxfT; sðun0 Þ; sðv n0 Þg, where n0 sufficiently large, then we have
0 ¼ lim xðun Þ ¼ lim ½xðun Þ Cðun Þxðsðun ÞÞ þ Cðun Þxðsðun ÞÞ ¼ lim ½xðun Þ Cðun Þxðsðun ÞÞ þ lim ½Cðun Þxðsðun ÞÞ P l n!1
n!1
n!1
n!1
and
g ¼ n!1 lim xðv n Þ ¼ lim ½xðv n Þ Cðv n Þxðsðv n ÞÞ þ Cðv n Þxðsðv n ÞÞ ¼ lim ½xðv n Þ Cðv n Þxðsðv n ÞÞ þ lim ½Cðv n Þxðsðv n ÞÞ n!1 n!1 n!1 6 l þ C g; which mean that ð1 CÞg 6 l 6 0, it follows from 0 < C < 1 and g P 0 that g ¼ 0. This shows limt!1 xðtÞ ¼ 0. If 0 ¼ lim inf t!1 jCðtÞj < lim supt!1 jCðtÞj ¼ C < 1, then it follows that
lim inf CðtÞ 6 0; t!1
lim sup CðtÞ 6 C < 1: t!1
Furthermore, we have
l ¼ n!1 lim ½xðun Þ Cðun Þxðsðun ÞÞ 6 C g and
l ¼ n!1 lim ½xðv n Þ Cðv n Þxðsðv n ÞÞ P g which imply that g ¼ 0, and so limt!1 xðtÞ ¼ 0. If 0 6 lim inf t!1 jCðtÞj ¼ lim supt!1 jCðtÞj ¼ C < 1, that is, the limit limt!1 jCðtÞj exists with 0 6 limt!1 jCðtÞj ¼ C < 1, then it can be proved in a similar way to Theorem 2.1 from [1]. Therefore, we conclude that limt!1 xðtÞ ¼ 0, and so the proof of Theorem 3.1 is complete. h Theorem 3.2. Let ðH1 Þ-ðH2 Þ hold. Assume that sðt k Þ is not an impulsive point for all k 2 Zþ , and
lim supjCðtÞj ¼ C < 1; t!1
lim ak ¼ 0:
ð3:13Þ
k!1
Suppose that there exist positive constants k and r 2 ð0; bn such that
lim sup Q 1 ðtÞ þ lim sup Q 2 ðtÞ 6 k < t!1
t!1
1 2C M
ð3:14Þ
and for sufficiently large t, n X Pi ðt=bi Þ i¼1
t
– 0;
ð3:15Þ
where
Q 1 ðtÞ ¼
n Z X i¼1
Q 2 ðtÞ ¼
P i ðs=bi Þ s ds; bi t t
n Z X i¼1
bi t rt
ð3:16Þ
Pi ðs=bi Þ ds: sgnðbi rÞ s
ð3:17Þ
Then every oscillatory solution of system (1.3) tends to zero as t ! 1. Proof. Let xðtÞ be an oscillatory solution of system (1.3). We first show that xðtÞ is bounded. Otherwise, xðtÞ is unbounded which implies that there exists a positive integer N sufficiently large such that limt!1 supsðtN Þ6s6t jxðsÞj ¼ 1 and
sup jxðsÞj ¼ sðtN Þ6s6t
sup jxðsÞj; tN =b1 6s6t
t P maxfsðt N Þ; t N =b1 g:
Set
yðtÞ ¼ xðtÞ CðtÞxðsðtÞÞ
n Z X i¼1
rt
bi t
Pi ðs=bi Þ f ðxðsÞÞds þ HðtÞ aðtÞ; s
where HðtÞ; aðtÞ are as in (1.4) and (3.5). By some simple calculation, we get for t – tk ; t – tk =bi ; i 2 K; k 2 Zþ ,
F. Jiang, J. Sun / Applied Mathematics and Computation 219 (2013) 9906–9913 n X Pi ðrt=bi Þ f ðxðrtÞÞ y0 ðtÞ ¼ t i¼1
9911
ð3:18Þ
and for t ¼ t k ; k ¼ N þ 1; N þ 2; . . .
