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On some new integral inequalities of Gronwall–Bellman–Bihari type with delay for discontinuous functions and their applications
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ScienceDirect Indagationes Mathematicae xx (xxxx) xxx–xxx www.elsevier.com/locate/indag
On some new integral inequalities of Gronwall–Bellman–Bihari type with delay for discontinuous functions and their applications✩ Q1
Xiaohong Liu a,∗ , Lihong Zhang b , Praveen Agarwal c , Guotao Wang b a School of Mathematical Sciences, Xiamen University, Xiamen, Fujian 361005, People’s Republic of China b School of Mathematics and Computer Science, Shanxi Normal University, Linfen, Shanxi 041004,
People’s Republic of China c Department of Mathematics, Anand International College of Engineering, Rajasthan, Jaipur 303012, India
Received 31 October 2014; received in revised form 26 June 2015; accepted 5 July 2015 Communicated by H. Woerdeman
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Abstract
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In this article, some new explicit bounds on solutions to a class of new nonlinear integral inequalities of Gronwall–Bellman–Bihari type with delay for discontinuous functions are established. These inequalities generalize and improve some former famous results about inequalities, and which provide an excellent tool to discuss the qualitative and quantitative properties for solutions to some nonlinear differential and integral equations. To illustrate our results, we present an example to show estimated solutions for an impulsive differential system. c 2015 Published by Elsevier B.V. on behalf of Royal Dutch Mathematical Society (KWG). ⃝
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Keywords: Integral inequalities; Discontinuous functions; Gronwall–Bellman–Bihari type inequalities; Impulsive differential systems
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✩ This work was supported by the Natural Science Foundation for Young Scientists of Shanxi Province, China (2012021002-3). ∗ Corresponding author. Tel.: +86 15959262015. E-mail addresses: [email protected] (X. Liu), [email protected] (L. Zhang), [email protected] (P. Agarwal), [email protected] (G. Wang).
http://dx.doi.org/10.1016/j.indag.2015.07.001 c 2015 Published by Elsevier B.V. on behalf of Royal Dutch Mathematical Society (KWG). 0019-3577/⃝
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1. Introduction
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Q2
During the past years, a number of efforts have been devoted to the integral inequalities. The celebrated results given by Gronwall, Bellman, Bihari and their linear and nonlinear generalizations in the case of continuous and discontinuous functions provide a fundamental role in the study of many qualitative properties of differential and integral equations, which we can find in [3,6,4,11,12,15–17,2,1,10,9]. For some new development on this topic, see [5,8,13,18,14,19,7]. In 2007, Jiang and Meng [8] discussed the following delay integral inequalities: t p x (t) ≤ ρ(t) + π(t) f (s)x q (s) + h(s)x r (σ (s)) ds t0
9
with the initial condition:
10
s(t) = φ(t) t ∈ R+
11
φ(σ (t)) ≤ ρ(t) p
1
12
t ∈ R+ σ (t) ≤ 0
and x (t) ≤ ρ(t) + π(t) p
13
t
f (s)x q (s) + L s, x(σ (s))
t0 14 15 16 17 18 19
20
with the same initial condition. Under the assumption of functions x(t), ρ(t), π(t), f (t), h(t) ∈ C(R + , R + ), authors obtained explicit bounds on unknown function x(t) of the above inequalities. The purpose of this paper is to give explicit bounds to some new nonlinear integral inequalities of Gronwall–Bellman–Bihari type with delay for discontinuous functions. Precisely, we consider the following nonlinear integral inequalities: t x p (t) ≤ ρ(t) + π(t) f (s)x q (s) + h(s)x r (σ (s)) ds + ai x m (ti − 0) to
t0 21
22
and x p (t) ≤ ρ(t) + π(t)
t
f (s)x q (s) + L s, x(σ (s)) + ai x m (ti − 0).
t0
to
25
Our results develop and improve certain results which were proved by Jiang and Meng [8]. At the end of the article, an example of application is presented to show estimated solutions for an impulsive differential system.
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2. Main results
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Lemma 2.1 ([8]). Assume that a ≥ 0, p ≥ q ≥ 0 and p ̸= 0, then
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q 28
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ap ≤
q q−p p p − q qp K a+ K p p
for any K > 0.
