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Some new delay integral inequalities and their applications
Applied Mathematics and Computation 208 (2009) 231–237
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Some new delay integral inequalities and their applications q Zhiling Yuan a, Xingwei Yuan b, Fanwei Meng c,*, Hongxia Zhang c a
School of Science Jiangnan University, Wuxi 214122, Jiangshu, People’s Republic of China Key and Open Laboratory of Marine and Estuarine Fisheries, Ministry of Agriculture, East China Sea Fisheries Research Institute, Chinese Academy of Fishery Sciences, Shanghai 200090, People’s Republic of China c Department of Mathematics, Qufu Normal University, Qufu 273165, Shandong, People’s Republic of China b
a r t i c l e
i n f o
a b s t r a c t The main objective of this paper is to establish some new delay integral inequalities, which provide explicit bounds on unknown functions. The inequalities given here can be used as tools in the qualitative theory of certain delay differential equations and delay integral equations. Ó 2008 Elsevier Inc. All rights reserved.
Keywords: Integral equation Integral inequalities Bounded
1. Introduction The integral inequalities involving functions of one and more than one independent variables, which provide explicit bounds on unknown function play a fundamental role in the development of the theory of differential equation and integral equations. Many retarded inequalities have been discovered see [1–9], Others results may be found in [10–13]. Recently Jiang and Meng [10] presented a retarded inequality which has very well characters. In this paper, we establish some new inequality, the results of which generalize the results of Jiang and Meng [6] and Pachpatte [12]. 2. Main results In what follows, R denotes the set of real numbers and Rþ ¼ ½0; þ1Þ is the given subset of R, and CðM; SÞ denotes the class of all continuous functions defined on set M with range in the set S. The following lemmas are useful in our main results. Lemma 2.1 [6]. Assume that a P 0; p P q P 0, and p–0, then q q qp p q qp ap 6 K p a þ K ; p p
for any K > 0. Lemma 2.2 [13]. Let u; a; b 2 ðCðRþ ; Rþ Þ and a be nondecreasing, if
uðtÞ 6 aðtÞ þ
Z
t
bðsÞuðsÞds;
0
q
This research was supported by the NNSF of China (10771118). * Corresponding author. E-mail addresses: [email protected] (Z. Yuan), [email protected] (X. Yuan), [email protected] (F. Meng).
0096-3003/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2008.11.043
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Z. Yuan et al. / Applied Mathematics and Computation 208 (2009) 231–237
then
uðtÞ 6 aðtÞexp
Z
t
bðsÞds:
0
Theorem 1. Assume that xðtÞ; aðtÞ; bðtÞ; f ðtÞ; gðtÞ 2 CðRþ ; Rþ Þ. If aðtÞ; bðtÞ are nondecreasing in Rþ , and xðtÞ satisfies the following form of delay integral inequality
xðtÞp 6 aðtÞ þ bðtÞ
Z t Z s f ðsÞxq ðrðsÞÞ þ gðsÞxr ðsÞ þ hðsÞxm ðsÞds ds; 0
t 2 Rþ ;
ð1Þ
0
with the initial condition
t 2 ½a; 0;
xðtÞ ¼ /ðtÞ;
1
/ðrðtÞÞ 6 aðtÞp
f or t 2 Rþ with
ð2Þ
rðtÞ 6 0;
where p–0; p P q P 0; p P r P 0; p P m P 0; p; q; r are constants,
rðtÞ 2 CðRþ ; RÞ; rðtÞ 6 t,
1 < a ¼ inf frðtÞ; t 2 Rþ g 6 0; and /ðtÞ 2 Cð½a; 0; Rþ Þ, then
1p Z t xðtÞ 6 aðtÞ þ bðtÞAðtÞexp BðsÞds ;
ð3Þ
0
for any K > 0, where
Z s Z t q qp p q qp r rp p r pr m mp p m mp f ðsÞ K p aðsÞ þ hðsÞ K þ gðs; tÞ K p aðsÞ þ K þ K p aðsÞ þ K ds ds; p p p p p p 0 0 Z t rp mp q qp r m hðsÞbðsÞ K p ds; t 2 Rþ : BðtÞ ¼ K p f ðtÞ þ K p gðtÞ bðtÞ þ p p p 0 AðtÞ ¼
ð4Þ ð5Þ
Proof. For any fixed positive number T we define a function zðtÞ by
zðtÞ ¼
aðTÞ þ bðtÞ
Z t
f ðsÞxq ðrðsÞÞ þ gðsÞxr ðsÞ þ
Z
0
s
1p hðsÞxm ðsÞds ds ;
t 2 Rþ ;
ð6Þ
0
it is easy to see that zðtÞ is a nonnegative and nondecreasing and
xðtÞ 6 zðtÞ;
t 2 ½0; T:
Therefore, for t 2 Rþ with
ð7Þ
rðtÞ P 0, we have
xðrðtÞÞ 6 zðrðtÞÞ 6 zðtÞ;
t 2 ½0; T:
ð8Þ
On the other hand, using the initial condition (2), for t 2 Rþ with 1 p
1 p
xðrðtÞÞ ¼ /ðrðtÞÞ 6 aðtÞ 6 aðTÞ 6 zðtÞ;
rðtÞ 6 0, we have
t 2 ½0; T:
ð9Þ
Combining (8) and (9), we obtain
xðrðtÞÞ 6 zðtÞ;
t 2 ½0; T:
ð10Þ
It follows from (6) and (10) that
zp ðtÞ 6 aðTÞ þ bðtÞ
