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Some new integral inequalities and their applications in studying the stability of nonlinear integro-differential equations with time delay
J. Math. Anal. Appl. 377 (2011) 853–862
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Some new integral inequalities and their applications in studying the stability of nonlinear integro-differential equations with time delay ✩ Lianzhong Li a,b,∗ , Fanwei Meng b , Peijun Ju a a b
School of Mathematical and System Sciences, Taishan University, Tai’an, Shandong 271021, People’s Republic of China School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, People’s Republic of China
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 3 August 2010 Available online 8 December 2010 Submitted by Steven G. Krantz
In this paper, we generalize some integral inequalities to more general situations, and the inequalities of Pachpatte type are corollaries of our’s. We establish bounds on the solutions, and we show the usefulness of our results in investigating the asymptotic behavior and the stability on the solutions of integral equations, differential equations and integrodifferential equations with time delay. © 2010 Elsevier Inc. All rights reserved.
Keywords: Integral inequality Integral equation Differential equation Integro-differential equation Uniform stability
1. Introduction In a paper published in 1981, Gripenberg [5] studied the qualitative behavior of solutions of the equation
t
x(t ) = k p (t ) −
A (t − s)x(s) ds 0
t f (t ) +
B (t − s)x(s) ds .
(1.1)
0
This equation arises in the study of the spread of an infectious disease that does not induce permanent immunity. For detailed meanings of the various functions arising in (1.1), see [5] and also [1,3,4,6], for more results, see [2,7–10,13] and the references therein. In [5], the author studied the existence of a unique bounded continuous and nonnegative solution of (1.1) for t ∈ R+ under appropriate assumptions on A and B, and also obtained sufficient conditions for the convergence of the solution to a limit when t → ∞. Over the years integral inequalities have become a major tool in the analysis of various differential and integral equations that occur in nature or are built by man. Pachpatte [11] gave a new integral inequality and studied the boundedness, asymptotic behavior and growth of the solutions of Eq. (1.1) using the inequality. We list the inequality as follows. Theorem A (Pachpatte). (See [11].) Let u , f , g be real-valued nonnegative continuous functions defined on R+ , and c 1 , c 2 be nonnegative constants. If
✩ This research was supported by the National Natural Science Foundation of China under Grant No. 10771118 and the Natural Science Foundation of Shandong Province under Grant No. ZR2009AM011. Corresponding author at: School of Mathematical and System Sciences, Taishan University, Tai’an, Shandong 271021, People’s Republic of China. E-mail addresses: [email protected] (L. Li), [email protected] (F. Meng).
*
0022-247X/$ – see front matter doi:10.1016/j.jmaa.2010.12.002
©
2010 Elsevier Inc. All rights reserved.
854
L. Li et al. / J. Math. Anal. Appl. 377 (2011) 853–862
t
u (t ) c 1 + and c 1 c 2
t 0
t
c2 +
f (s)u (s) ds 0
g (s)u (s) ds 0
R (s) Q (s) ds < 1 for all t ∈ R+ , then
c 1 c 2 Q (t )
u (t )
1 − c1 c2
t 0
R (s) Q (s) ds
,
t ∈ R+ ,
where
t R (t ) =
f (t ) g (s) + f (s) g (t ) ds,
0
t
Q (t ) = exp
c 1 g (s) + c 2 f (s) ds .
0
Very recently, Lipovan observed the following integral inequality: Theorem B (Lipovan). (See [12].) Let a ∈ C (R+ × R+ , R+) with (t , s) → ∂t a(t , s) ∈ C (R+ × R+ , R+ ), k, ω ∈ C (R+ , R+ ) be non∞ decreasing functions with k(0) > 0, ω(t ) > 0 for t > 0 and 1 ωdt(t ) = ∞. Assume in addition that α ∈ C 1 (R+ , R+ ) is non-decreasing and α (t ) t for t 0. If u ∈ C (R+ , R+ ) satisfies
α (t ) u (t ) k(t ) +
a(t , s)ω u (s) ds,
t 0,
0
then
u (t ) Ω where Ω(t ) :=
−1
t 1
Ω k(t ) +
α (t )
t 0,
a(t , s) ds , 0
1 ω(s) ds, t > 0.
Aside from the various physical meanings of the functions arising in (1.1), we believe that equations like (1.1) are of great interest in their right and that further investigations of the qualitative behaviors of their solutions even under the usual hypotheses on the functions in (1.1) are much more interesting. In this paper, we establish some new inequalities, and the inequalities of Pachpatte type are corollaries of our’s. These furnish a handy tool for the study of qualitative as well as quantitative properties of solutions of equations like (1.1). We illustrate this by applying our new inequalities to study the boundedness properties, asymptotic behavior and the stability of the solutions of integral equations, differential equations and integro-differential equations with time delay. 2. Integral inequalities 2.1. The nonlinear case Theorem 2.1. Let L , K , M , N be nonnegative constants, a, h, f , g ∈ C (R+ × R+ , R+ ) with (t , s) → ∂t a(t , s), ∂t h(t , s), ∂t f (t , s), ∂t g (t , s) ∈ C (R+ × R+ , R+ ). Assume in addition that b, ω ∈ C (R+ , R+ ), α ∈ C 1 (R+ , R+ ) are non-decreasing functions and ω(t ) > 0, α (t ) t for t 0. If u ∈ C (R+ , R+ ) satisfies
α (t )
La(t , s)ω u (s) ds +
u (t ) b(t ) + and ω(b(t ))
t 0
0
t
K h(t , s)ω u (s) ds +
0
α (t )
M f (t , s)ω u (s) ds
0
t
N g (t , s)ω u (s) ds 0
R (s) Q (s) ds < 1 for all t ∈ R+ , then
u (t ) ω where
−1
ω(b(t )) Q (t ) 1 − ω(b(t ))
t 0
R (s) Q (s) ds
,
t ∈ S,
(2.2)
L. Li et al. / J. Math. Anal. Appl. 377 (2011) 853–862
α (t ) Q (t ) = exp
La(t , s) ds +
R (t ) =
K h(t , s) ds , 0
α (t )
d
t
0
t
M f (t , s) ds
dt
855
N g (t , s) ds
0
,
0
t ∈ S is chosen in such a way that
∈ Dom(ω−1 ), ω−1 is the inverse of ω .