yðt þk Þ yðtk Þ 6 0: When t P t N =b1 , we have
jyðtÞj P jxðtÞj CjxðsðtÞÞj
n Z X
bi t rt
i¼1
Pi ðs=bi Þ jf ðxðsÞÞjds jHðtÞj jaðtÞj sgnðbi rÞ s
P jxðtÞj sup jxðsÞj½C þ MQ 2 ðtÞ jHðtÞj jaðtÞj sðtN Þ6s6t
and so
"
#
sup jyðsÞj P sup jxðsÞj 1 C M sup Q 2 ðsÞ sðtN Þ6s6t
t N =b1 6s6t
tN =b1 6s6t
sup jaðsÞj:
sup jHðsÞj t N =b1 6s6t
ð3:19Þ
t N =b1 6s6t
It follows from (3.14) that ½1 C MsuptN =b1 6s6t Q 2 ðsÞ > 0, thus we have lim supt!1 jyðtÞj ¼ 1 by (H2) and (3.13). On the other hand, it follows from (H1) and (3.18) that y0 ðtÞ is oscillatory, so there exists a n P maxfsðtN Þ=b1 ; t N =b1 ; t N =b21 g such that
jyðnÞj ¼
sup jyðsÞj;
and y0 ðnÞ ¼ 0:
t N =b1 6s6n
Which together with (3.18) mean that xðrnÞ ¼ 0. By integrating both sides of (3.18) from rn to n, we obtain
yðnÞ ¼ yðrnÞ
Z
n
n X Pi ðsr=bi Þ
s
rn i¼1
6 CðrnÞxðsðnrÞÞ
n Z X
½yðt þk Þ yðtk Þ
rn
Pi ðs=bi Þ f ðxðsÞÞds þ HðrnÞ aðrnÞ: s
bi rn
i¼1
X
f ðxðsrÞÞds þ
Which implies that
jyðnÞj 6
sup jxðsÞjfC þ MQ 1 ðrnÞg þ jHðrnÞj þ jaðrnÞj:
ð3:20Þ
sðtN Þ6s6n
From (3.19) and (3.20), we get
"
!#
1 þ 2C þ M Q 1 ðrnÞ þ
sup Q 2 ðsÞ
þ
t N =b1 6s6n
jHðrnÞj þ jaðrnÞj suptN =b1 6s6n jHðsÞj þ suptN =b1 6s6n jaðsÞj P 0: þ supsðtN Þ6s6n jxðsÞj supsðtN Þ6s6n jxðsÞj
Letting n ! 1 and noting that limn!1 supsðtN Þ6s6n jxðsÞj ¼ 1, then by (H2) and (3.13), (3.14), we get
1 þ 2C þ Mk P 0; which contradicts with k < 12C , and so xðtÞ is bounded. M In the following, we shall prove that l ¼ lim supt!1 jxðtÞj ¼ 0. For this purpose, we again analysis the function yðtÞ defined above,
yðtÞ ¼ xðtÞ CðtÞxðsðtÞÞ þ
n Z X i¼1
tbi
rt
Pi ðs=bi Þ f ðxðsÞÞds þ HðtÞ aðtÞ: s
ð3:21Þ
From the preceding discussion, we know that yðtÞ is bounded, and for sufficiently large t,
jyðtÞj P jxðtÞj CjxðsðtÞÞj MQ 2 ðtÞ sup jxðsÞj jHðtÞj jaðtÞj: t N =b1 6s6t
Thus it follows from (H2) and (3.13) that,
b ¼ lim supjyðtÞj P lð1 CÞ M l lim supQ 2 ðtÞ ¼ lð1 C M lim supQ 2 ðtÞÞ: t!1
t!1
ð3:22Þ
t!1
On the other hand, for t – t k ; t – tk =bi ; i 2 K; k 2 Zþ , n X Pi ðtr=bi Þ f ðxðtrÞÞ y0 ðtÞ ¼ t i¼1
and y0 ðtÞ is oscillatory by (H1). Whence, there exists a sequence fnm g such that
ð3:23Þ
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F. Jiang, J. Sun / Applied Mathematics and Computation 219 (2013) 9906–9913
lim nm ¼ 1;
lim jyðnm Þj ¼ b;
m!1
m!1
y0 ðnm Þ ¼ 0:
Moreover, (3.23) and (H1) imply that xðrnm Þ ¼ 0; m ¼ 1; 2; . . . . Integrating (3.23) from rnm to nm and by a similar discussion to (3.20), we obtain
sup jxðsÞj½C þ MQ 1 ðrnm Þ þ jHðrnm Þj þ jaðrnm Þj:
jyðnm Þj 6
b1 nm 6s6nm
Which combine with (H2) and (3.13) imply that
b 6 lðC þ M lim supQ 1 ðtÞÞ: t!1
This together with (3.22) yield
l½1 C M lim supQ 2 ðtÞ 6 l½C þ M lim supQ 1 ðtÞ; t!1
t!1
that is,
l 1 þ 2C þ M lim supQ 1 ðtÞ þ lim supQ 2 ðtÞ t!1
P 0:
t!1
It follows from (3.14) that
lð1 þ 2C þ MkÞ P 0: l P 0, we have l ¼ 0. Therefore limt!1 xðtÞ ¼ 0, and so the proof of Theorem 3.2 is complete. h
Since k < 12C and M 4. Example
Consider impulsive neutral delay differential equation in Euler form
8 < :
1
1 2ðln1t1Þ ½xðtÞ CðtÞx et 0 þ 4ðlntt1Þ x et þ 2t x 2et ¼ t12 ; t P t0 ¼ e; t – tk ;
þ
1
xðk Þ xðkÞ ¼ ð1Þk k ;
ð4:1Þ
t k ¼ k ¼ 1; 2; . . . ;
where
( ½t
t 2 ð2k; 2k þ 1; k ¼ 0; 1; 2; . . . ;
;
4k ½t 4k2
CðtÞ ¼
; t 2 ð2k 1; 2k; k ¼ 1; 2; . . . 1
1 It is easy to see that sðtÞ ¼ et ; P1 ðtÞ ¼ 2 ln11t1 ; P 2 ðtÞ ¼ 4ðln1t1Þ ; f ðxÞ ¼ x; hðtÞ ¼ t12 , and 0 < b1 ¼ 2e < b2 ¼ 1e < 1; ak ¼ ð1Þk k . ð 2 Þ 1 On the other hand, by some simple calculation, we have lim supt!1 CðtÞ ¼ 2 ; lim inf t!1 CðtÞ ¼ 0, and 1 1 limt!1 aþ k ¼ limt!1 ak ¼ 0. When choose M ¼ 1; r ¼ b2 ¼ e and k ¼ 3, we claim that every non-oscillatory solution of Eq. (4.1) tends to zero as t ! 1. In fact, it is easily verify that ðH1 Þ-ðH3 Þ and (3.1) hold. In addition, for t P e, we have
P1
t b1
t Z
1
e
þ
P2
t b2
t
¼
3 P 0; 4t ln t
3 dt ¼ 1 4t ln t
and for large enough t,
Z
t e
t 2e
t dt 1 ¼ ln ln tjet ! 0 2e 2t ln t 2
by L’Hospital’s rule. Therefore by Theorem 3.1, we have that every non-oscillatory solution of Eq. (4.1) tends zero as t ! 1. However, we should be noted that the same conclusion cannot be obtained by Theorem 2.1 from [7], because limt!1 CðtÞ ¼ jpj (jpj < 1 is a constant) is not satisfied. Acknowledgements This work is supported by the NSF of China under grants 61174039, 11171085, and by the Fundamental Research Funds for the Central Universities of China.