Z t
f ðsÞzq ðsÞ þ gðsÞzr ðsÞ þ
Z
0
s
hðsÞzm ðsÞds ds;
t 2 Rþ :
ð11Þ
0
Taking t ¼ T in (11), we obtain
zp ðTÞ 6 aðTÞ þ bðTÞ
Z
T
f ðsÞzq ðsÞ þ gðsÞzr ðsÞ þ
Z
0
s
hðsÞzm ðsÞds ds:
ð12Þ
0
Noting that T 2 Rþ was arbitrary, from (12), we have
zp ðtÞ 6 aðtÞ þ bðtÞ
Z t Z s f ðsÞzq ðsÞ þ gðsÞzr ðsÞ þ hðsÞzm ðsÞds ds; 0
t 2 Rþ :
ð13Þ
0
Similarly, we obtain
xðtÞ 6 zðtÞ t 2 Rþ :
ð14Þ
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Z. Yuan et al. / Applied Mathematics and Computation 208 (2009) 231–237
Define a function uðtÞ by
uðtÞ ¼
Z t
Z
f ðsÞzq ðsÞ þ gðsÞzr ðsÞ þ
0
s
hðsÞzm ðsÞds ds:
ð15Þ
0
Then (13) can be restated as
zp ðtÞ 6 aðtÞ þ bðtÞuðtÞ; t 2 Rþ :
ð16Þ
Using Lemma 1, from (16) for any K > 0, we easily obtain q q qp p q qp zq ðtÞ 6 ðaðtÞ þ bðtÞuðtÞÞp 6 K p ðaðtÞ þ bðtÞuðtÞÞ þ K ; p p r r rp p r pr zr ðtÞ 6 ðaðtÞ þ bðtÞuðtÞÞp 6 K p ðaðtÞ þ bðtÞpðtÞÞ þ K ; p p m m mp p m mp zm ðtÞ 6 ðaðtÞ þ bðtÞuðtÞÞ p 6 K p ðaðtÞ þ bðtÞpðtÞÞ þ K : p p
ð17Þ ð18Þ ð19Þ
It follows (15), (17), (18) and (19) that
q qp p q qp r rp p r pr K p ðaðsÞ þ bðsÞuðsÞÞ þ K þ gðsÞ K p ðaðsÞ þ bðsÞuðsÞÞ þ K p p p p 0 Z s Z t m mp p m mp hðsÞ BðsÞuðsÞds; t 2 Rþ ; þ K p ðaðsÞ þ bðsÞuðsÞÞ þ K ds ds 6 AðtÞ þ p p 0 0
uðtÞ 6
Z
t
f ðsÞ
ð20Þ
where AðtÞ; BðtÞ are defined as in (4) and (5), it is easy to find that AðtÞ is nonnegative, continuous and nondecreasing for t 2 Rþ . By Lemma 2 we obtain
uðtÞ 6 AðtÞexp
Z
t
BðsÞds:
ð21Þ
0
From (16) and (21), it follows that
1p Z t zðtÞ 6 aðtÞ þ bðtÞAðtÞexp BðsÞds :
ð22Þ
0
Thus, the desired inequality (3) follows from (14) and (22). This completes the proof of Theorem 1.
h
Remark. If we take hðtÞ ¼ 0, then the inequalities established in Theorem 1 reduce to [6, Theorem 2.3]. Theorem 2. Assume that p; q; m; xðtÞ; aðtÞ; f ðtÞ; gðtÞ; hðtÞ; rðtÞ are defined as Theorem 1. If xðtÞ satisfies the integral inequality
xðtÞp 6 aðtÞ þ
Z
t
f ðsÞxp ðsÞds þ 0
Z t
gðsÞxq ðrðsÞÞ þ
0
Z
s
hðsÞxm ðsÞds ds;
0
ðx; yÞ 2 R2þ ;
ð23Þ
with the initial condition (2) in Theorem 1, then
" xðtÞ 6 BðtÞ aðtÞ þ FðtÞexp
1p # Z t Z s q qp m mp gðsÞBr ðsÞ K p þ hðsÞBm ðsÞ K p ds ds ; p p 0 0
ð24Þ
for any K > 0, where
1p Z t BðtÞ ¼ exp f ðsÞ ds dt ; 0 Z s Z s p q qp q qp p m mp m mp gðsÞBq ðsÞ hðsÞBm ðsÞ K þ K p aðsÞ þ K þ K p aðsÞ : FðtÞ ¼ p p p p 0 0
ð25Þ ð26Þ
Proof. Fixing any positive number T we define a function zðtÞ by
zðtÞ ¼
1p Z t Z t Z s aðTÞ þ f ðsÞxp ðsÞds þ gðsÞxq ðrðsÞÞds þ hðsÞxm ðsÞds ds : 0
0
ð27Þ
0
Using the similar to the proof Theorem 1, we easily obtain that zðtÞ is a nonnegative and nondecreasing and
xðtÞ 6 zðtÞ; t 2 ½0; T; xðrðtÞÞ 6 zðtÞ; t 2 ½0; T;
ð28Þ ð29Þ
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Z. Yuan et al. / Applied Mathematics and Computation 208 (2009) 231–237
and
zp ðtÞ 6 aðtÞ þ
Z
t
f ðsÞzp ðsÞds þ
0
Z t Z s gðsÞzq ðsÞ þ hðsÞzm ðsÞds ds; 0
0
ðx; yÞ 2 R2þ :
ð30Þ
Define a function uðtÞ by
uðtÞ ¼ aðtÞ þ v ðtÞ;
ð31Þ
where
v ðtÞ ¼
Z t Z s gðsÞzq ðsÞ þ hðsÞzm ðsÞds ds: 0
ð32Þ
0
Then from (30) can be restated as
zp ðtÞ 6 uðtÞ þ
Z
t
f ðsÞzp ðsÞds;
t 2 Rþ :
ð33Þ
0
It is easy to see that zðtÞ is nonnegative, continuous and nondecreasing function for t 2 Rþ , respectively, therefore using Lemma 2 to (33), we obtain
zp ðtÞ 6 uðtÞexp
Z
t
f ðsÞds;
t 2 Rþ ;
0
i.e 1
zðtÞ 6 BðtÞ½aðtÞ þ v ðtÞp :
ð34Þ
Using Lemma 1, for any K > 0, it follows from (34) that
q qp p q pq K p ðaðtÞ þ v ðtÞÞ þ K ; p p mp m p m mp m zm ðtÞ 6 B ðtÞ K p ðaðtÞ þ v ðtÞÞ þ K : p p zq ðtÞ 6 Bq ðtÞ
ð35Þ ð36Þ
Combining (32), (35) and (36) we obtain
Z s q qp p q qp m mp p m mp gðsÞBq ðsÞ K p ðaðsÞ þ v ðsÞÞ þ hðsÞBm ðsÞ K þ K p ðaðsÞ þ v ðsÞÞ þ K ds ds p p p p 0 0 Z s Z t q qp m mp gðsÞBq ðsÞ K p þ hðsÞBm ðsÞ K p ds v ðsÞds: 6 FðtÞ þ p p 0 0
v ðtÞ 6
Z t
ð37Þ
Obviously, FðtÞ is nonnegative,continuous and nondecreasing for t 2 Rþ . Using Lemma 2, from (37), we have
v ðtÞ 6 FðtÞexp
Z t 0
q qp gðsÞBq ðsÞ K p þ p
Z
s
hðsÞBm ðsÞ
0
m mp K p ds ds: p
ð38Þ
It follows from (34) and (38), that
1p Z t Z s q qp m mp zðtÞ 6 BðtÞ aðtÞ þ FðtÞexp gðsÞBq ðsÞ K p þ hðsÞBm ðsÞ K p ds p p 0 0 Therefore, the desired inequality (24) follows from (28), (29) and (39). The proof is complete.