ω(b(t )) Q (t ) t 1−ω(b(t )) 0 R (s) Q (s) ds
Proof. Let T ∈ S, T 0 be fixed and denote
α (t )
La(t , s)ω u (s) ds +
x(t ) = 0
t
α (t )
K h(t , s)ω u (s) ds +
M f (t , s)ω u (s) ds
0
0
x (t ) = La t , α (t ) ω u α (t ) α (t ) +
α , ω imply that x is non-decreasing on R+ . Hence, for
0
0
+ M f t , α (t ) ω u α (t ) α (t ) +
α (t )
t
M ∂t f (t , s)ω u (s) ds
N g (t , s)ω u (s) ds
0
+ N g (t , t )ω u (t ) +
t 0,
t L ∂t a(t , s)ω u (s) ds + K h(t , t )ω u (t ) + K ∂t h(t , s)ω u (s) ds
α (t )
N g (t , s)ω u (s) ds, 0
then u (t ) b(t ) + x(t ). Our assumptions on L , K , M , N , a, h, f , g and t ∈ [0, T ], by calculations we have
t
0
t
α (t )
N ∂t g (t , s)ω u (s) ds
M f (t , s)ω u (s) ds
0
0
α (t ) t d d ω b α (t ) + x α (t ) La(t , s) ds + ω b(t ) + x(t ) K h(t , s) ds dt
dt
0
0
d + ω b α (t ) + x α (t ) ω b(t ) + x(t )
t
α (t ) M f (t , s) ds
dt 0
d + ω b α (t ) + x α (t ) ω b(t ) + x(t )
N g (t , s) ds 0
α (t )
t N g (t , s) ds
dt 0
M f (t , s) ds 0
α (t ) t d ω b( T ) + x(t ) La(t , s) ds + K h(t , s) ds
dt
0
d + ω b( T ) + x(t ) 2
0
α (t ) M f (t , s) ds
dt
t
0
N g (t , s) ds . 0
Suppose b(0) > 0 (if b(0) = 0, carry out the following arguments with b(t ) + ε instead of b(t ), where small constant, and subsequently pass to the limit as ε → 0 to complete the proof) then we get
x (t )
ω2 [b( T ) + x(t )]
−
1
d
ω[b( T ) + x(t )] dt
α (t ) La(t , s) ds + 0
t K h(t , s) ds 0
d
ε > 0 is an arbitrary
α (t )
t
M f (t , s) ds
dt 0
N g (t , s) ds .
0
(2.3)
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L. Li et al. / J. Math. Anal. Appl. 377 (2011) 853–862
Let
z(t ) =
α (t )
1
ω[b( T ) + x(t )]
q(t ) =
,
0
d dt
R (t ) =
K h(t , s) ds ,
d
α (t )
t
M f (t , s) ds
dt
0
then (2.3) is equivalent to
K h(t , s) ds, 0
t
La(t , s) ds +
z (t ) + z(t )
La(t , s) ds + 0
α (t ) Q (t ) = exp
t
0
N g (t , s) ds , 0
q(t ) − R (t ).
Multiplying the above inequality by eq(t ) = Q (t ), we get
d dt
z(t ) Q (t ) − Q (t ) R (t ).
Consider now the integral on the interval [0, t ] to obtain
t z(t ) Q (t ) z(0) −
0 t T,
Q (s) R (s) ds, 0
so,
z(t ) =
1
ω[b( T ) + x(t )]
1 − ω(b( T ))
Q (s) R (s) ds
1 Q (t )
t 0
Q (s) R (s) ds
ω(b( T )) Q (t )
ω(b( T ))
ω b( T ) + x( T )
−
ω(b( T ))
0
= for 0 t T . Let t = T , since
t
1
T 0
Q (s) R (s) ds < 1, we have
ω(b( T )) Q ( T ) 1 − ω(b( T ))
T 0
Q (s) R (s) ds
,
ω is non-decreasing on R+ , then
ω(b( T )) Q ( T ) T 1 − ω(b( T )) 0 Q (s) R (s) ds
considering that the function
b( T ) + x( T ) ω−1
for T ∈ S. Since T 0 was arbitrarily chosen, considering u (t ) b(t ) + x(t ), we get (2.2), the proof is completed.
2
Different choices of L , M , N , K , α , a, h, f , g in Theorem 2.1, can derive many forms of inequalities, since the proofs are similar, we give the following corollaries without proof, the details are left to the readers. Corollary 2.1. Let L , M , N , b, α , a, f , g , ω be as in Theorem 2.1. If u ∈ C (R+ , R+ ) satisfies
α (t )
La(t , s)ω u (s) ds +
u (t ) b(t ) + 0
and ω(b(t ))
t 0
α (t )
M f (t , s)ω u (s) ds
0
α (t )
N g (t , s)ω u (s) ds 0
R (s) Q (s) ds < 1 for all t ∈ R+ , then
u (t ) ω−1 where Q (t ) = e
α (t ) 0
ω(b(t )) Q (t ) 1 − ω(b(t ))
La(t ,s) ds
t 0
, R (t ) =
R (s) Q (s) ds
α (t ) d [( 0 dt
,
t∈S
α (t )
M f (t , s) ds)(
0
(2.4)
N g (t , s) ds)].
L. Li et al. / J. Math. Anal. Appl. 377 (2011) 853–862
857
Corollary 2.2. Let M , N , α , f , g , ω be as in Theorem 2.1. Assume in addition that k, p ∈ C (R+ , R+ ), (k · p ) is non-decreasing on R+ . If u ∈ C (R+ , R+ ) satisfies
α (t )
M f (t , s)ω u (s) ds
u (t ) k(t ) + and ω[k(t ) p (t )]
u (t ) ω−1
N g (t , s)ω u (s) ds ,
α (t ) p (t ) +
0
t 0
t 0,
(2.5)
0
R (s) Q (s) ds < 1 for all t ∈ R+ , then
ω[k(t ) p (t )] Q (t ) 1 − ω[k(t ) p (t )]
t 0
t ∈ S,
,
R (s) Q (s) ds
(2.6)
where
R (t ) =
d
α (t )
α (t )
M f (t , s) ds
dt 0
Q (t ) = exp
α (t )
N g (t , s) ds
,
0
Mp (t ) f (t , s) + Nk(t ) g (t , s) ds .