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Theorem 2.1. If x(t)(t ≥ t0 ≥ 0) is a nonnegative piecewise continuous function with discontinuities of the 1st kind in the points ti (t0 < t1 < t2 < · · · , limi→∞ ti = ∞) and satisfies the following form of inequality: x p (t) ≤ ρ(t) + π(t)
t
ai x m (ti − 0) f (s)x q (s) + h(s)x r (σ (s)) ds +
(2.1)
1 2 3
4
to
t0
where ρ(t), π(t) ≥ 1 are both non-decreasing functions at t ≥ t0 , h(s), f (s) ∈ C(R+ , R+ ), σ (s) ≤ s, lim|s|→∞ σ (s) = ∞, ai ≥ 0, m > 0, p ≥ q ≥ 0, p ≥ r ≥ 0. Then, for t ≥ t0 , the following estimates hold: t 1 t0 Q(s)ds p ρ(t)π(t)A(t)e , t ∈ I0 = [t0 , t1 ] t 1 Q(s)ds p , t ∈ I1 = [t1 , t2 ] ρ(t)π(t)[M2 (t) + A1 L 2 (t)]e t1 ti s Q(τ )dτ x(t) ≤ ρ(t)π(t) M (t) + Q(s) M (s) + A L (s) e ti−1 ds i i i−1 i t i−1 1 t m t p ti−1 Q(s)ds p ti Q(s)ds e , +Ai (Mi (t) + Ai−1 L i (t))e t ∈ Ii = [ti , ti−1 ]
(2.2)
where
5 6 7
8
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A(t) = 1 +
t
f (s)
t0 m p
p − r rp p − q qp K + h(s) K ds, p p
m
Ai = ai ρ (ti − 0)π p (ti − 0), q q− p r r−p Q(t) = K p f (t)π(t) + K p h(t)π(t), p p t1 s Q(τ )dτ M2 (t) = A(t) + Q(s)A(s)e t0 ds,
t m Q(s)ds p , L 2 (t) = A(t)e t0
(2.3)
10
t0
Mi (t) = Mi−1 (t) +
ti
s Q(τ )dτ Q(s) Mi−1 (s) + Ai−2 L i−1 (s) e ti−1 ds
(i > 2),
ti−1 t m Q(s)ds p L i (t) = (Mi−1 (t) + Ai−2 L i−1 (t))e ti−2
(i > 2).
Proof. Being ρ(t) ≥ 1, we have t x p (t) x q (s) x r (σ (s)) ≤ 1 + π(t) f (s) + h(s) ds + ai x m (ti − 0). ρ(t) ρ(t) ρ(t) t0 to
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Considering ρ(t) a non-decreasing function, for s ≤ t, we get ρ(s) ≤ ρ(t), then: x p (t) ≤ 1 + π(t) ρ(t)
t t0
f (s)
x q (s) x r (σ (s)) + h(s) ds + ai x m (ti − 0). ρ(s) ρ(s) to
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(2.4)
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By setting u(t) =
x p (t) ρ(t)
u(t) ≤ π(t) 1 +
2
, we can see u(t0 ) = 1 and there is
t t0
3
ω(t) = 1 +
t
f (s)
t0
8
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It is obvious that x p (t) ≤ π(t)ω(t). ρ(t) This means that x p (t) ≤ ρ(t)π(t)ω(t). The above inequality implies the estimation for x(t) such as 1 p x(t) ≤ ρ(t)π(t)ω(t)
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13
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1 p x(ti − 0) ≤ ρ(ti − 0)π(ti − 0)ω(ti − 0) .
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By Lemma 2.1, for any K > 0, we obtain
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q p−q q q q− p p x q (t) ≤ ρ(t)π(t)ω(t) ≤ K p ρ(t)π(t)ω(t) + Kp p p r p−r r r r−p p x r (t) ≤ ρ(t)π(t)ω(t) ≤ K p ρ(t)π(t)ω(t) + K p. p p It follows from (2.7) and (2.8) that t q q− p p−q q Kp ω(t) ≤ 1 + f (s) K p π(s)ω(s) + p p t0 r r−p p−r r + h(s) K p π(s)ω(s) + K p ds p p m m p p + ai ρ(ti − 0)π(ti − 0) ω(ti − 0) to
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(2.7)
and
Noting σ (s) ≤ s, then 1 1 p p x σ (s) ≤ ρ σ (s) π σ (s) ω σ (s) ≤ ρ(s)π(s)ω(s) .