ð39Þ h
Theorem 3. Assume that xðtÞ; aðtÞ; bðtÞ; f ðtÞ; hðtÞ; rðtÞ are defined as in Theorem 1 in Rþ , and xðtÞ satisfies the following form of integral inequality
xp ðtÞ 6 aðtÞ þ bðtÞ
Z t Z s f ðsÞxq ðsÞ þ Lðs; xðrðsÞÞÞ þ hðsÞxm ðsÞds ds; 0
t 2 Rþ ;
ð40Þ
0
where p P q > 0; p P m > 0; p P 1 be constants, and L; M 2 CðR2þ ; Rþ Þ satisfying
0 6 Lðt; xÞ Lðt; yÞ 6 Mðt; yÞðx yÞ;
ð41Þ
for x P y P 0 then inequality (40) with the initial condition (2), implies
Z t 1p xðtÞ 6 aðtÞ þ bðtÞEðtÞexp GðsÞds ; 0
ð42Þ
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Z. Yuan et al. / Applied Mathematics and Computation 208 (2009) 231–237
where
! Z s Z t Z t p q qp q qp p m mp m mp p 1 1p aðsÞ f ðsÞ hðsÞ L s; K þ K p aðsÞ þ K þ K p aðsÞ ds ds þ K þ p1 ds; p p p p p 0 0 0 pK p ! Z t q qp m mp p 1 1p aðtÞ bðtÞ GðtÞ ¼ f ðtÞbðtÞ K p þ hðsÞbðsÞ K p ds þ M t; ; K þ p1 p1 p p p 0 pK p pK p EðtÞ ¼
ð43Þ ð44Þ
for some K > 0. Proof. Define a function zðtÞ by
zðtÞ ¼
Z t
f ðsÞxq ðsÞ þ Lðs; xðrðsÞÞÞ þ
0
Z
s
hðsÞxm ðsÞds ds:
ð45Þ
0
Then we can easily see that zðtÞ is nonnegative and nondecreasing function with zð0Þ ¼ 0 and (40) can be restated as 1
xðtÞ 6 ðaðtÞ þ bðtÞzðtÞÞp :
ð46Þ
By Lemma 2.1, we have
xðtÞ 6
p 1 1p 1 1p K þ ðaðtÞ þ bðtÞzðtÞÞ K p : p p
Therefore, for t 2 Rþ with
ð47Þ
rðtÞ P 0, we obtain 1p
xðrðtÞÞ 6
p 1 1p K p p 1 1p 1 1p K þ ðaðrðtÞÞ þ bðrðtÞÞzðrðtÞÞ K þ ðaðtÞ þ bðtÞzðtÞÞ K p ; 6 p p p p
and for t 2 Rþ with
ð48Þ
rðtÞ P 0, using the initial condition (2) and (47), we yield 1p
1
xðrðtÞÞ ¼ /ðrðtÞÞ 6 ðaðtÞÞp 6
p 1 1p K p K þ ðaðtÞ þ bðtÞzðtÞÞ : p p
ð49Þ
From (48) and (49), there is 1p
xðrðtÞÞ 6
p 1 1p K p K þ ðaðtÞ þ bðtÞzðtÞÞ ; p p
t 2 Rþ :
ð50Þ
It follows from (45), (46) and (50)
Z t Z s Z t q m p 1 1p 1 1p f ðsÞðaðsÞ þ bðsÞzðsÞÞp þ hðsÞðaðsÞ þ bðsÞzðsÞÞ p ds ds þ L s; K þ ðaðsÞ þ bðsÞzðsÞ K p ds p p 0 0 0 Z s Z t mp m p q pq q qp p m m f ðsÞ hðsÞ K þ K p ðaðsÞ þ bðsÞzðsÞÞ þ K p þ K p ðaðsÞ þ bðsÞzðsÞÞ ds ds 6 p p p p 0 0 ! ! ! Z t Z t Z t p 1 1p aðsÞ bðsÞ p 1 1p aðsÞ p 1 1p aðsÞ L s; L s; L s; K þ p1 þ p1 zðsÞ ds K þ p1 ds þ K þ p1 þ p p p 0 0 0 pK p pK p pK p pK p Z s Z t p q pq q qp p m mp m mp f ðsÞ hðsÞ K þ K p ðaðsÞ þ bðsÞzðsÞÞ þ K þ K p ðaðsÞ þ bðsÞzðsÞÞ ds ds 6 p p p p 0 0 ! ! Z t Z t p 1 1p aðsÞ bðsÞ p 1 1p aðsÞ þ L s; zðsÞds þ L s; K þ p1 K þ p1 ds p1 p p 0 0 pK p pK p pK p Z t GðsÞzðsÞds; 6 EðtÞ þ
zðtÞ 6
ð51Þ
0
where GðtÞ; EðtÞ are defined as (43), (44). It is easy to see that GðtÞ is nonnegative, continuous and nondecreasing for t 2 Rþ . In view of lemma 2 from (51), we have
zðtÞ 6 EðtÞexp
Z
t
GðtÞds;
t 2 Rþ :
ð52Þ
0
Hence the desired relation (42) immediately follows from (46) and (52). This completes the proof.