0
In fact (2.5) is equivalent to
u (t ) k(t ) p (t ) +
α (t )
Mp (t ) f (t , s) + Nk(t ) g (t , s) ω u (s) ds +
0
α (t )
M f (t , s)ω u (s) ds
0
α (t )
N g (t , s)ω u (s) ds. 0
Now applying Corollary 2.1 one can easily get (2.6). Corollary 2.3. Let M , N , k, p , α , ω, f , g be as in Corollary 2.2. If u ∈ C (R+ , R+ ) satisfies
α (t )
M f (t , s)ω u (s) ds
u (t ) k(t ) + and ω[k(t ) p (t )]
u (t ) ω−1
t p (t ) +
0
t 0
N g (t , s)ω u (s) ds
0
R (s) Q (s) ds < 1 for all t ∈ R+ , then
ω[k(t ) p (t )] Q (t ) 1 − ω[k(t ) p (t )]
t 0
R (s) Q (s) ds
t ∈ S,
,
(2.7)
where
R (t ) =
d
α (t )
t
M f (t , s) ds
dt 0
N g (t , s) ds
,
0
α (t ) Q (t ) = exp
t
Mp (t ) f (t , s) ds + 0
Nk(t ) g (t , s) ds . 0
Let f (t , s) = a(t ) f (s) and (or) g (t , s) = b(t ) g (s), one can derive many corollaries, we list only one of them, the others are left to the readers. Corollary 2.4. Let M , N , k, p , α , ω be as in Corollary 2.2. Assume in addition that f , g ∈ C (R+ , R+ ) and a, v ∈ C 1 (R+ , R+ ). If u ∈ C (R+ , R+ ) satisfies
α (t )
Ma(t ) f (s)ω u (s) ds
u (t ) k(t ) + and ω[k(t ) p (t )]
t 0
t p (t ) +
0
R (s) Q (s) ds < 1 for all t ∈ R+ , then
0
N v (t ) g (s)ω u (s) ds
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L. Li et al. / J. Math. Anal. Appl. 377 (2011) 853–862
u (t ) ω−1
ω[k(t ) p (t )] Q (t ) 1 − ω[k(t ) p (t )]
t 0
t ∈ S,
,
R (s) Q (s) ds
(2.8)
where
R (t ) =
α (t )
d
t
Ma(t ) f (s) ds
dt
N v (t ) g (s) ds
0
0
α (t ) Q (t ) = exp
,
t
Mp (t )a(t ) f (s) ds + 0
Nk(t ) v (t ) g (s) ds . 0
2.2. The linear case In the nonlinear case, let ω(u ) = u in the theorems and corollaries. We get the results of linear case, since the proofs are similar, we give some of the corollaries without proof, the others are left to the readers. Corollary 2.5. Let L , K , M , N , b, α , a, h, f , g be as in Theorem 2.1. If u ∈ C (R+ , R+ ) satisfies
α (t ) u (t ) b(t ) +
and b(t )
t 0
t
La(t , s)u (s) ds + 0
α (t ) K h(t , s)u (s) ds +
0
t
M f (t , s)u (s) ds 0
N g (t , s)u (s) ds 0
R (s) Q (s) ds < 1 for all t ∈ R+ , then
b(t ) Q (t )
u (t )
1 − b(t )
t 0
R (s) Q (s) ds
t 0,
,
(2.9)
where
α (t ) Q (t ) = exp
La(t , s) ds + 0
R (t ) =
K h(t , s) ds , 0
α (t )
d
t
t
M f (t , s) ds
dt
N g (t , s) ds
0
.
0
Corollary 2.6. Let L , M , N , b, α , a, f , g be as in Theorem 2.1. If u ∈ C (R+ , R+ ) satisfies
α (t ) u (t ) b(t ) +
and b(t )
t 0
α (t )
La(t , s)u (s) ds +
α (t )
M f (t , s)u (s) ds
0
0
N g (t , s)u (s) ds 0
R (s) Q (s) ds < 1 for all t ∈ R+ , then
u (t )
b(t ) Q (t ) 1 − b(t )
where Q (t ) = e
α (t ) 0
t 0
R (s) Q (s) ds
La(t ,s) ds
, R (t ) =
,
t 0,
α (t ) d [( 0 dt
(2.10)
α (t )
M f (t , s) ds)(
0
N g (t , s) ds)].
Corollary 2.7. Let M , N , k, p , α , f , g be as in Corollary 2.2. If u ∈ C (R+ , R+ ) satisfies
u (t ) k(t ) +
and k(t ) p (t )
t
u (t )
0
α (t )
α (t ) p (t ) +
M f (t , s)u (s) ds 0
N g (t , s)u (s) ds 0
R (s) Q (s) ds < 1 for all t ∈ R+ , then
k(t ) p (t ) Q (t ) 1 − k(t ) p (t )
t 0
R (s) Q (s) ds
,
t 0,
(2.11)
L. Li et al. / J. Math. Anal. Appl. 377 (2011) 853–862
859
where
R (t ) =
d
α (t )
α (t )
M f (t , s) ds
dt
N g (t , s) ds
0
Q (t ) = exp
,
0
α (t )
Mp (t ) f (t , s) + Nk(t ) g (t , s) ds .
0
Corollary 2.8. Let M , N , k, p , α , f , g be as in Corollary 2.2. If u ∈ C (R+ , R+ ) satisfies
α (t )
u (t ) k(t ) +
p (t ) +
M f (t , s)u (s) ds 0
and k(t ) p (t )
t 0
N g (t , s)u (s) ds 0
R (s) Q (s) ds < 1 for all t ∈ R+ , then
k(t ) p (t ) Q (t )
u (t )
t
1 − k(t ) p (t )
t 0
t 0,
,
R (s) Q (s) ds
(2.12)
where
R (t ) =
d
α (t )
t
M f (t , s) ds
dt 0
N g (t , s) ds
,
0
α (t ) Q (t ) = exp
t
Mp (t ) f (t , s) ds +
Nk(t ) g (t , s) ds .
0
0
Corollary 2.9. Let M , N , k, p , a, v , f , g be as in Corollary 2.4. If u ∈ C (R+ , R+ ) satisfies
t
u (t ) k(t ) +
Ma(t ) f (s)u (s) ds
p (t ) +
0
and k(t ) p (t )
t 0
u (t )
t N v (t ) g (s)u (s) ds 0
R (s) Q (s) ds < 1 for all t ∈ R+ , then
k(t ) p (t ) Q (t ) 1 − k(t ) p (t )
t 0
R (s) Q (s) ds
,
t 0,
(2.13)
where
R (t ) =
d
t
t Ma(t ) f (s) ds
dt 0
t Q (t ) = exp
N v (t ) g (s) ds
,
0
Mp (t )a(t ) f (s) + Nk(t ) v (t ) g (s) ds .
0
Remark 1. Different choices of L , K , M , N and k, p , α can give many different inequalities. Obviously, our results generalize many results obtained before, for example, let k(t ) ≡ c 1 , p (t ) ≡ c 2 , Ma(t ) ≡ 1, Nb(t ) ≡ 1, then our Corollary 2.9 reduces to Pachpatte’s Theorem A [11]. Let M = 1, N = 0, k(t ) ≡ k, p (t ) ≡ 1, α (t ) = t, f (t , s) = h(s), then our Corollary 2.7 reduces to the famous Gronwall’s inequality and our Corollary 2.2 reduces to the famous Bihari’s inequality. At the same time our inequalities generalize Lipovan’s results [12], for example, let L = 1, K = M = N = 0, then our Theorem 2.1 reduces to Lipovan’s Theorem B [12].
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L. Li et al. / J. Math. Anal. Appl. 377 (2011) 853–862
3. Applications In this section we present some applications to study the boundedness property, asymptotic behavior and the stability of the solutions of integral equations, differential equations and integro-differential equations with time delay. Proposition 1. Suppose all the assumptions in Corollary 2.7 hold, u ∈ C (R+ , R+ ) is a solution of the integral equation
α (t )
u (t ) = k(t ) +
p (t ) +
M f (t , s)u (s) ds
α (t )
0
N g (t , s)u (s) ds ,
t 0.