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(2.6)
u(t) ≤ π(t)ω(t).
10
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x q (s) x r (σ (s)) ai x m (ti − 0). + h(s) ds + ρ(s) ρ(s) to
Using (2.6) in (2.5), we obtain
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(2.5)
Denote
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x q (s) x r (σ (s)) m ai x (ti − 0) . f (s) ds + + h(s) ρ(s) ρ(s) to
= 1+
t0
− q qp p − r rp K f (s) + K h(s) ds p p
(2.8)
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t r r−p q q−p p f (s)π(s) + K p h(s)π(s) ω(s)ds K + p t0 p m m p p ai ρ(ti − 0)π(ti − 0) + ω(ti − 0) .
1
(2.9)
2
to
Set
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t p − q qp p − r rp A(t) = 1 + K f (s) + K h(s) ds p p t0 q q−p p r r −p p Q(t) = K f (t)π(t) + K h(t)π(t) p p m p Ai = ai ρ(ti − 0)π(ti − 0) . Then (2.9) can be restated as t m Q(s)ω(s)ds + Ai [ω(ti − 0)] p . ω(t) ≤ A(t) +
(2.10)
4
5
(2.11)
6
to
t0
Let us consider Ii = [ti , ti+1 ](i = 0, 1 · · ·). For t ∈ I0 , t t Q(s)ds ω(t) ≤ A(t) + Q(s)ω(s)ds ≤ A(t)e t0 .
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(2.12)
9
t0
It is obvious that t 1 Q(s)ds p . x(t) ≤ ρ(t)π(t)A(t)e t0
10
11
Now let us consider the next interval I1 . For t ∈ I1 , we have t m p ω(t) ≤ A(t) + Q(s)ω(s)ds + A1 ω(t1 − 0) t0 t1
s
≤ A(t) +
Q(s)A(s)e
t0
Q(τ )dτ
t
ds +
t0
12
13
t m Q(s)ds p Q(s)ω(s)ds + A1 A(t)e t0
14
t1 t1
≤ A(t) +
s
Q(s)A(s)e
t0
Q(τ )dτ
t m t Q(s)ds p Q(s)ds ds + A1 A(t)e t0 e t1 .
(2.13)
15
t0
By setting M2 (t) = A(t) +
16
t1
s
Q(s)A(s)e
t0
Q(τ )dτ
ds
17
t0 t m Q(s)ds p L 2 (t) = A(t)e t0 .
Therefor for the interval I1 , we see t 1 Q(s)ds p x(t) ≤ ρ(t)π(t)[M2 (t) + A1 L 2 (t)]e t1 .
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Consider the next interval I2 , we have t m p ω(t) ≤ A(t) + Q(s)ω(s)ds + A2 ω(t2 − 0) t0 t1
3
s
≤ A(t) +
Q(s)A(s)e
t0
Q(τ )dτ
ds +
t0
t
+
4
t2
Q(s)[M2 (s) + A1 L 2 (s)]e
s t1
Q(τ )dτ
ds
t1
t m Q(s)ds p Q(s)ω(s)ds + A2 [M2 (t) + A1 L 2 (t)]e t1
t2 5
≤ M2 (t) +
t2
Q(s)[M2 (s) + A1 L 2 (s)]e
s t1
Q(τ )dτ
ds
t1
t
+
6
t m Q(s)ds p . Q(s)ω(s)ds + A2 [M2 (t) + A1 L 2 (t)]e t1
(2.14)
t2 7
8
Thus, we have ω(t) ≤
M2 (t) +
t2
s Q(τ )dτ Q(s) M2 (s) + A1 L 2 (s) e t1 ds
t1
9
10 11 12
13 14
15
+ A2
m t p t1 Q(s)ds M2 (t) + A1 L 2 (t) e
t
e
t2
17
.