h
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Z. Yuan et al. / Applied Mathematics and Computation 208 (2009) 231–237
3. Some applications In this section, we present some immediate applications of our results. Example 1. Consider the delay differential equation
Z t pðp 1Þu0 ðtÞ ¼ H t; xðtÞ; xðrðtÞÞ; xm ðsÞds ;
t 2 Rþ ;
ð53Þ
0
with the initial condition
xðtÞ ¼ /ðtÞ; t 2 ½a; 0; 1
j/ðrðtÞÞj 6 jCjp
for t 2 Rþ ; with
ð54Þ
rðtÞ 6 0;
where H 2 CðR4þ ; RÞ; C ¼ xp ð0Þ; p P m > 0 are constants,
rðtÞ 2 CðRþ ; RÞ; rðtÞ 6 t,
1 < a ¼ inf frðtÞ; t 2 Rþ g 6 0; and /ðtÞ 2 Cð½a; 0; Rþ Þ. Assume that
jHðt; x; y; zÞj 6 f ðtÞjxjq þ gðtÞjyjr þ jzj;
ðt; x; y; zÞ 2 R4þ ;
ð55Þ
where f ðtÞ; gðtÞ; q; r are defined as in Theorem 1. Let xðtÞ is a solution of Eq. (53), satisfying initial condition (54), integrating (53)from 0 to t, by (54) and (55) we have inequality
jxðtÞjp 6 jCj þ
Z t Z s f ðsÞjxðsÞjq þ gðsÞjxððrðsÞÞjr þ xm ðsÞds ds; 0
ð56Þ
0
with the initial condition (54). Now using Theorem 1, we get a explicit bound on the solution of Eq. (53)
1p Z t jxðtÞj 6 jCj þ AðtÞexp BðsÞds : 0
For any K > 0, where AðtÞ; BðtÞ are defined as in Theorem 1, with aðtÞ ¼ jCj; bðtÞ ¼ hðtÞ ¼ 1. Example 2. Consider the following integral equation
Z t Z s xp ðtÞ ¼ T t; Nðs; xðsÞ; xðrðsÞÞ; xm ðsÞds ds ; 0
t 2 Rþ ;
ð57Þ
0
with initial condition (2) , where T 2 CðRþ R; RÞ; N 2 CðRþ R3 ; RÞ, satisfying
jTðt; uÞj 6 aðtÞ þ bðtÞjuj;
ð58Þ
jNðt; x; y; zÞj 6 f ðtÞjxjq þ Lðt; jyjÞ þ jzj;
ð59Þ
and
with a(t), b(t) – constant, f ðtÞ; L are defined as in Theorem 3, then
1p Z t jxðtÞj 6 aðtÞ þ bðtÞEðtÞexp GðsÞds ;
ð60Þ
0
where EðtÞ; GðtÞ are defined as in Theorem 3. In fact, by assumptions of Eq. (57) with (2), we have
Z t Z t Z s Z s jxp ðtÞj ¼ T t; Nðs; xðsÞ; xðrðsÞÞ; xm ðsÞdsÞds 6 aðtÞ þ bðtÞ Nðs; xðsÞ; xðrðsÞÞ; xm ðsÞdsÞds 0
6 aðtÞ þ bðtÞ
Z t 0
0
q
f ðsÞjxðsÞj þ Lðs; jxðrðsÞÞjÞ þ
Z
0
s
jx ðsÞjds ds;
0
m
0
with (2) in Theorem 3, it follows from Theorem 3 immediately (60) is valid. Acknowledgements The authors thank the referee for his corrections to the original manuscript. References [1] W.N. Li, M.A. Han, F.W. Meng, Some new delay integral inequality and their applications, J. Comput. Appl. Math. 180 (2005) 191–200. [2] Fanewi Meng, Wei Nian Li, On some new integral inequalities and their applications, Appl. Math. Comput. 148 (2004) 381–392.
ð61Þ
Z. Yuan et al. / Applied Mathematics and Computation 208 (2009) 231–237
237
[3] Hongxia Zhang, Fanwei Meng, Integral inequalities in two independent variables for retarded Volterra equations, Appl. Math. Comput. 199 (1) (2008) 90–98. [4] O. Lipovan, A retarded integral inequality and its applications, J. Math. Anal. Appl. 285 (2003) 436–443. [5] Xueqin Zhao, Fanwei Meng, Fenghua Yang, On some new integral inequality and their applications, Indian J. Pure Appl. Math. 38 (5) (2007) 403–414. [6] Fangcui Jiang, Fanwei Meng, Explicit bounds on some new nonlinear integral inequality with delay, J. Comput. Appl. Math. 205 (2007) 479–486. [7] O. Lipovan, Integral inequalities for retarded Volterra equations, J. Math. Anal. Appl. 322 (2006) 349–358. [8] B.G. Pachpatte, Explicit bounds on certain integral inequalities, J. Math. Anal. Appl. 267 (2002) 48–61. [9] B.G. Pachpatte, A note on certain integral inequalities with delay, Period. Math. Hungar. 31 (1995) 234–299. [10] B.G. Pachpatte, On a certain inequality arising in the theory of differential equations, J. Math. Anal. Appl. 182 (1994) 143–157. [11] B.G. Pachpatte, On some new nonlinear retarded integral inequalities, J. Inequal. Pure Appl. Math. 5 (2004) (Article 80). [12] B.G. Pachpatte, Bounds on certain integral inequalities, J. Inequal. Pure Appl. Math. 3 (3) (2002). Article 47. [13] B.G. Pachpatte, Inequalities for Differential and Integral Equations, Academic Press, New York, 1998.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Some new delay integral inequalities and their applications q Zhiling Yuan a, Xingwei Yuan b, Fanwei Meng c,*, Hongxia Zhang c a
School of Science Jiangnan University, Wuxi 214122, Jiangshu, People’s Republic of China Key and Open Laboratory of Marine and Estuarine Fisheries, Ministry of Agriculture, East China Sea Fisheries Research Institute, Chinese Academy of Fishery Sciences, Shanghai 200090, People’s Republic of China c Department of Mathematics, Qufu Normal University, Qufu 273165, Shandong, People’s Republic of China b
a r t i c l e
i n f o
a b s t r a c t The main objective of this paper is to establish some new delay integral inequalities, which provide explicit bounds on unknown functions. The inequalities given here can be used as tools in the qualitative theory of certain delay differential equations and delay integral equations. Ó 2008 Elsevier Inc. All rights reserved.