0
If |k(t ) p (t )| K 1 on R+ , moreover
L = lim
α (t )
t →∞
Mp (t ) f (t , s) + Nk(t ) g (t , s) ds < ∞,
0
α (t ) K 2 = lim
α (t ) N g (t , s) ds < ∞,
M f (t , s) ds
t →∞ 0
0 L
where K 1 , L , K 2 are nonnegative constants and K 1 K 2 e L < 1, then u is bounded on R+ , in fact, |u (t )| 1−KK1 eK e L for t 0. 1 2 Proposition 2. Suppose all the assumptions in Proposition 1 hold except that the condition “|k(t ) p (t )| K 1 on R+ ” is replaced by “|k(t ) p (t )| K 1 e −t on R+ .” Then u (t ) → 0 as t → ∞. Proof of the Propositions. The Propositions follow immediately from Corollary 2.7, we omit the details. Moreover, according to our theorems, one can give many bounded results and asymptotic results analogue to the Propositions, they are left to the readers. 2 Now we will show that our results are useful in studying the stability of solutions to certain differential equations with time delay. These applications are given as examples. Example 1. Consider the nonlinear differential equation with time delay
x (t ) = f t , x(t ) + g t , x t − τ (t ) ,
t 0,
(3.14)
where f , g ∈ C (R+ × R, R), f (t , 0) = g (t , 0) = 0 for t 0, increasing diffeomorphism of R+ and
f (t , x) a(t )|x|,
g (t , x) b(t )|x|,
τ ∈ C 1 (R+ , R+ ), and τ (t ) t on R+ . If α (t ) = t − τ (t ) is an
(t , x) ∈ (R+ , R)
for a(t ), b(t ) ∈ C (R+ , R+ ), moreover,
∞ L=
∞
a(s) ds < ∞,
M=
0
b(s) ds < ∞ 0
for t 0, where L , M are nonnegative constants and of (3.14) is uniformly stable on R+ .
α −1 is the inverse of the diffeomorphism α , then the trivial solution
Proof. In fact, if x(t ) is a solution of (3.14) which satisfies x(t 0 ) = x0 , then x(t ) satisfies
t x(t ) = x0 +
t
f s, x(s) ds + t0
t = x0 + t0
g s, x
α (s) ds
t0
f s, x(s) ds +
α (t )
α (t 0 )
g (α −1 (r ), x(r ))
α (α −1 (r ))
dr ,
0 t0 t < ∞
L. Li et al. / J. Math. Anal. Appl. 377 (2011) 853–862
861
after performing the change of variables r = α (s) in the second integral. Our hypotheses on f , g guarantee that
x(t ) |x0 | +
t
α (t )
a(u )x(u ) du +
0
b(α −1 (u ))|x(u )|
|α (α −1 (u ))|
0
du ,
then from our Corollary 2.5 we deduce that
x(t ) |x0 |e
t 0
a(u ) du
α (t )
e
0
b(α −1 (u )) du |α (α −1 (u ))|
= |x0 |e
t
0 [a(u )+b (u )] du
|x0 |e L + M ,
0 t 0 t < ∞.
Then we conclude that the trivial solution of (3.14) is uniformly stable on R+ .
2
Example 2. Consider the nonlinear integro-differential equation with time delay
x (t ) = f t , x(t ) + g t , x t − τ (t ) + f t , x(t )
t
g s, x s − τ (s)
t 0,
ds,
(3.15)
0
where f , g , τ , α are as in Example 1. Suppose all the assumptions in Example 1 hold. Then the trivial solution of (3.15) is uniformly stable on R+ . Proof. In fact, if x(t ) is a solution of (3.15) which satisfies x(t 0 ) = x0 , then x(t ) satisfies
t x(t ) = x0 +
t
f u , x(u ) ds +
t0
g s, x α (s) ds +
t0
t = x0 +
f u , x(u ) ds +
t0
f u , x(u )
t0
α (t )
t
g (α −1 (r ), x(r ))
α (α −1 (r ))
u
g s, x α (s) ds du
0
t dr +
f u , x(u )
α(u )
t0
α (t 0 )
g (α −1 (r ), x(r ))
α (α −1 (r ))
dr du
0
for 0 t 0 t < ∞ after performing the change of variables r = α (s) at some intermediate step. Our hypotheses on f , g guarantee that
x(t ) |x0 | +
t
a(u )x(u ) du +
0
0
t |x0 | +
α (t )
a(u )x(u ) du +
0
α (t ) 0
b(α −1 (u ))|x(u )|
|α (α −1 (u ))| b(α −1 (u ))|x(u )|
|α (α −1 (u ))|
t du +
a(u )x(u )
α (t )
0
t du +
b(α −1 (r ))|x(r )|
|α (α −1 (r ))|
0
a(u )x(u ) du
0
α (t ) 0
R (t ) =
t
α (t ) a(u ) du
dt 0
0
t Q (t ) = exp
α (t ) a(u ) du +
0
0
b(α −1 (u ))
|α (α −1 (u ))| b(α −1 (u ))
|α (α −1 (u ))|
du
=
d
|α (α −1 (u ))|
t a(u ) du
dt
du
t
t = exp
0
a(u ) + b(u ) du ,
0
then from our Corollary 2.5 we deduce that
x(t ) =
|x0 |e L + M |x0 | Q (t ) t t 1 − |x0 | 0 Q (u ) R (u ) du 1 − |x0 |e L + M 0 R (u ) du 1 − |x0 |e L + M
|x0 |e L + M u α (u ) b(α −1 (s)) d [( 0 a(s) ds)( 0 0 du |α (α −1 (s))|
t
=
|x0 |e L + M t t 1 − |x0 |e L + M 0 a(s) ds 0 b(s) ds
|x0 |e L + M 1 − |x0 | LMe L + M
b(u ) du
0
ds)] du
dr du
b(α −1 (u ))|x(u )|
Let
d
,
du .
862
L. Li et al. / J. Math. Anal. Appl. 377 (2011) 853–862
for |x0 | LMe L + M < 1 and 0 t 0 t < ∞. So, for min{q,
(1−qLMe
L+M
e L+M
)ε
}, then |x(t )| < 2
is uniformly stable on R+ .
(1−qLMe
L+M
e L+M
)ε
×
ε > 0 and t 0 0, let 0 < q < L+M
e 1−qLMe L + M
1 LMe L + M
be a constant, choose δ =
= ε for |x0 | < δ . This proves that the trivial solution of (3.15)
Acknowledgment The authors are very grateful to the referee for his/her valuable suggestions.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
N.T.J. Bailey, The Mathematical Theory of Infectious Diseases and Its Applications, 2nd ed., Hafner, New York, 1975. R. Bellman, K.L. Cooke, Differential–Difference Equations, Academic Press, New York, 1963. O. Diekmann, Run for your life – A note on the asymptotic speed of propagation of an epidemic, J. Differential Equations 33 (1979) 58–73. G. Gripenberg, Periodic solutions of an epidemic model, J. Math. Biol. 10 (1980) 271–280. G. Gripenberg, On some epidemic models, Quart. Appl. Math. 39 (1981) 317–327. P. Waltma, Deterministic Threshold Models in the Theory of Epidemics, Lecture Notes in Biomathematics, vol. 1, Springer-Verlag, New York, 1974. B.G. Pachpatte, Stability and asymptotic behavior of perturbed nonlinear systems, J. Differential Equations 16 (1974) 14–25. B.G. Pachpatte, Perturbations of nonlinear systems of differential equations, J. Math. Anal. Appl. 51 (1975) 550–556. B.G. Pachpatte, On discrete inequalities related to Gronwall’s inequality, Proc. Indian Acad. Sci. Math. Sci. Part A 85 (1977) 26–40. B.G. Pachpatte, On some new discrete inequalities and their applications to a class of sum–difference equations, An. Stiint. Univ. Al. I. Cuza Iasi 24 (1978) 315–326. [11] B.G. Pachpatte, On a new inequality suggested by the study of certain epidemic models, J. Math. Anal. Appl. 195 (1995) 638–644. [12] O. Lipovan, Integral inequalities for retarded Volterra equations, J. Math. Anal. Appl. 322 (2006) 349–358. [13] L. Li, F. Meng, L. He, Some generalized integral inequalities and their applications, J. Math. Anal. Appl. 372 (2010) 339–349.