(2.15)
1 p Taking into account x(t) is such that x(t) ≤ ρ(t)π(t)ω(t) , ∀t ≥ t0 , from inequalities (2.15), we can see for t ∈ I2 estimates (2.2) are valid for x(t). Consequently, by using a similar procedure for t ∈ Ii , we complete the proof of Theorem 2.1. Theorem 2.2. Assume that x(t), ρ(t), π(t), σ (t) are defined in Theorem 2.1 and x(t) satisfies the following form of integral inequality t f (s)x q (s) + L s, x(σ (s)) + ai x m (ti − 0) (2.16) x p (t) ≤ ρ(t) + π(t) t0
16
Q(s)ds
to
2 , R ) satisfying where p ≥ q ≥ 1, m ≥ 0 and L , M ∈ (R+ +
0 ≤ L(t, x) − L(t, y) ≤ M(t, y)(x − y)
18
for x ≥ y ≥ 0, then the inequality (3.1) with the initial condition in Theorem 2.1 implies
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t 1 ˜ t0 Q(s)ds p ˜ ρ(t)π(t) A(t)e , t ∈ I0 = [t0 , t1 ] t ˜ 1 Q(s)ds p ρ(t)π(t) M˜ 2 (t) + A˜1 L˜2 (t) e t1 , t ∈ I1 = [t1 , t2 ] ti s ˜ ˜ i (t) + ˜ ˜ i (s) + Ai−1 ˜ L˜i (s) e ti−1 Q(τ )dτ ds x(t) ≤ ρ(t)π(t) M Q(s) M ti−1 1 t m t ˜ p ˜ ti−1 Q(s)ds p t1 Q(s)ds ˜ ˜ ˜ ˜ + A ( M (t) + A L (t))e e , i i i−1 i t ∈ Ii = [ti , ti−1 ]
(2.17)
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where
1
t p − q qp p − r rp ˜ =1+ K f (s) + K h(s) ds, A(t) p p t0 m p A˜i = ai ρ(ti − 0)π(ti − 0) q q− p r r−p ˜ Q(t) = K p f (t)π(t) + K p h(t)π(t) p p t1 s ˜ t0 Q(τ )dτ ˜ + ˜ A(s)e ˜ M˜ 2 (t) = A(t) Q(s) ds
(2.18)
t m ˜ t0 Q(s)ds p ˜ L˜2 (t) = A(t)e
2
t0
˜ (t) + M˜ i (t) = Mi−1
ti
s ˜ ˜ ˜ (s) + Ai−2 ˜ L i−1 ˜ (s) e ti−1 Q(τ )dτ ds Q(s) Mi−1
ti−1
˜ (t) + Ai−2 ˜ L i−1 ˜ (t))e L˜i (t) = ( Mi−1
t ti−2
˜ Q(s)ds
m p
.
Proof. Being ρ(t) ≥ 1 a non-decreasing function, the following inequality holds: t x q (s) x p (t) ≤ 1 + π(t) f (s) + L s, x(σ (s)) ds + ai x m (ti − 0) ρ(t) ρ(s) t0 to
3
4
(2.19)
Denote
5
Q4
ω(t) ˜ =1+
t
f (s)
t0
x q (s) ρ(s)
+ L(s, x(σ (s))) ds + ai x m (ti − 0)
6
7
to
it is obvious that 1 p x(t) ≤ ρ(t)π(t)ω(t) ˜
8
(2.20)
1 p x(ti − 0) ≤ ρ(ti − 0)π(ti − 0)ω(t ˜ i − 0) .