Keywords: Integral equation Integral inequalities Bounded
1. Introduction The integral inequalities involving functions of one and more than one independent variables, which provide explicit bounds on unknown function play a fundamental role in the development of the theory of differential equation and integral equations. Many retarded inequalities have been discovered see [1–9], Others results may be found in [10–13]. Recently Jiang and Meng [10] presented a retarded inequality which has very well characters. In this paper, we establish some new inequality, the results of which generalize the results of Jiang and Meng [6] and Pachpatte [12]. 2. Main results In what follows, R denotes the set of real numbers and Rþ ¼ ½0; þ1Þ is the given subset of R, and CðM; SÞ denotes the class of all continuous functions defined on set M with range in the set S. The following lemmas are useful in our main results. Lemma 2.1 [6]. Assume that a P 0; p P q P 0, and p–0, then q q qp p q qp ap 6 K p a þ K ; p p
for any K > 0. Lemma 2.2 [13]. Let u; a; b 2 ðCðRþ ; Rþ Þ and a be nondecreasing, if
uðtÞ 6 aðtÞ þ
Z
t
bðsÞuðsÞds;
0
q
This research was supported by the NNSF of China (10771118). * Corresponding author. E-mail addresses: [email protected] (Z. Yuan), [email protected] (X. Yuan), [email protected] (F. Meng).
0096-3003/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2008.11.043
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Z. Yuan et al. / Applied Mathematics and Computation 208 (2009) 231–237
then
uðtÞ 6 aðtÞexp
Z
t
bðsÞds:
0
Theorem 1. Assume that xðtÞ; aðtÞ; bðtÞ; f ðtÞ; gðtÞ 2 CðRþ ; Rþ Þ. If aðtÞ; bðtÞ are nondecreasing in Rþ , and xðtÞ satisfies the following form of delay integral inequality
xðtÞp 6 aðtÞ þ bðtÞ
Z t Z s f ðsÞxq ðrðsÞÞ þ gðsÞxr ðsÞ þ hðsÞxm ðsÞds ds; 0
t 2 Rþ ;
ð1Þ
0
with the initial condition
t 2 ½a; 0;
xðtÞ ¼ /ðtÞ;
1
/ðrðtÞÞ 6 aðtÞp
f or t 2 Rþ with
ð2Þ
rðtÞ 6 0;
where p–0; p P q P 0; p P r P 0; p P m P 0; p; q; r are constants,
rðtÞ 2 CðRþ ; RÞ; rðtÞ 6 t,
1 < a ¼ inf frðtÞ; t 2 Rþ g 6 0; and /ðtÞ 2 Cð½a; 0; Rþ Þ, then
1p Z t xðtÞ 6 aðtÞ þ bðtÞAðtÞexp BðsÞds ;
ð3Þ
0
for any K > 0, where
Z s Z t q qp p q qp r rp p r pr m mp p m mp f ðsÞ K p aðsÞ þ hðsÞ K þ gðs; tÞ K p aðsÞ þ K þ K p aðsÞ þ K ds ds; p p p p p p 0 0 Z t rp mp q qp r m hðsÞbðsÞ K p ds; t 2 Rþ : BðtÞ ¼ K p f ðtÞ þ K p gðtÞ bðtÞ þ p p p 0 AðtÞ ¼
ð4Þ ð5Þ
Proof. For any fixed positive number T we define a function zðtÞ by
zðtÞ ¼
aðTÞ þ bðtÞ
Z t
f ðsÞxq ðrðsÞÞ þ gðsÞxr ðsÞ þ
Z
0
s
1p hðsÞxm ðsÞds ds ;
t 2 Rþ ;
ð6Þ
0
it is easy to see that zðtÞ is a nonnegative and nondecreasing and
xðtÞ 6 zðtÞ;
t 2 ½0; T:
Therefore, for t 2 Rþ with
ð7Þ
rðtÞ P 0, we have
xðrðtÞÞ 6 zðrðtÞÞ 6 zðtÞ;
t 2 ½0; T:
ð8Þ
On the other hand, using the initial condition (2), for t 2 Rþ with 1 p
1 p
xðrðtÞÞ ¼ /ðrðtÞÞ 6 aðtÞ 6 aðTÞ 6 zðtÞ;
rðtÞ 6 0, we have
t 2 ½0; T:
ð9Þ
Combining (8) and (9), we obtain
xðrðtÞÞ 6 zðtÞ;
t 2 ½0; T:
ð10Þ
It follows from (6) and (10) that
zp ðtÞ 6 aðTÞ þ bðtÞ
Z t
f ðsÞzq ðsÞ þ gðsÞzr ðsÞ þ
Z
0
s
hðsÞzm ðsÞds ds;
t 2 Rþ :
ð11Þ
0
Taking t ¼ T in (11), we obtain
zp ðTÞ 6 aðTÞ þ bðTÞ
Z
T
f ðsÞzq ðsÞ þ gðsÞzr ðsÞ þ
Z
0
s
hðsÞzm ðsÞds ds:
ð12Þ
0
Noting that T 2 Rþ was arbitrary, from (12), we have
zp ðtÞ 6 aðtÞ þ bðtÞ
Z t Z s f ðsÞzq ðsÞ þ gðsÞzr ðsÞ þ hðsÞzm ðsÞds ds; 0
t 2 Rþ :
ð13Þ
0
Similarly, we obtain
xðtÞ 6 zðtÞ t 2 Rþ :
ð14Þ
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Z. Yuan et al. / Applied Mathematics and Computation 208 (2009) 231–237
Define a function uðtÞ by
uðtÞ ¼
Z t
Z