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Some new integral inequalities and their applications in studying the stability of nonlinear integro-differential equations with time delay ✩ Lianzhong Li a,b,∗ , Fanwei Meng b , Peijun Ju a a b
School of Mathematical and System Sciences, Taishan University, Tai’an, Shandong 271021, People’s Republic of China School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, People’s Republic of China
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 3 August 2010 Available online 8 December 2010 Submitted by Steven G. Krantz
In this paper, we generalize some integral inequalities to more general situations, and the inequalities of Pachpatte type are corollaries of our’s. We establish bounds on the solutions, and we show the usefulness of our results in investigating the asymptotic behavior and the stability on the solutions of integral equations, differential equations and integrodifferential equations with time delay. © 2010 Elsevier Inc. All rights reserved.
Keywords: Integral inequality Integral equation Differential equation Integro-differential equation Uniform stability
1. Introduction In a paper published in 1981, Gripenberg [5] studied the qualitative behavior of solutions of the equation
t
x(t ) = k p (t ) −
A (t − s)x(s) ds 0
t f (t ) +
B (t − s)x(s) ds .
(1.1)
0
This equation arises in the study of the spread of an infectious disease that does not induce permanent immunity. For detailed meanings of the various functions arising in (1.1), see [5] and also [1,3,4,6], for more results, see [2,7–10,13] and the references therein. In [5], the author studied the existence of a unique bounded continuous and nonnegative solution of (1.1) for t ∈ R+ under appropriate assumptions on A and B, and also obtained sufficient conditions for the convergence of the solution to a limit when t → ∞. Over the years integral inequalities have become a major tool in the analysis of various differential and integral equations that occur in nature or are built by man. Pachpatte [11] gave a new integral inequality and studied the boundedness, asymptotic behavior and growth of the solutions of Eq. (1.1) using the inequality. We list the inequality as follows. Theorem A (Pachpatte). (See [11].) Let u , f , g be real-valued nonnegative continuous functions defined on R+ , and c 1 , c 2 be nonnegative constants. If
✩ This research was supported by the National Natural Science Foundation of China under Grant No. 10771118 and the Natural Science Foundation of Shandong Province under Grant No. ZR2009AM011. Corresponding author at: School of Mathematical and System Sciences, Taishan University, Tai’an, Shandong 271021, People’s Republic of China. E-mail addresses: [email protected] (L. Li), [email protected] (F. Meng).
*
0022-247X/$ – see front matter doi:10.1016/j.jmaa.2010.12.002
©
2010 Elsevier Inc. All rights reserved.
854
L. Li et al. / J. Math. Anal. Appl. 377 (2011) 853–862
t
u (t ) c 1 + and c 1 c 2
t 0
t
c2 +
f (s)u (s) ds 0
g (s)u (s) ds 0
R (s) Q (s) ds < 1 for all t ∈ R+ , then
c 1 c 2 Q (t )
u (t )
1 − c1 c2
t 0
R (s) Q (s) ds
,
t ∈ R+ ,
where
t R (t ) =
f (t ) g (s) + f (s) g (t ) ds,
0
t
Q (t ) = exp
c 1 g (s) + c 2 f (s) ds .
0
Very recently, Lipovan observed the following integral inequality: Theorem B (Lipovan). (See [12].) Let a ∈ C (R+ × R+ , R+) with (t , s) → ∂t a(t , s) ∈ C (R+ × R+ , R+ ), k, ω ∈ C (R+ , R+ ) be non∞ decreasing functions with k(0) > 0, ω(t ) > 0 for t > 0 and 1 ωdt(t ) = ∞. Assume in addition that α ∈ C 1 (R+ , R+ ) is non-decreasing and α (t ) t for t 0. If u ∈ C (R+ , R+ ) satisfies
α (t ) u (t ) k(t ) +
a(t , s)ω u (s) ds,
t 0,
0
then
u (t ) Ω where Ω(t ) :=
−1
t 1
Ω k(t ) +
α (t )
t 0,
a(t , s) ds , 0
1 ω(s) ds, t > 0.
Aside from the various physical meanings of the functions arising in (1.1), we believe that equations like (1.1) are of great interest in their right and that further investigations of the qualitative behaviors of their solutions even under the usual hypotheses on the functions in (1.1) are much more interesting. In this paper, we establish some new inequalities, and the inequalities of Pachpatte type are corollaries of our’s. These furnish a handy tool for the study of qualitative as well as quantitative properties of solutions of equations like (1.1). We illustrate this by applying our new inequalities to study the boundedness properties, asymptotic behavior and the stability of the solutions of integral equations, differential equations and integro-differential equations with time delay. 2. Integral inequalities 2.1. The nonlinear case Theorem 2.1. Let L , K , M , N be nonnegative constants, a, h, f , g ∈ C (R+ × R+ , R+ ) with (t , s) → ∂t a(t , s), ∂t h(t , s), ∂t f (t , s), ∂t g (t , s) ∈ C (R+ × R+ , R+ ). Assume in addition that b, ω ∈ C (R+ , R+ ), α ∈ C 1 (R+ , R+ ) are non-decreasing functions and ω(t ) > 0, α (t ) t for t 0. If u ∈ C (R+ , R+ ) satisfies
α (t )
La(t , s)ω u (s) ds +
u (t ) b(t ) + and ω(b(t ))
t 0
0
t
K h(t , s)ω u (s) ds +
0
α (t )
M f (t , s)ω u (s) ds
0
t
N g (t , s)ω u (s) ds 0
R (s) Q (s) ds < 1 for all t ∈ R+ , then
u (t ) ω where
−1
ω(b(t )) Q (t ) 1 − ω(b(t ))
t 0
R (s) Q (s) ds
,
t ∈ S,
(2.2)
L. Li et al. / J. Math. Anal. Appl. 377 (2011) 853–862
α (t ) Q (t ) = exp
La(t , s) ds +
R (t ) =
K h(t , s) ds , 0
α (t )
d
t
0
t
M f (t , s) ds
dt
855
N g (t , s) ds
0
,
0
t ∈ S is chosen in such a way that
∈ Dom(ω−1 ), ω−1 is the inverse of ω .