F. Jiang, J. Sun / Applied Mathematics and Computation 219 (2013) 9906–9913
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References [1] K.Z. Guan, J.H. Shen, Asymptotic behavior of solutions of a first-order impulsive neutral differential equation in Euler form, Appl. Math. Lett. 24 (2011) 1218–1224. [2] J.H. Shen, Y.J. Liu, Asymptotic behavior of solutions of nonlinear neutral differential equations with impulses, J. Math. Anal. Appl. 322 (2007) 179–189. [3] G.P. Wei, J.H. Shen, Asymptotic behavior for a class of nonlinear impulsive neutral delay differential equations, J. Math. Phys. 30 (2010) 753–763. [4] A. Zhao, J. Yan, Asymptotic behavior of solutions of impulsive delay differential equations, J. Math. Anal. Appl. 201 (1996) 943–954. [5] F.F. Jiang, J.H. Shen, Asymptotic behaviors of nonlinear neutral impulsive delay differential equations with forced term, Kodai. Math. J. 35 (2012) 126– 137. [6] X.Z. Liu, J.H. Shen, Asymptotic behavior of solutions of impulsive neutral differential equations, Appl. Math. Lett. 12 (1999) 51–58. [7] F.F. Jiang, J.H. Shen, Asymptotic behavior of solutions for a nonlinear differential equation with constant impulsive jumps, Acta Math. Hungar. 138 (2013) 1–14. [8] F.F. Jiang, J.H. Shen, On the asymptotic stability for impulsive functional differential equations, Acta Math. Hungar. 134 (2012) 307–321. [9] B. Zhang, Asymptotic stability in functional differential equations by Lyapunov functionals, Trans. Am. Math. Soc. 347 (1995) 1375–1382. [10] K.Z. Guan, J.H. Shen, On first-order neutral differential equations of Euler form with unbounded delay, Appl. Math. Comput. 189 (2007) 1419–1427. [11] K.Z. Guan, J.H. Shen, Oscillation criteria for a first-order impulsive neutral differential equations of Euler form, Comput. Math. Appl. 58 (2009) 670–677. [12] C.X. Li, J.P. Shi, J.T. Sun, Stability of impulsive stochastic differential delay systems and its application to impulsive stochastic neural networks, Nonlinear Anal. - Theory Methods Appl. 74 (2011) 3099–3111. [13] L.J. Chen, J.T. Sun, Nonlinear boundary value problem of first order impulsive functional differential equations, J. Math. Anal. Appl. 318 (2006) 726–741. [14] H. Chen, J.T. Sun, Stability analysis for coupled systems with time delay on networks, Phys. A 391 (2012) 528–534.
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Asymptotic behavior of neutral delay differential equation of euler form with constant impulsive jumps Fangfang Jiang, Jitao Sun ⇑ Department of Mathematics, Tongji University, Shanghai 200092, China
a r t i c l e
i n f o
Keywords: Asymptotic behavior Nonlinear neutral differential equation Unbounded delay Constant impulsive jumps Euler form
a b s t r a c t In this paper, we investigate the asymptotic behavior of solutions for a class of forced nonlinear neutral differential equation in first-order Euler form with constant impulsive jumps and unbounded delay. Several new sufficient conditions are given to guarantee that every non-oscillatory/oscillatory solution of system tends to zero as t ! 1. Our conclusions improve and generalize some related existing results in the literatures. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction It is well known that there are two basic methods in studying the asymptotic behavior of solutions of neutral delay differential equation with/without impulsive perturbations. One direction is by the construction of Lyapunov (functions) functionals, see [1–5]. In [1], Guan and Shen investigated the following impulsive neutral delay differential equation in Euler form
(
½xðtÞ CðtÞxðatÞ0 þ PðtÞ xðbtÞ ¼ 0; t P t 0 > 0; t – t k ; t Rt xðt þk Þ ¼ bk xðtk Þ þ ð1 bk Þ btkk Pðs=bÞ xðsÞds; k 2 Zþ s
ð1:1Þ
and obtained that every solution of system (1.1) tends to a constant as t ! 1. The other direction is by considering the asymptotic behavior of non-oscillatory and oscillatory solutions respectively, for example, see [6,7]. Such as in [6], the authors studied the asymptotic behavior of every non-oscillatory/ oscillatory solution for a forced nonlinear neutral differential equation with impulses and bounded delay
8 n X > < ½xðtÞ pxðt sÞ0 þ qi ðtÞf ðxðt ri ÞÞ ¼ hðtÞ; t – t k ; i¼1 > : þ k 2 Zþ : xðt k Þ xðt k Þ ¼ bk xðt k Þ;
ð1:2Þ