f ðsÞzq ðsÞ þ gðsÞzr ðsÞ þ
0
s
hðsÞzm ðsÞds ds:
ð15Þ
0
Then (13) can be restated as
zp ðtÞ 6 aðtÞ þ bðtÞuðtÞ; t 2 Rþ :
ð16Þ
Using Lemma 1, from (16) for any K > 0, we easily obtain q q qp p q qp zq ðtÞ 6 ðaðtÞ þ bðtÞuðtÞÞp 6 K p ðaðtÞ þ bðtÞuðtÞÞ þ K ; p p r r rp p r pr zr ðtÞ 6 ðaðtÞ þ bðtÞuðtÞÞp 6 K p ðaðtÞ þ bðtÞpðtÞÞ þ K ; p p m m mp p m mp zm ðtÞ 6 ðaðtÞ þ bðtÞuðtÞÞ p 6 K p ðaðtÞ þ bðtÞpðtÞÞ þ K : p p
ð17Þ ð18Þ ð19Þ
It follows (15), (17), (18) and (19) that
q qp p q qp r rp p r pr K p ðaðsÞ þ bðsÞuðsÞÞ þ K þ gðsÞ K p ðaðsÞ þ bðsÞuðsÞÞ þ K p p p p 0 Z s Z t m mp p m mp hðsÞ BðsÞuðsÞds; t 2 Rþ ; þ K p ðaðsÞ þ bðsÞuðsÞÞ þ K ds ds 6 AðtÞ þ p p 0 0
uðtÞ 6
Z
t
f ðsÞ
ð20Þ
where AðtÞ; BðtÞ are defined as in (4) and (5), it is easy to find that AðtÞ is nonnegative, continuous and nondecreasing for t 2 Rþ . By Lemma 2 we obtain
uðtÞ 6 AðtÞexp
Z
t
BðsÞds:
ð21Þ
0
From (16) and (21), it follows that
1p Z t zðtÞ 6 aðtÞ þ bðtÞAðtÞexp BðsÞds :
ð22Þ
0
Thus, the desired inequality (3) follows from (14) and (22). This completes the proof of Theorem 1.
h
Remark. If we take hðtÞ ¼ 0, then the inequalities established in Theorem 1 reduce to [6, Theorem 2.3]. Theorem 2. Assume that p; q; m; xðtÞ; aðtÞ; f ðtÞ; gðtÞ; hðtÞ; rðtÞ are defined as Theorem 1. If xðtÞ satisfies the integral inequality
xðtÞp 6 aðtÞ þ
Z
t
f ðsÞxp ðsÞds þ 0
Z t
gðsÞxq ðrðsÞÞ þ
0
Z
s
hðsÞxm ðsÞds ds;
0
ðx; yÞ 2 R2þ ;
ð23Þ
with the initial condition (2) in Theorem 1, then
" xðtÞ 6 BðtÞ aðtÞ þ FðtÞexp
1p # Z t Z s q qp m mp gðsÞBr ðsÞ K p þ hðsÞBm ðsÞ K p ds ds ; p p 0 0
ð24Þ
for any K > 0, where
1p Z t BðtÞ ¼ exp f ðsÞ ds dt ; 0 Z s Z s p q qp q qp p m mp m mp gðsÞBq ðsÞ hðsÞBm ðsÞ K þ K p aðsÞ þ K þ K p aðsÞ : FðtÞ ¼ p p p p 0 0
ð25Þ ð26Þ
Proof. Fixing any positive number T we define a function zðtÞ by
zðtÞ ¼
1p Z t Z t Z s aðTÞ þ f ðsÞxp ðsÞds þ gðsÞxq ðrðsÞÞds þ hðsÞxm ðsÞds ds : 0
0
ð27Þ
0
Using the similar to the proof Theorem 1, we easily obtain that zðtÞ is a nonnegative and nondecreasing and
xðtÞ 6 zðtÞ; t 2 ½0; T; xðrðtÞÞ 6 zðtÞ; t 2 ½0; T;
ð28Þ ð29Þ
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Z. Yuan et al. / Applied Mathematics and Computation 208 (2009) 231–237
and
zp ðtÞ 6 aðtÞ þ
Z
t
f ðsÞzp ðsÞds þ
0
Z t Z s gðsÞzq ðsÞ þ hðsÞzm ðsÞds ds; 0
0
ðx; yÞ 2 R2þ :
ð30Þ
Define a function uðtÞ by
uðtÞ ¼ aðtÞ þ v ðtÞ;
ð31Þ
where
v ðtÞ ¼
Z t Z s gðsÞzq ðsÞ þ hðsÞzm ðsÞds ds: 0
ð32Þ
0
Then from (30) can be restated as
zp ðtÞ 6 uðtÞ þ
Z
t
f ðsÞzp ðsÞds;
t 2 Rþ :
ð33Þ
0
It is easy to see that zðtÞ is nonnegative, continuous and nondecreasing function for t 2 Rþ , respectively, therefore using Lemma 2 to (33), we obtain
zp ðtÞ 6 uðtÞexp
Z
t
f ðsÞds;
t 2 Rþ ;
0
i.e 1
zðtÞ 6 BðtÞ½aðtÞ þ v ðtÞp :
ð34Þ
Using Lemma 1, for any K > 0, it follows from (34) that
q qp p q pq K p ðaðtÞ þ v ðtÞÞ þ K ; p p mp m p m mp m zm ðtÞ 6 B ðtÞ K p ðaðtÞ þ v ðtÞÞ þ K : p p zq ðtÞ 6 Bq ðtÞ
ð35Þ ð36Þ
Combining (32), (35) and (36) we obtain
Z s q qp p q qp m mp p m mp gðsÞBq ðsÞ K p ðaðsÞ þ v ðsÞÞ þ hðsÞBm ðsÞ K þ K p ðaðsÞ þ v ðsÞÞ þ K ds ds p p p p 0 0 Z s Z t q qp m mp gðsÞBq ðsÞ K p þ hðsÞBm ðsÞ K p ds v ðsÞds: 6 FðtÞ þ p p 0 0
v ðtÞ 6
Z t
ð37Þ
Obviously, FðtÞ is nonnegative,continuous and nondecreasing for t 2 Rþ . Using Lemma 2, from (37), we have
v ðtÞ 6 FðtÞexp
Z t 0
q qp gðsÞBq ðsÞ K p þ p
Z
s
hðsÞBm ðsÞ
0
m mp K p ds ds: p
ð38Þ
It follows from (34) and (38), that
1p Z t Z s q qp m mp zðtÞ 6 BðtÞ aðtÞ þ FðtÞexp gðsÞBq ðsÞ K p þ hðsÞBm ðsÞ K p ds p p 0 0 Therefore, the desired inequality (24) follows from (28), (29) and (39). The proof is complete.