ω(b(t )) Q (t ) t 1−ω(b(t )) 0 R (s) Q (s) ds
Proof. Let T ∈ S, T 0 be fixed and denote
α (t )
La(t , s)ω u (s) ds +
x(t ) = 0
t
α (t )
K h(t , s)ω u (s) ds +
M f (t , s)ω u (s) ds
0
0
x (t ) = La t , α (t ) ω u α (t ) α (t ) +
α , ω imply that x is non-decreasing on R+ . Hence, for
0
0
+ M f t , α (t ) ω u α (t ) α (t ) +
α (t )
t
M ∂t f (t , s)ω u (s) ds
N g (t , s)ω u (s) ds
0
+ N g (t , t )ω u (t ) +
t 0,
t L ∂t a(t , s)ω u (s) ds + K h(t , t )ω u (t ) + K ∂t h(t , s)ω u (s) ds
α (t )
N g (t , s)ω u (s) ds, 0
then u (t ) b(t ) + x(t ). Our assumptions on L , K , M , N , a, h, f , g and t ∈ [0, T ], by calculations we have
t
0
t
α (t )
N ∂t g (t , s)ω u (s) ds
M f (t , s)ω u (s) ds
0
0
α (t ) t d d ω b α (t ) + x α (t ) La(t , s) ds + ω b(t ) + x(t ) K h(t , s) ds dt
dt
0
0
d + ω b α (t ) + x α (t ) ω b(t ) + x(t )
t
α (t ) M f (t , s) ds
dt 0
d + ω b α (t ) + x α (t ) ω b(t ) + x(t )
N g (t , s) ds 0
α (t )
t N g (t , s) ds
dt 0
M f (t , s) ds 0
α (t ) t d ω b( T ) + x(t ) La(t , s) ds + K h(t , s) ds
dt
0
d + ω b( T ) + x(t ) 2
0
α (t ) M f (t , s) ds
dt
t
0
N g (t , s) ds . 0
Suppose b(0) > 0 (if b(0) = 0, carry out the following arguments with b(t ) + ε instead of b(t ), where small constant, and subsequently pass to the limit as ε → 0 to complete the proof) then we get
x (t )
ω2 [b( T ) + x(t )]
−
1
d
ω[b( T ) + x(t )] dt
α (t ) La(t , s) ds + 0
t K h(t , s) ds 0
d
ε > 0 is an arbitrary
α (t )
t
M f (t , s) ds
dt 0
N g (t , s) ds .
0
(2.3)
856
L. Li et al. / J. Math. Anal. Appl. 377 (2011) 853–862
Let
z(t ) =
α (t )
1
ω[b( T ) + x(t )]
q(t ) =
,
0
d dt
R (t ) =
K h(t , s) ds ,
d
α (t )
t
M f (t , s) ds
dt
0
then (2.3) is equivalent to
K h(t , s) ds, 0
t
La(t , s) ds +
z (t ) + z(t )
La(t , s) ds + 0
α (t ) Q (t ) = exp
t
0
N g (t , s) ds , 0
q(t ) − R (t ).
Multiplying the above inequality by eq(t ) = Q (t ), we get
d dt
z(t ) Q (t ) − Q (t ) R (t ).
Consider now the integral on the interval [0, t ] to obtain
t z(t ) Q (t ) z(0) −
0 t T,
Q (s) R (s) ds, 0
so,
z(t ) =
1
ω[b( T ) + x(t )]
1 − ω(b( T ))
Q (s) R (s) ds
1 Q (t )
t 0
Q (s) R (s) ds
ω(b( T )) Q (t )
ω(b( T ))
ω b( T ) + x( T )
−
ω(b( T ))
0
= for 0 t T . Let t = T , since
t
1
T 0
Q (s) R (s) ds < 1, we have
ω(b( T )) Q ( T ) 1 − ω(b( T ))
T 0
Q (s) R (s) ds
,
ω is non-decreasing on R+ , then
ω(b( T )) Q ( T ) T 1 − ω(b( T )) 0 Q (s) R (s) ds
considering that the function
b( T ) + x( T ) ω−1
for T ∈ S. Since T 0 was arbitrarily chosen, considering u (t ) b(t ) + x(t ), we get (2.2), the proof is completed.
2
Different choices of L , M , N , K , α , a, h, f , g in Theorem 2.1, can derive many forms of inequalities, since the proofs are similar, we give the following corollaries without proof, the details are left to the readers. Corollary 2.1. Let L , M , N , b, α , a, f , g , ω be as in Theorem 2.1. If u ∈ C (R+ , R+ ) satisfies
α (t )
La(t , s)ω u (s) ds +
u (t ) b(t ) + 0
and ω(b(t ))
t 0
α (t )
M f (t , s)ω u (s) ds
0
α (t )
N g (t , s)ω u (s) ds 0
R (s) Q (s) ds < 1 for all t ∈ R+ , then
u (t ) ω−1 where Q (t ) = e
α (t ) 0
ω(b(t )) Q (t ) 1 − ω(b(t ))
La(t ,s) ds
t 0
, R (t ) =
R (s) Q (s) ds
α (t ) d [( 0 dt
,
t∈S
α (t )
M f (t , s) ds)(
0
(2.4)
N g (t , s) ds)].
L. Li et al. / J. Math. Anal. Appl. 377 (2011) 853–862
857
Corollary 2.2. Let M , N , α , f , g , ω be as in Theorem 2.1. Assume in addition that k, p ∈ C (R+ , R+ ), (k · p ) is non-decreasing on R+ . If u ∈ C (R+ , R+ ) satisfies
α (t )
M f (t , s)ω u (s) ds
u (t ) k(t ) + and ω[k(t ) p (t )]
u (t ) ω−1
N g (t , s)ω u (s) ds ,
α (t ) p (t ) +
0
t 0
t 0,
(2.5)
0
R (s) Q (s) ds < 1 for all t ∈ R+ , then
ω[k(t ) p (t )] Q (t ) 1 − ω[k(t ) p (t )]
t 0
t ∈ S,
,
R (s) Q (s) ds
(2.6)
where
R (t ) =
d
α (t )
α (t )
M f (t , s) ds
dt 0
Q (t ) = exp
α (t )
N g (t , s) ds
,
0
Mp (t ) f (t , s) + Nk(t ) g (t , s) ds .