For the related stability results of impulsive functional differential equations, and the corresponding oscillatory theory on neutral differential equation in Euler form, the reader may want to consult [8–14] and the references therein. To the best of our knowledge, there is rarely result concerning the asymptotic behavior of non-oscillatory and/ or oscillatory solutions of impulsive neutral differential system. Especially, only paper [7] studies the asymptotic behavior of solutions for a nonlinear neutral delay differential equation with constant impulsive jumps. Moreover, there is hardly any result concerning the asymptotic behavior of solutions for a class of neutral differential equation in Euler form with constant
⇑ Corresponding author. E-mail address: [email protected] (J. Sun). 0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.04.022
F. Jiang, J. Sun / Applied Mathematics and Computation 219 (2013) 9906–9913
9907
impulsive jumps. Therefore, inspired by the ideas from [1,7], in present paper, we consider the asymptotic behavior of every non-oscillatory/ oscillatory solution for the following forced nonlinear neutral differential equation in first-order Euler form with constant impulsive jumps and unbounded delay
8 n X > Pi ðtÞ < ½xðtÞ CðtÞxðsðtÞÞ0 þ f ðxðbi tÞÞ ¼ hðtÞ; t P t0 > 0; t – t k ; t i¼1 > : þ k 2 Zþ : xðtk Þ xðtk Þ ¼ ak ;
ð1:3Þ
It should be noted that in previous references, such as [1,6,7], the condition limt!1 jCðtÞj ¼ C < 1 is satisfied, while in our work, we do not require limt!1 jCðtÞj exists. In fact, we obtain our results just require lim supt!1 jCðtÞj < 1. So the obtained conclusions further improve and generalize some related existing results. In addition, it is worth mentioning that in system (1.3), we cannot simply and directly apply the above two methods to derive sufficient conditions such that every non-oscillatory/ oscillatory solution of system (1.3) tends to zero as t ! 1. Therefore, for the purpose of desirable results, we introduce the function HðtÞ of the form
(R1 Rt1
HðtÞ ¼
tk
hðsÞds;
t 2 ðt k ; tkþ1 ;
ð1:4Þ
hðsÞds þ aþk1 ; t ¼ t k ; k 2 Zþ ;
S where aþ f0g, and a0 ¼ 0. k ¼ maxfak ; 0g; k 2 Zþ In system (1.3), sðtÞ is monotone increasing for t > t0 and sðt0 Þ 6 t0 , 0 < bi < 1 satisfies b1 < b2 < < bn ; i 2 K; CðtÞ; Pi ðtÞ; hðtÞ 2 PCðR; RÞ; f 2 CðR; RÞ, where K , f1; 2; ; ng; R denotes the family of all real numbers, PCðR; RÞ denotes the set of all functions u : R ! R such that u is continuous everywhere except at some points þ t k ; k 2 Zþ , and the limits uðt þ k Þ ¼ limt!t uðtÞ; uðt k Þ ¼ limt!t k uðtÞ exist with uðt k Þ ¼ uðt k Þ; the sequence ft k g; k 2 Zþ is impulk
sive time which satisfying 0 < t 0 < t k < tkþ1 " 1 as k ! 1; fak g; k 2 Zþ is a constant impulsive perturbations sequence, where Zþ denotes the set of all positive integers. The rest of the paper is organized as follows. In the next section, we present some preliminaries. In Section 3, we give and prove our main results by a technique of construction. Finally, in Section 4, as an application of our results, we present an example to illustrate the usefulness of the obtained results. 2. Preliminaries In connection with the nonlinear function f, the forced term hðtÞ, and the impulsive perturbations sequence fak g in system (1.3), we assume that the following assumptions hold. (H1) there exists a constant M > 0 such that
jf ðxÞj 6 Mjxj;
x 2 R;
xf ðxÞ > 0 for x – 0;
(H2) for all t P t0 > 0, the integration
GðtÞ ¼
Z
1
hðsÞds is convergent;
t þ (H3) sðt k Þ is not an impulsive point for all k 2 Zþ , and limk!1 aþ k ¼ 0, where ak ¼ maxfak ; 0g.
With system (1.3), one associates with an initial value condition
xt0 ¼ uðdÞ;
n
d 2 ½q; 1;
ð2:1Þ
o where q ¼ min t0 ; b1 ; xt0 ¼ xðdt 0 Þ for q 6 d 6 1 and u 2 PCð½q; 1; RÞ , fu : ½q; 1 ! R : u is continuous everywhere exþ þ cept at some points tk ; k 2 Zþ , and uðt k Þ ¼ limt!t k uðtÞ, uðt k Þ ¼ limt!t uðtÞ exist with uðt k Þ ¼ uðt k Þg. sðt0 Þ
k
One can easily show the global existence and uniqueness of the solution of the initial value problem (1.3) and (2.1). In the following, we give two relevant definitions. Definition 2.1. A function xðtÞ is said to be a solution of system (1.3) satisfying the initial value condition (2.1), if (1) xðtÞ ¼ uðt=t 0 Þ for qt0 6 t 6 t0 , and xðtÞ is continuous for t P t0 ; t – tk ; k 2 Zþ ; (2) xðtÞ CðtÞxðsðtÞÞ is continuously differentiable for t > t0 ; t – tk =bi ; i 2 K; k 2 Zþ and satisfies system (1.3); (3) xðtþ k Þ and xðt k Þ exist with xðt k Þ ¼ xðt k Þ for all k 2 Zþ , and satisfies system (1.3).