ð39Þ h
Theorem 3. Assume that xðtÞ; aðtÞ; bðtÞ; f ðtÞ; hðtÞ; rðtÞ are defined as in Theorem 1 in Rþ , and xðtÞ satisfies the following form of integral inequality
xp ðtÞ 6 aðtÞ þ bðtÞ
Z t Z s f ðsÞxq ðsÞ þ Lðs; xðrðsÞÞÞ þ hðsÞxm ðsÞds ds; 0
t 2 Rþ ;
ð40Þ
0
where p P q > 0; p P m > 0; p P 1 be constants, and L; M 2 CðR2þ ; Rþ Þ satisfying
0 6 Lðt; xÞ Lðt; yÞ 6 Mðt; yÞðx yÞ;
ð41Þ
for x P y P 0 then inequality (40) with the initial condition (2), implies
Z t 1p xðtÞ 6 aðtÞ þ bðtÞEðtÞexp GðsÞds ; 0
ð42Þ
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Z. Yuan et al. / Applied Mathematics and Computation 208 (2009) 231–237
where
! Z s Z t Z t p q qp q qp p m mp m mp p 1 1p aðsÞ f ðsÞ hðsÞ L s; K þ K p aðsÞ þ K þ K p aðsÞ ds ds þ K þ p1 ds; p p p p p 0 0 0 pK p ! Z t q qp m mp p 1 1p aðtÞ bðtÞ GðtÞ ¼ f ðtÞbðtÞ K p þ hðsÞbðsÞ K p ds þ M t; ; K þ p1 p1 p p p 0 pK p pK p EðtÞ ¼
ð43Þ ð44Þ
for some K > 0. Proof. Define a function zðtÞ by
zðtÞ ¼
Z t
f ðsÞxq ðsÞ þ Lðs; xðrðsÞÞÞ þ
0
Z
s
hðsÞxm ðsÞds ds:
ð45Þ
0
Then we can easily see that zðtÞ is nonnegative and nondecreasing function with zð0Þ ¼ 0 and (40) can be restated as 1
xðtÞ 6 ðaðtÞ þ bðtÞzðtÞÞp :
ð46Þ
By Lemma 2.1, we have
xðtÞ 6
p 1 1p 1 1p K þ ðaðtÞ þ bðtÞzðtÞÞ K p : p p
Therefore, for t 2 Rþ with
ð47Þ
rðtÞ P 0, we obtain 1p
xðrðtÞÞ 6
p 1 1p K p p 1 1p 1 1p K þ ðaðrðtÞÞ þ bðrðtÞÞzðrðtÞÞ K þ ðaðtÞ þ bðtÞzðtÞÞ K p ; 6 p p p p
and for t 2 Rþ with
ð48Þ
rðtÞ P 0, using the initial condition (2) and (47), we yield 1p
1
xðrðtÞÞ ¼ /ðrðtÞÞ 6 ðaðtÞÞp 6
p 1 1p K p K þ ðaðtÞ þ bðtÞzðtÞÞ : p p
ð49Þ
From (48) and (49), there is 1p
xðrðtÞÞ 6
p 1 1p K p K þ ðaðtÞ þ bðtÞzðtÞÞ ; p p
t 2 Rþ :
ð50Þ
It follows from (45), (46) and (50)
Z t Z s Z t q m p 1 1p 1 1p f ðsÞðaðsÞ þ bðsÞzðsÞÞp þ hðsÞðaðsÞ þ bðsÞzðsÞÞ p ds ds þ L s; K þ ðaðsÞ þ bðsÞzðsÞ K p ds p p 0 0 0 Z s Z t mp m p q pq q qp p m m f ðsÞ hðsÞ K þ K p ðaðsÞ þ bðsÞzðsÞÞ þ K p þ K p ðaðsÞ þ bðsÞzðsÞÞ ds ds 6 p p p p 0 0 ! ! ! Z t Z t Z t p 1 1p aðsÞ bðsÞ p 1 1p aðsÞ p 1 1p aðsÞ L s; L s; L s; K þ p1 þ p1 zðsÞ ds K þ p1 ds þ K þ p1 þ p p p 0 0 0 pK p pK p pK p pK p Z s Z t p q pq q qp p m mp m mp f ðsÞ hðsÞ K þ K p ðaðsÞ þ bðsÞzðsÞÞ þ K þ K p ðaðsÞ þ bðsÞzðsÞÞ ds ds 6 p p p p 0 0 ! ! Z t Z t p 1 1p aðsÞ bðsÞ p 1 1p aðsÞ þ L s; zðsÞds þ L s; K þ p1 K þ p1 ds p1 p p 0 0 pK p pK p pK p Z t GðsÞzðsÞds; 6 EðtÞ þ
zðtÞ 6
ð51Þ
0
where GðtÞ; EðtÞ are defined as (43), (44). It is easy to see that GðtÞ is nonnegative, continuous and nondecreasing for t 2 Rþ . In view of lemma 2 from (51), we have
zðtÞ 6 EðtÞexp
Z
t
GðtÞds;
t 2 Rþ :
ð52Þ
0
Hence the desired relation (42) immediately follows from (46) and (52). This completes the proof.