0
In fact (2.5) is equivalent to
u (t ) k(t ) p (t ) +
α (t )
Mp (t ) f (t , s) + Nk(t ) g (t , s) ω u (s) ds +
0
α (t )
M f (t , s)ω u (s) ds
0
α (t )
N g (t , s)ω u (s) ds. 0
Now applying Corollary 2.1 one can easily get (2.6). Corollary 2.3. Let M , N , k, p , α , ω, f , g be as in Corollary 2.2. If u ∈ C (R+ , R+ ) satisfies
α (t )
M f (t , s)ω u (s) ds
u (t ) k(t ) + and ω[k(t ) p (t )]
u (t ) ω−1
t p (t ) +
0
t 0
N g (t , s)ω u (s) ds
0
R (s) Q (s) ds < 1 for all t ∈ R+ , then
ω[k(t ) p (t )] Q (t ) 1 − ω[k(t ) p (t )]
t 0
R (s) Q (s) ds
t ∈ S,
,
(2.7)
where
R (t ) =
d
α (t )
t
M f (t , s) ds
dt 0
N g (t , s) ds
,
0
α (t ) Q (t ) = exp
t
Mp (t ) f (t , s) ds + 0
Nk(t ) g (t , s) ds . 0
Let f (t , s) = a(t ) f (s) and (or) g (t , s) = b(t ) g (s), one can derive many corollaries, we list only one of them, the others are left to the readers. Corollary 2.4. Let M , N , k, p , α , ω be as in Corollary 2.2. Assume in addition that f , g ∈ C (R+ , R+ ) and a, v ∈ C 1 (R+ , R+ ). If u ∈ C (R+ , R+ ) satisfies
α (t )
Ma(t ) f (s)ω u (s) ds
u (t ) k(t ) + and ω[k(t ) p (t )]
t 0
t p (t ) +
0
R (s) Q (s) ds < 1 for all t ∈ R+ , then
0
N v (t ) g (s)ω u (s) ds
858
L. Li et al. / J. Math. Anal. Appl. 377 (2011) 853–862
u (t ) ω−1
ω[k(t ) p (t )] Q (t ) 1 − ω[k(t ) p (t )]
t 0
t ∈ S,
,
R (s) Q (s) ds
(2.8)
where
R (t ) =
α (t )
d
t
Ma(t ) f (s) ds
dt
N v (t ) g (s) ds
0
0
α (t ) Q (t ) = exp
,
t
Mp (t )a(t ) f (s) ds + 0
Nk(t ) v (t ) g (s) ds . 0
2.2. The linear case In the nonlinear case, let ω(u ) = u in the theorems and corollaries. We get the results of linear case, since the proofs are similar, we give some of the corollaries without proof, the others are left to the readers. Corollary 2.5. Let L , K , M , N , b, α , a, h, f , g be as in Theorem 2.1. If u ∈ C (R+ , R+ ) satisfies
α (t ) u (t ) b(t ) +
and b(t )
t 0
t
La(t , s)u (s) ds + 0
α (t ) K h(t , s)u (s) ds +
0
t
M f (t , s)u (s) ds 0
N g (t , s)u (s) ds 0
R (s) Q (s) ds < 1 for all t ∈ R+ , then
b(t ) Q (t )
u (t )
1 − b(t )
t 0
R (s) Q (s) ds
t 0,
,
(2.9)
where
α (t ) Q (t ) = exp
La(t , s) ds + 0
R (t ) =
K h(t , s) ds , 0
α (t )
d
t
t
M f (t , s) ds
dt
N g (t , s) ds
0
.
0
Corollary 2.6. Let L , M , N , b, α , a, f , g be as in Theorem 2.1. If u ∈ C (R+ , R+ ) satisfies
α (t ) u (t ) b(t ) +
and b(t )
t 0
α (t )
La(t , s)u (s) ds +
α (t )
M f (t , s)u (s) ds
0
0
N g (t , s)u (s) ds 0
R (s) Q (s) ds < 1 for all t ∈ R+ , then
u (t )
b(t ) Q (t ) 1 − b(t )
where Q (t ) = e
α (t ) 0
t 0
R (s) Q (s) ds
La(t ,s) ds
, R (t ) =
,
t 0,
α (t ) d [( 0 dt
(2.10)
α (t )
M f (t , s) ds)(
0
N g (t , s) ds)].
Corollary 2.7. Let M , N , k, p , α , f , g be as in Corollary 2.2. If u ∈ C (R+ , R+ ) satisfies
u (t ) k(t ) +
and k(t ) p (t )
t
u (t )
0
α (t )
α (t ) p (t ) +
M f (t , s)u (s) ds 0
N g (t , s)u (s) ds 0
R (s) Q (s) ds < 1 for all t ∈ R+ , then
k(t ) p (t ) Q (t ) 1 − k(t ) p (t )
t 0
R (s) Q (s) ds
,
t 0,
(2.11)
L. Li et al. / J. Math. Anal. Appl. 377 (2011) 853–862
859
where
R (t ) =
d
α (t )
α (t )
M f (t , s) ds
dt
N g (t , s) ds
0
Q (t ) = exp
,
0
α (t )
Mp (t ) f (t , s) + Nk(t ) g (t , s) ds .
0
Corollary 2.8. Let M , N , k, p , α , f , g be as in Corollary 2.2. If u ∈ C (R+ , R+ ) satisfies
α (t )
u (t ) k(t ) +
p (t ) +
M f (t , s)u (s) ds 0
and k(t ) p (t )
t 0
N g (t , s)u (s) ds 0
R (s) Q (s) ds < 1 for all t ∈ R+ , then
k(t ) p (t ) Q (t )
u (t )
t
1 − k(t ) p (t )
t 0
t 0,
,
R (s) Q (s) ds
(2.12)
where
R (t ) =
d
α (t )
t
M f (t , s) ds
dt 0
N g (t , s) ds
,
0
α (t ) Q (t ) = exp
t
Mp (t ) f (t , s) ds +
Nk(t ) g (t , s) ds .
0
0
Corollary 2.9. Let M , N , k, p , a, v , f , g be as in Corollary 2.4. If u ∈ C (R+ , R+ ) satisfies
t
u (t ) k(t ) +
Ma(t ) f (s)u (s) ds
p (t ) +
0
and k(t ) p (t )
t 0
u (t )
t N v (t ) g (s)u (s) ds 0
R (s) Q (s) ds < 1 for all t ∈ R+ , then
k(t ) p (t ) Q (t ) 1 − k(t ) p (t )
t 0
R (s) Q (s) ds
,
t 0,
(2.13)
where
R (t ) =
d
t
t Ma(t ) f (s) ds
dt 0
t Q (t ) = exp
N v (t ) g (s) ds
,
0
Mp (t )a(t ) f (s) + Nk(t ) v (t ) g (s) ds .
0
Remark 1. Different choices of L , K , M , N and k, p , α can give many different inequalities. Obviously, our results generalize many results obtained before, for example, let k(t ) ≡ c 1 , p (t ) ≡ c 2 , Ma(t ) ≡ 1, Nb(t ) ≡ 1, then our Corollary 2.9 reduces to Pachpatte’s Theorem A [11]. Let M = 1, N = 0, k(t ) ≡ k, p (t ) ≡ 1, α (t ) = t, f (t , s) = h(s), then our Corollary 2.7 reduces to the famous Gronwall’s inequality and our Corollary 2.2 reduces to the famous Bihari’s inequality. At the same time our inequalities generalize Lipovan’s results [12], for example, let L = 1, K = M = N = 0, then our Theorem 2.1 reduces to Lipovan’s Theorem B [12].