Definition 2.2. A solution xðtÞ of system (1.3) is said to be eventually positive (negative) if it is positive (negative) for all sufficiently large t. It is said to be oscillatory if it is neither eventually positive nor eventually negative. Otherwise, it is said to be non-oscillatory.
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F. Jiang, J. Sun / Applied Mathematics and Computation 219 (2013) 9906–9913
3. Main results Theorem 3.1. Let ðH1 Þ-ðH3 Þ hold. Assume that for any n > 0, there exists a constant g > 0 such that
jf ðxÞj P g for jxj P n:
ð3:1Þ
Suppose
lim sup jCðtÞj ¼ C < 1
ð3:2Þ
t!1
and n X Pi ðt=bi Þ
t
i¼1
P 0;
Z
1
t0
n X Pi ðt=bi Þ
t
i¼1
dt ¼ 1
ð3:3Þ
and for sufficiently large t, there exists a constant k > 0 such that
XZ bi
rt
bi t
þ X Z bi t Pi ðs=bi Þ Pi ðs=bi Þ 1C ds þ ds 6 k < ; s s M b >r rt
ð3:4Þ
i
þ n o n o iÞ iÞ iÞ iÞ where r 2 ð0; bn , and Pi ðs=b ¼ max Pi ðs=b ; 0 , Pi ðs=b ¼ max Pi ðs=b ; 0 . Then every non-oscillatory solution of system s s s s (1.3) tends to zero as t ! 1. Proof. Choose a positive integer N sufficiently large such that there exists an integer m large enough which satisfying
sðtm Þ > tN and (3.4) holds for t P tN , where N is the largest subscript of satisfying sðtm Þ > tN . Let xðtÞ be a non-oscillatory solution of system (1.3). Without loss of generality, we will assume that xðtÞ is eventually positive solution, the case where xðtÞ is eventually negative is similar and we here omit it. Let xðtÞ > 0 for t P tN . For all t P t N , define
aðtÞ ¼
aþNt ; t > tNþ1 ; 0;
ð3:5Þ
t 2 ½t N ; tNþ1 ;
where N t corresponds to the largest subscript of impulsive points in the interval ðtN ; tÞ. Set
yðtÞ ¼ xðtÞ CðtÞxðsðtÞÞ
n Z X i¼1
rt
bi t
Pi ðs=bi Þ f ðxðsÞÞds þ HðtÞ aðtÞ; s
ð3:6Þ
where HðtÞ is as in (1.4). When t – tk , we choose Dt sufficiently small such that there is no impulsive point in the interval ðt; t þ DtÞ, then
a0 ðtÞ ¼ lim
Dt!0
aðt þ DtÞ aðtÞ Dt
¼ 0;
t – tk :
From this and (3.6), ðH2 Þ-ðH3 Þ, we see that for t – t k ; t – t k =bi ; i 2 K; k 2 Zþ ,
y0 ðtÞ ¼ ½xðtÞ CðtÞxðsðtÞÞ0
n X Pi ðrt=bi Þ
t
i¼1
f ðxðrtÞÞ þ
n X Pi ðtÞ i¼1
n X Pi ðrt=bi Þ f ðxðbi tÞÞ hðtÞ ¼ f ðxðrtÞÞ t t i¼1
ð3:7Þ
and for t ¼ tk ; k ¼ N þ 1; N þ 2; ,
Hðtþk Þ Hðtk Þ ¼ aþk1 :
ð3:8Þ
Moreover, when k ¼ N þ 1,
yðtþNþ1 Þ yðtNþ1 Þ ¼ aNþ1 aðt þN Þ aðt þNþ1 Þ 6 0; when k ¼ N þ 2; N þ 3; . . ., it follows from (H3), (3.5) and (3.8) that
yðtþk Þ yðt k Þ ¼ xðtþk Þ xðtk Þ þ Hðtþk Þ Hðt k Þ aðtþk Þ þ aðtk Þ ¼ ak aþk1 aþk þ aþk1 6 0: Which together with (3.7) imply that yðtÞ is nonincreasing on trN ; 1 . Let L ¼ limt!1 yðtÞ, we claim that L 2 R. Otherwise, L ¼ 1, then xðtÞ is unbounded. In fact, if xðtÞ is bounded, it follows from (3.6) and (H1) that for some constant G > 0,
yðtÞ P xðtÞ CxðsðtÞÞ G
" XZ bi
rt bi t
# þ X Z bi t Pi ðs=bi Þ Pi ðs=bi Þ ds þ ds þ HðtÞ aðtÞ: s s b >r rt i
Furthermore, it follows from ðH2 Þ-ðH3 Þ and (3.4) that L > 1. This is a contradiction and so xðtÞ is unbounded.