h
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Z. Yuan et al. / Applied Mathematics and Computation 208 (2009) 231–237
3. Some applications In this section, we present some immediate applications of our results. Example 1. Consider the delay differential equation
Z t pðp 1Þu0 ðtÞ ¼ H t; xðtÞ; xðrðtÞÞ; xm ðsÞds ;
t 2 Rþ ;
ð53Þ
0
with the initial condition
xðtÞ ¼ /ðtÞ; t 2 ½a; 0; 1
j/ðrðtÞÞj 6 jCjp
for t 2 Rþ ; with
ð54Þ
rðtÞ 6 0;
where H 2 CðR4þ ; RÞ; C ¼ xp ð0Þ; p P m > 0 are constants,
rðtÞ 2 CðRþ ; RÞ; rðtÞ 6 t,
1 < a ¼ inf frðtÞ; t 2 Rþ g 6 0; and /ðtÞ 2 Cð½a; 0; Rþ Þ. Assume that
jHðt; x; y; zÞj 6 f ðtÞjxjq þ gðtÞjyjr þ jzj;
ðt; x; y; zÞ 2 R4þ ;
ð55Þ
where f ðtÞ; gðtÞ; q; r are defined as in Theorem 1. Let xðtÞ is a solution of Eq. (53), satisfying initial condition (54), integrating (53)from 0 to t, by (54) and (55) we have inequality
jxðtÞjp 6 jCj þ
Z t Z s f ðsÞjxðsÞjq þ gðsÞjxððrðsÞÞjr þ xm ðsÞds ds; 0
ð56Þ
0
with the initial condition (54). Now using Theorem 1, we get a explicit bound on the solution of Eq. (53)
1p Z t jxðtÞj 6 jCj þ AðtÞexp BðsÞds : 0
For any K > 0, where AðtÞ; BðtÞ are defined as in Theorem 1, with aðtÞ ¼ jCj; bðtÞ ¼ hðtÞ ¼ 1. Example 2. Consider the following integral equation
Z t Z s xp ðtÞ ¼ T t; Nðs; xðsÞ; xðrðsÞÞ; xm ðsÞds ds ; 0
t 2 Rþ ;
ð57Þ
0
with initial condition (2) , where T 2 CðRþ R; RÞ; N 2 CðRþ R3 ; RÞ, satisfying
jTðt; uÞj 6 aðtÞ þ bðtÞjuj;
ð58Þ
jNðt; x; y; zÞj 6 f ðtÞjxjq þ Lðt; jyjÞ þ jzj;
ð59Þ
and
with a(t), b(t) – constant, f ðtÞ; L are defined as in Theorem 3, then
1p Z t jxðtÞj 6 aðtÞ þ bðtÞEðtÞexp GðsÞds ;
ð60Þ
0
where EðtÞ; GðtÞ are defined as in Theorem 3. In fact, by assumptions of Eq. (57) with (2), we have
Z t Z t Z s Z s jxp ðtÞj ¼ T t; Nðs; xðsÞ; xðrðsÞÞ; xm ðsÞdsÞds 6 aðtÞ þ bðtÞ Nðs; xðsÞ; xðrðsÞÞ; xm ðsÞdsÞds 0
6 aðtÞ þ bðtÞ
Z t 0
0
q
f ðsÞjxðsÞj þ Lðs; jxðrðsÞÞjÞ þ
Z
0
s
jx ðsÞjds ds;
0
m
0
with (2) in Theorem 3, it follows from Theorem 3 immediately (60) is valid. Acknowledgements The authors thank the referee for his corrections to the original manuscript. References [1] W.N. Li, M.A. Han, F.W. Meng, Some new delay integral inequality and their applications, J. Comput. Appl. Math. 180 (2005) 191–200. [2] Fanewi Meng, Wei Nian Li, On some new integral inequalities and their applications, Appl. Math. Comput. 148 (2004) 381–392.
ð61Þ
Z. Yuan et al. / Applied Mathematics and Computation 208 (2009) 231–237
237
[3] Hongxia Zhang, Fanwei Meng, Integral inequalities in two independent variables for retarded Volterra equations, Appl. Math. Comput. 199 (1) (2008) 90–98. [4] O. Lipovan, A retarded integral inequality and its applications, J. Math. Anal. Appl. 285 (2003) 436–443. [5] Xueqin Zhao, Fanwei Meng, Fenghua Yang, On some new integral inequality and their applications, Indian J. Pure Appl. Math. 38 (5) (2007) 403–414. [6] Fangcui Jiang, Fanwei Meng, Explicit bounds on some new nonlinear integral inequality with delay, J. Comput. Appl. Math. 205 (2007) 479–486. [7] O. Lipovan, Integral inequalities for retarded Volterra equations, J. Math. Anal. Appl. 322 (2006) 349–358. [8] B.G. Pachpatte, Explicit bounds on certain integral inequalities, J. Math. Anal. Appl. 267 (2002) 48–61. [9] B.G. Pachpatte, A note on certain integral inequalities with delay, Period. Math. Hungar. 31 (1995) 234–299. [10] B.G. Pachpatte, On a certain inequality arising in the theory of differential equations, J. Math. Anal. Appl. 182 (1994) 143–157. [11] B.G. Pachpatte, On some new nonlinear retarded integral inequalities, J. Inequal. Pure Appl. Math. 5 (2004) (Article 80). [12] B.G. Pachpatte, Bounds on certain integral inequalities, J. Inequal. Pure Appl. Math. 3 (3) (2002). Article 47. [13] B.G. Pachpatte, Inequalities for Differential and Integral Equations, Academic Press, New York, 1998.