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L. Li et al. / J. Math. Anal. Appl. 377 (2011) 853–862
3. Applications In this section we present some applications to study the boundedness property, asymptotic behavior and the stability of the solutions of integral equations, differential equations and integro-differential equations with time delay. Proposition 1. Suppose all the assumptions in Corollary 2.7 hold, u ∈ C (R+ , R+ ) is a solution of the integral equation
α (t )
u (t ) = k(t ) +
p (t ) +
M f (t , s)u (s) ds
α (t )
0
N g (t , s)u (s) ds ,
t 0.
0
If |k(t ) p (t )| K 1 on R+ , moreover
L = lim
α (t )
t →∞
Mp (t ) f (t , s) + Nk(t ) g (t , s) ds < ∞,
0
α (t ) K 2 = lim
α (t ) N g (t , s) ds < ∞,
M f (t , s) ds
t →∞ 0
0 L
where K 1 , L , K 2 are nonnegative constants and K 1 K 2 e L < 1, then u is bounded on R+ , in fact, |u (t )| 1−KK1 eK e L for t 0. 1 2 Proposition 2. Suppose all the assumptions in Proposition 1 hold except that the condition “|k(t ) p (t )| K 1 on R+ ” is replaced by “|k(t ) p (t )| K 1 e −t on R+ .” Then u (t ) → 0 as t → ∞. Proof of the Propositions. The Propositions follow immediately from Corollary 2.7, we omit the details. Moreover, according to our theorems, one can give many bounded results and asymptotic results analogue to the Propositions, they are left to the readers. 2 Now we will show that our results are useful in studying the stability of solutions to certain differential equations with time delay. These applications are given as examples. Example 1. Consider the nonlinear differential equation with time delay
x (t ) = f t , x(t ) + g t , x t − τ (t ) ,
t 0,
(3.14)
where f , g ∈ C (R+ × R, R), f (t , 0) = g (t , 0) = 0 for t 0, increasing diffeomorphism of R+ and
f (t , x) a(t )|x|,
g (t , x) b(t )|x|,
τ ∈ C 1 (R+ , R+ ), and τ (t ) t on R+ . If α (t ) = t − τ (t ) is an
(t , x) ∈ (R+ , R)
for a(t ), b(t ) ∈ C (R+ , R+ ), moreover,
∞ L=
∞
a(s) ds < ∞,
M=
0
b(s) ds < ∞ 0
for t 0, where L , M are nonnegative constants and of (3.14) is uniformly stable on R+ .
α −1 is the inverse of the diffeomorphism α , then the trivial solution
Proof. In fact, if x(t ) is a solution of (3.14) which satisfies x(t 0 ) = x0 , then x(t ) satisfies
t x(t ) = x0 +
t
f s, x(s) ds + t0
t = x0 + t0
g s, x
α (s) ds
t0
f s, x(s) ds +
α (t )
α (t 0 )
g (α −1 (r ), x(r ))
α (α −1 (r ))
dr ,
0 t0 t < ∞
L. Li et al. / J. Math. Anal. Appl. 377 (2011) 853–862
861
after performing the change of variables r = α (s) in the second integral. Our hypotheses on f , g guarantee that
x(t ) |x0 | +
t
α (t )
a(u )x(u ) du +
0
b(α −1 (u ))|x(u )|
|α (α −1 (u ))|
0
du ,
then from our Corollary 2.5 we deduce that
x(t ) |x0 |e
t 0
a(u ) du
α (t )
e
0
b(α −1 (u )) du |α (α −1 (u ))|
= |x0 |e
t
0 [a(u )+b (u )] du
|x0 |e L + M ,
0 t 0 t < ∞.
Then we conclude that the trivial solution of (3.14) is uniformly stable on R+ .
2
Example 2. Consider the nonlinear integro-differential equation with time delay
x (t ) = f t , x(t ) + g t , x t − τ (t ) + f t , x(t )
t
g s, x s − τ (s)
t 0,
ds,
(3.15)
0
where f , g , τ , α are as in Example 1. Suppose all the assumptions in Example 1 hold. Then the trivial solution of (3.15) is uniformly stable on R+ . Proof. In fact, if x(t ) is a solution of (3.15) which satisfies x(t 0 ) = x0 , then x(t ) satisfies
t x(t ) = x0 +
t
f u , x(u ) ds +
t0
g s, x α (s) ds +
t0
t = x0 +
f u , x(u ) ds +
t0
f u , x(u )
t0
α (t )
t
g (α −1 (r ), x(r ))
α (α −1 (r ))
u
g s, x α (s) ds du
0
t dr +
f u , x(u )
α(u )
t0
α (t 0 )
g (α −1 (r ), x(r ))
α (α −1 (r ))
dr du
0
for 0 t 0 t < ∞ after performing the change of variables r = α (s) at some intermediate step. Our hypotheses on f , g guarantee that
x(t ) |x0 | +
t
a(u )x(u ) du +
0
0
t |x0 | +
α (t )
a(u )x(u ) du +
0
α (t ) 0
b(α −1 (u ))|x(u )|
|α (α −1 (u ))| b(α −1 (u ))|x(u )|
|α (α −1 (u ))|
t du +
a(u )x(u )
α (t )
0
t du +
b(α −1 (r ))|x(r )|
|α (α −1 (r ))|
0
a(u )x(u ) du
0
α (t ) 0
R (t ) =
t
α (t ) a(u ) du
dt 0
0
t Q (t ) = exp
α (t ) a(u ) du +
0
0
b(α −1 (u ))
|α (α −1 (u ))| b(α −1 (u ))
|α (α −1 (u ))|
du
=
d
|α (α −1 (u ))|
t a(u ) du
dt
du
t
t = exp
0
a(u ) + b(u ) du ,
0
then from our Corollary 2.5 we deduce that
x(t ) =
|x0 |e L + M |x0 | Q (t ) t t 1 − |x0 | 0 Q (u ) R (u ) du 1 − |x0 |e L + M 0 R (u ) du 1 − |x0 |e L + M
|x0 |e L + M u α (u ) b(α −1 (s)) d [( 0 a(s) ds)( 0 0 du |α (α −1 (s))|
t
=
|x0 |e L + M t t 1 − |x0 |e L + M 0 a(s) ds 0 b(s) ds
|x0 |e L + M 1 − |x0 | LMe L + M
b(u ) du
0
ds)] du
dr du
b(α −1 (u ))|x(u )|
Let
d
,
du .
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L. Li et al. / J. Math. Anal. Appl. 377 (2011) 853–862
for |x0 | LMe L + M < 1 and 0 t 0 t < ∞. So, for min{q,
(1−qLMe
L+M
e L+M
)ε
}, then |x(t )| < 2
is uniformly stable on R+ .
(1−qLMe
L+M
e L+M
)ε
×
ε > 0 and t 0 0, let 0 < q < L+M
e 1−qLMe L + M
1 LMe L + M
be a constant, choose δ =
= ε for |x0 | < δ . This proves that the trivial solution of (3.15)
Acknowledgment The authors are very grateful to the referee for his/her valuable suggestions.
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