9
By Lemma 2.1, we obtain q p−q q p x q (t) ≤ ρ(t)π(t)ω(t) ˜ ρ(t)π(t)ω(t) ˜ + ≤ p p
10
11
and
12
1 p−1 1 p x(t) ≤ ρ(t)π(t)ω(t) ˜ ≤ ρ(t)π(t)ω(t) ˜ + . p p
13
Then there is
Q5
q p−q ρ(s)π(s)ω(s) ˜ π(s)ω(s) ˜ + ds + L s, + p p p t0 t0 m m p p + ai ρ(ti − 0)π(ti − 0) ω(t ˜ i − 0)
ω(t) ˜ ≤ 1+
t
f (s)
to
t
p − 1 p
ds
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t q ρ(s)π(s)ω(s) ˜ p−q p − 1 ˜ + ds + L s, + ds f (s) π(s)ω(s) p p p p t0 t0 t t p − 1 p − 1 − ds + ds L s, L s, p p t0 t0 m m p + ai ρ(ti − 0)π(ti − 0) [ω(t ˜ i − 0)] p
t
= 1+
1
2
3
to
t q p−q p − 1 ρ(s)π(s)ω(s) ˜ ≤ 1+ f (s) π(s)ω(s) ˜ + ds ds + M s, p p p p t0 t0 m m p p + ai ρ(ti − 0)π(ti − 0) ω(t ˜ i − 0)
4
5
to
t p − 1 ρ(s)π(s) p−q q = 1+ f (s) ds + f (s)π(s) + M s, ω(s)ds ˜ p p p t0 t0 p m m p + ai ρ(ti − 0)π(ti − 0) [ω(t ˜ i − 0)] p . (2.21)
6
7
to
8
9
10
11
12
13
Setting in the last inequality t p−q ˜ A(t) = 1 + f (s)ds p t0 m p A˜i = ai ρ(ti − 0)π(ti − 0) p − 1 ρ(t)π(t) q ˜ Q(t) = f (t)π(t) + M t, p p p we obtain ˜ + ω(t) ˜ ≤ A(t)
t t0
˜ ω(s)ds Q(s) ˜ +
to
m p A˜i ω(t ˜ i − 0) .
14
Thus, we get the similar form of inequality (2.11) in Theorem 2.1, we complete the proof.
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3. Example
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17
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As an application, when p = 1, let us consider the following impulsive system: dy = u(y, t) + v(y, t) t ̸= ti dt 1y| = Ii (y) +t=ti y(t0 ) = y0 when t ≥ t0 ≥ 0, limi→∞ ti = ∞, ti−1 < ti , i = 1, 2, . . . . Suppose that u(y, t), v(y, t) and Ii (y) satisfy the nonlinearity conditions: (a) ∥u(y, t)∥ ≤ f (t)∥y∥q f : R+ → R+ ; (b) ∥v(y, t)∥ ≤ h(t)∥y∥r h : R+ → R+ ; (c) ∥Ii (y)∥ ≤ ai ∥y∥m , ai = const. > 0, m > 0.
(2.22)
(3.1)
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Denote y(t) = x(t, t0 , y0 ) the solution for (3.1), so that y(t) = y(t, t0 , y0 )(y(t0 ) = y0 ), it is obvious that its integro-sum representation is the following: t t y(t, t0 , y0 ) = y0 + u(s, y(s, t0 , y0 ))ds + v(s, y(s, t0 , y0 ))ds t0 t0 Ii (y(ti − 0, t0 , y0 )). +
1 2
3
4
to
Thus, (3.1) is equivalent to the above equation. By using conditions (a), (b) and (c), it is easy to have: t t ∥y(t, t0 , y0 )∥ ≤ ∥y0 ∥ + f (s)∥y(s, t0 , y0 )∥ds + h(s)∥y(s, t0 , y0 )∥ t t0 0 + ai ∥y(ti − 0, t0 , y0 )∥m .