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F. Jiang, J. Sun / Applied Mathematics and Computation 219 (2013) 9906–9913
On the other hand, since xðtÞ is unbounded and limt!1 yðtÞ ¼ 1, we can choose a t P maxftN =b1 ; sðt m Þg (sufficiently large if necessary) such that yðt Þ Hðt Þ þ aðt Þ < 0 and xðt Þ ¼ maxfxðtÞ : maxftm ; t N g 6 t 6 t g. Thus, it follows from (3.4) that
0 > yðt Þ Hðt Þ þ aðt Þ
þ X Z bi t P i ðs=bi Þ Pi ðs=bi Þ xðsÞds M xðsÞds þ Hðt Þ aðt Þ Hðt Þ þ aðt Þ s s bi
P xðt Þ Cxðsðt ÞÞ M
XZ
rt
i
i
Which is a contradiction and so L 2 R. By integrating both sides of (3.7) from tN =b1 to t, we obtain that
Z
n X Pi ðrs=bi Þ
t
s
t N =b1 i¼1
f ðxðsrÞÞds ¼
Z
t
y0 ðsÞds ¼ yðt N =b1 Þ yðtÞ þ tN =b1
X
½yðt þk Þ yðtk Þ < yðtN =b1 Þ L:
ð3:9Þ
t N =b1
(3.9) together with (3.3) imply that
f ðxðtÞÞ 2 L1 ð½t N =b1 ; 1Þ; RÞ; and then lim inf t!1 f ðxðtÞÞ ¼ 0. We claim that
lim inf xðtÞ ¼ 0:
ð3:10Þ
t!1
Let fSm g satisfies 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 exists a subsequence fSmk g of fSm g such that xðSmk Þ P 2c for k sufficiently large. But due to (3.1), we have f ðxðSmk ÞÞ P gc for some gc > 0 and sufficiently large k. Which yields a contradiction because of limk!1 f ðxðSmk ÞÞ ¼ 0. Therefore, (3.10) holds. From (3.9), we obtain
Z
1
t0
n X Pi ðrs=bi Þ i¼1
s
f ðxðsrÞÞds < 1:
ð3:11Þ
Set
zðtÞ ¼ yðtÞ þ
n Z X
rt
bi t
i¼1
Pi ðs=bi Þ f ðxðsÞÞds HðtÞ þ aðtÞ: s
By the preceding proofs, one can see that xðtÞ is bounded. From ðH2 Þ-ðH3 Þ and (3.11), we have
limzðtÞ ¼ l exists:
t!1
Which together with (3.6) field that
lim½xðtÞ CðtÞxðsðtÞÞ ¼ l:
ð3:12Þ
t!1
If 0 < lim inf t!1 jCðtÞj < lim supt!1 jCðtÞj ¼ C < 1, one can see that CðtÞ is eventually positive or eventually negative. Otherwise, there is a sequence fsk g with sk ! 1 as k ! 1 such that Cðsk Þ ¼ 0, and so limk!1 Cðsk Þ ¼ 0. Which is a contradiction. Thus, one can find a sufficiently large T such that
0 < jCðtÞj < 1;
for all t > T:
Let
g ¼ lim sup xðtÞ: t!1
Which together with lim inf t!1 xðtÞ ¼ 0 imply that there exist two sequences fun g and fv n g with un ! 1; v n ! 1 as n ! 1, such that
limxðun Þ ¼ 0;
t!1
limxðv n Þ ¼ g:
t!1
On the other hand, it follows that there exists an integer n0 sufficiently large such that sðun Þ P tN ; sðv n Þ P tN for all n P n0 . We from the following two possible cases to discuss. Case 1. 1 < CðtÞ < 0 for t > maxfT; sðun0 Þ; sðv n0 Þg, where n0 sufficiently large, then we have
l ¼ n!1 lim ½xðun Þ Cðun Þxðsðun ÞÞ 6 lim supxðun Þ þ lim sup½Cðun Þxðsðun ÞÞ 6 C g n!1
and
n!1
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F. Jiang, J. Sun / Applied Mathematics and Computation 219 (2013) 9906–9913
l ¼ n!1 lim ½xðv n Þ Cðv n Þxðsðv n ÞÞ P lim inf xðv n Þ þ lim inf ½Cðv n Þxðsðv n ÞÞ P g; n!1 n!1 which mean that C g P g, it follows from g P 0 and 0 < C < 1 that g ¼ 0. This shows limt!1 xðtÞ ¼ 0. Case 2. 0 < CðtÞ < 1 for t > maxfT; sðun0 Þ; sðv n0 Þg, where n0 sufficiently large, then we have
0 ¼ lim xðun Þ ¼ lim ½xðun Þ Cðun Þxðsðun ÞÞ þ Cðun Þxðsðun ÞÞ ¼ lim ½xðun Þ Cðun Þxðsðun ÞÞ þ lim ½Cðun Þxðsðun ÞÞ P l n!1
n!1
n!1
n!1
and
g ¼ n!1 lim xðv n Þ ¼ lim ½xðv n Þ Cðv n Þxðsðv n ÞÞ þ Cðv n Þxðsðv n ÞÞ ¼ lim ½xðv n Þ Cðv n Þxðsðv n ÞÞ þ lim ½Cðv n Þxðsðv n ÞÞ n!1 n!1 n!1 6 l þ C g; which mean that ð1 CÞg 6 l 6 0, it follows from 0 < C < 1 and g P 0 that g ¼ 0. This shows limt!1 xðtÞ ¼ 0. If 0 ¼ lim inf t!1 jCðtÞj < lim supt!1 jCðtÞj ¼ C < 1, then it follows that
lim inf CðtÞ 6 0; t!1
lim sup CðtÞ 6 C < 1: t!1
Furthermore, we have
l ¼ n!1 lim ½xðun Þ Cðun Þxðsðun ÞÞ 6 C g and
l ¼ n!1 lim ½xðv n Þ Cðv n Þxðsðv n ÞÞ P g which imply that g ¼ 0, and so limt!1 xðtÞ ¼ 0. If 0 6 lim inf t!1 jCðtÞj ¼ lim supt!1 jCðtÞj ¼ C < 1, that is, the limit limt!1 jCðtÞj exists with 0 6 limt!1 jCðtÞj ¼ C < 1, then it can be proved in a similar way to Theorem 2.1 from [1]. Therefore, we conclude that limt!1 xðtÞ ¼ 0, and so the proof of Theorem 3.1 is complete. h Theorem 3.2. Let ðH1 Þ-ðH2 Þ hold. Assume that sðt k Þ is not an impulsive point for all k 2 Zþ , and
lim supjCðtÞj ¼ C < 1; t!1
lim ak ¼ 0:
ð3:13Þ
k!1
Suppose that there exist positive constants k and r 2 ð0; bn such that
lim sup Q 1 ðtÞ þ lim sup Q 2 ðtÞ 6 k < t!1
t!1
1 2C M
ð3:14Þ
and for sufficiently large t, n X Pi ðt=bi Þ i¼1
t
– 0;
ð3:15Þ
where
Q 1 ðtÞ ¼
n Z X i¼1
Q 2 ðtÞ ¼
P i ðs=bi Þ s ds; bi t t
n Z X i¼1
bi t rt
ð3:16Þ
Pi ðs=bi Þ ds: sgnðbi rÞ s
ð3:17Þ
Then every oscillatory solution of system (1.3) tends to zero as t ! 1. Proof. Let xðtÞ be an oscillatory solution of system (1.3). We first show that xðtÞ is bounded. Otherwise, xðtÞ is unbounded which implies that there exists a positive integer N sufficiently large such that limt!1 supsðtN Þ6s6t jxðsÞj ¼ 1 and
sup jxðsÞj ¼ sðtN Þ6s6t
sup jxðsÞj; tN =b1 6s6t
t P maxfsðt N Þ; t N =b1 g:
Set
yðtÞ ¼ xðtÞ CðtÞxðsðtÞÞ
n Z X i¼1
rt
bi t
Pi ðs=bi Þ f ðxðsÞÞds þ HðtÞ aðtÞ; s
where HðtÞ; aðtÞ are as in (1.4) and (3.5). By some simple calculation, we get for t – tk ; t – tk =bi ; i 2 K; k 2 Zþ ,
F. Jiang, J. Sun / Applied Mathematics and Computation 219 (2013) 9906–9913 n X Pi ðrt=bi Þ f ðxðrtÞÞ y0 ðtÞ ¼ t i¼1
9911
ð3:18Þ
and for t ¼ t k ; k ¼ N þ 1; N þ 2; . . .
yðt þk Þ yðtk Þ 6 0: When t P t N =b1 , we have
jyðtÞj P jxðtÞj CjxðsðtÞÞj
n Z X
bi t rt
i¼1
Pi ðs=bi Þ jf ðxðsÞÞjds jHðtÞj jaðtÞj sgnðbi rÞ s
P jxðtÞj sup jxðsÞj½C þ MQ 2 ðtÞ jHðtÞj jaðtÞj sðtN Þ6s6t
and so
"
#
sup jyðsÞj P sup jxðsÞj 1 C M sup Q 2 ðsÞ sðtN Þ6s6t
t N =b1 6s6t
tN =b1 6s6t
sup jaðsÞj:
sup jHðsÞj t N =b1 6s6t
ð3:19Þ
t N =b1 6s6t
It follows from (3.14) that ½1 C MsuptN =b1 6s6t Q 2 ðsÞ > 0, thus we have lim supt!1 jyðtÞj ¼ 1 by (H2) and (3.13). On the other hand, it follows from (H1) and (3.18) that y0 ðtÞ is oscillatory, so there exists a n P maxfsðtN Þ=b1 ; t N =b1 ; t N =b21 g such that
jyðnÞj ¼
sup jyðsÞj;
and y0 ðnÞ ¼ 0:
t N =b1 6s6n
Which together with (3.18) mean that xðrnÞ ¼ 0. By integrating both sides of (3.18) from rn to n, we obtain
yðnÞ ¼ yðrnÞ
Z
n
n X Pi ðsr=bi Þ
s
rn i¼1
6 CðrnÞxðsðnrÞÞ
n Z X
½yðt þk Þ yðtk Þ
rn
Pi ðs=bi Þ f ðxðsÞÞds þ HðrnÞ aðrnÞ: s
bi rn
i¼1
X
f ðxðsrÞÞds þ
Which implies that
jyðnÞj 6
sup jxðsÞjfC þ MQ 1 ðrnÞg þ jHðrnÞj þ jaðrnÞj:
ð3:20Þ
sðtN Þ6s6n
From (3.19) and (3.20), we get
"
!#
1 þ 2C þ M Q 1 ðrnÞ þ
sup Q 2 ðsÞ
þ
t N =b1 6s6n
jHðrnÞj þ jaðrnÞj suptN =b1 6s6n jHðsÞj þ suptN =b1 6s6n jaðsÞj P 0: þ supsðtN Þ6s6n jxðsÞj supsðtN Þ6s6n jxðsÞj
Letting n ! 1 and noting that limn!1 supsðtN Þ6s6n jxðsÞj ¼ 1, then by (H2) and (3.13), (3.14), we get
1 þ 2C þ Mk P 0; which contradicts with k < 12C , and so xðtÞ is bounded. M In the following, we shall prove that l ¼ lim supt!1 jxðtÞj ¼ 0. For this purpose, we again analysis the function yðtÞ defined above,
yðtÞ ¼ xðtÞ CðtÞxðsðtÞÞ þ
n Z X i¼1
tbi
rt
Pi ðs=bi Þ f ðxðsÞÞds þ HðtÞ aðtÞ: s
ð3:21Þ
From the preceding discussion, we know that yðtÞ is bounded, and for sufficiently large t,
jyðtÞj P jxðtÞj CjxðsðtÞÞj MQ 2 ðtÞ sup jxðsÞj jHðtÞj jaðtÞj: t N =b1 6s6t
Thus it follows from (H2) and (3.13) that,
b ¼ lim supjyðtÞj P lð1 CÞ M l lim supQ 2 ðtÞ ¼ lð1 C M lim supQ 2 ðtÞÞ: t!1
t!1
ð3:22Þ
t!1
On the other hand, for t – t k ; t – tk =bi ; i 2 K; k 2 Zþ , n X Pi ðtr=bi Þ f ðxðtrÞÞ y0 ðtÞ ¼ t i¼1
and y0 ðtÞ is oscillatory by (H1). Whence, there exists a sequence fnm g such that
ð3:23Þ
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F. Jiang, J. Sun / Applied Mathematics and Computation 219 (2013) 9906–9913
lim nm ¼ 1;
lim jyðnm Þj ¼ b;
m!1
m!1
y0 ðnm Þ ¼ 0:
Moreover, (3.23) and (H1) imply that xðrnm Þ ¼ 0; m ¼ 1; 2; . . . . Integrating (3.23) from rnm to nm and by a similar discussion to (3.20), we obtain
sup jxðsÞj½C þ MQ 1 ðrnm Þ þ jHðrnm Þj þ jaðrnm Þj:
jyðnm Þj 6
b1 nm 6s6nm
Which combine with (H2) and (3.13) imply that
b 6 lðC þ M lim supQ 1 ðtÞÞ: t!1
This together with (3.22) yield
l½1 C M lim supQ 2 ðtÞ 6 l½C þ M lim supQ 1 ðtÞ; t!1
t!1
that is,
l 1 þ 2C þ M lim supQ 1 ðtÞ þ lim supQ 2 ðtÞ t!1
P 0:
t!1
It follows from (3.14) that
lð1 þ 2C þ MkÞ P 0: l P 0, we have l ¼ 0. Therefore limt!1 xðtÞ ¼ 0, and so the proof of Theorem 3.2 is complete. h
Since k < 12C and M 4. Example
Consider impulsive neutral delay differential equation in Euler form
8 < :
1
1 2ðln1t1Þ ½xðtÞ CðtÞx et 0 þ 4ðlntt1Þ x et þ 2t x 2et ¼ t12 ; t P t0 ¼ e; t – tk ;
þ
1
xðk Þ xðkÞ ¼ ð1Þk k ;
ð4:1Þ
t k ¼ k ¼ 1; 2; . . . ;
where
( ½t
t 2 ð2k; 2k þ 1; k ¼ 0; 1; 2; . . . ;
;
4k ½t 4k2
CðtÞ ¼
; t 2 ð2k 1; 2k; k ¼ 1; 2; . . . 1
1 It is easy to see that sðtÞ ¼ et ; P1 ðtÞ ¼ 2 ln11t1 ; P 2 ðtÞ ¼ 4ðln1t1Þ ; f ðxÞ ¼ x; hðtÞ ¼ t12 , and 0 < b1 ¼ 2e < b2 ¼ 1e < 1; ak ¼ ð1Þk k . ð 2 Þ 1 On the other hand, by some simple calculation, we have lim supt!1 CðtÞ ¼ 2 ; lim inf t!1 CðtÞ ¼ 0, and 1 1 limt!1 aþ k ¼ limt!1 ak ¼ 0. When choose M ¼ 1; r ¼ b2 ¼ e and k ¼ 3, we claim that every non-oscillatory solution of Eq. (4.1) tends to zero as t ! 1. In fact, it is easily verify that ðH1 Þ-ðH3 Þ and (3.1) hold. In addition, for t P e, we have
P1
t b1
t Z
1
e
þ
P2
t b2
t
¼
3 P 0; 4t ln t
3 dt ¼ 1 4t ln t
and for large enough t,
Z
t e
t 2e
t dt 1 ¼ ln ln tjet ! 0 2e 2t ln t 2
by L’Hospital’s rule. Therefore by Theorem 3.1, we have that every non-oscillatory solution of Eq. (4.1) tends zero as t ! 1. However, we should be noted that the same conclusion cannot be obtained by Theorem 2.1 from [7], because limt!1 CðtÞ ¼ jpj (jpj < 1 is a constant) is not satisfied. Acknowledgements This work is supported by the NSF of China under grants 61174039, 11171085, and by the Fundamental Research Funds for the Central Universities of China.
F. Jiang, J. Sun / Applied Mathematics and Computation 219 (2013) 9906–9913
9913
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