5 6
7
8
to
Set x(t) = ∥y(t, t0 , y0 )∥, we obtain the particular case of integro-sum inequality (2.1): t x(t) ≤ ∥y0 ∥ + f (s)x q (s) + h(s)x r (s) ds + ai x m (ti − 0)
9
10
to
t0
if in (2.1) p = 1, ρ(t) = ∥y0 ∥, π(t) = 1, τ (s) = s. Therefore, we get the estimated solutions for impulsive system:
11 12
t Q(s)ds ∥y0 ∥A(t)e t0 , t ∈ I0 = [t0 , t1 ] t t Q(s)ds ∥y0 ∥[M2 (t) + A , t ∈ I1 = [t1 , t2 ] 1 L 2 (t)]e 1 s Q(τ )dτ ti ∥y(t, t0 , y0 )∥ ≤ ∥y0 ∥ Mi (t) + Q(s) Mi (s) + Ai−1 L i (s) e ti−1 ds ti−1 t t m Q(s)ds Q(s)ds +Ai (Mi (t) + Ai−1 L i (t))e ti−1 e ti , t ∈ Ii = [ti , ti−1 ]
(3.2)
where
13
14
A(t) = 1 +
t
(1 − q)K q f (s) + (1 − r )K r h(s) ds,
f (t) + r K r −1 h(t), t1 s Q(τ )dτ ds, Q(s)A(s)e t0 M2 (t) = A(t) + Q(t) = q K
Ai = ai ∥y(t, t0 , y0 )∥,
t0 q−1
t Q(s)ds m , L 2 (t) = A(t)e t0
t0
Mi (t) = Mi−1 (t) +
ti
s Q(τ )dτ Q(s) Mi−1 (s) + Ai−2 L i−1 (s) e ti−1 ds
(3.3)
15
(i > 2),
ti−1 t Q(s)ds m L i (t) = (Mi−1 (t) + Ai−2 L i−1 (t))e ti−2
(i > 2).
References [1] R.P. Agarwal, Difference Equation and Inequalities: Theory, Methods and Applications, New York, 1992. [2] R. Bellan, K.L. Cooke, Differential-Difference Equations, Academic Press, New York, 1963.
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[3] R. Bellman, The stability of solutions of liner differential equations, Duke Math. J. 10 (1943) 643–647. [4] I. Bihari, A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations, Acta Math. Acad. Sci. Hungar. 7 (1965) 81–94. [5] A. Gallo, A.M. Piccirillo, About new analogies of Gronwall–Bellman–Bihari type inequalities for discontinuous functions and estimated solutions for impulsive differential systems, Nonlinear Anal. 67 (2007) 1550–1559. [6] T.H. Gronwall, Note on the derivatives with respect to a parameter of the solutions of a system of differential equations, Ann. of Math. 20 (1919) 292–296. [7] J. Gu, F. Meng, Some new nonlinear Volterra–Fredholm type dynamic integral inequalities on time scales, Appl. Math. Comput. 245 (2014) 235–242. [8] F. Jiang, F. Meng, Explicit bounds on some new nonlinear integral inequalities with delay, J. Comput. Appl. Math. 205 (2007) 479–486. [9] V. Lakshmikantham, D. Bainov, P.S. Simeonov, Theory of Impulsive Differential Equations, in: Ser. Modern Appl. Math., vol. 6, 1989. [10] V. Lakshmikantham, S. Leela, Differential and Integral Inequalities, Theory and Applications, Academic Press, New York, 1969. [11] W. Li, M. Han, F. Meng, Some new delay integral inequalities and their applications, J. Comput. Appl. Math. 180 (2005) 191–200. Q6 [12] O. Lipovan, A retarded integral inequality and its applications, J. Math. Anal. Appl. 285 (2003) 434–436. [13] Q. Ma, J. Peˇcari´c, Some new explicit bounds for weakly singular integral inequalities with applications to fractional differential and integral equations, J. Math. Anal. Appl. 341 (2008) 894–905. [14] Q. Ma, J. Wang, Xi. Ke, J. Peˇcari´c, On the boundedness of a class of nonlinear dynamic equations of second order, Appl. Math. Lett. 26 (11) (2013) 1099–1105. [15] Q. Ma, E.H. Yang, Some new Gronwall–Bellman–Bihari type integral inequalities with delay, Period. Math. Hungar. 44 (2002) 225–238. [16] B.G. Pachpatte, Inequalities for Differential and Integral Equations, Academic Press, New York, 1998. [17] B.G. Pachpatte, Explicit bounds on certain integral inequalities, J. Math. Anal. Appl. 267 (2002) 48–61. [18] Y. Sun, T. Hassan, Some nonlinear dynamic integral inequalities on time scales, Appl. Math. Comput. 220 (2013) 221–225. Q7 [19] B. Zheng, Some generalized Gronwall–Bellman type nonlinear delay integral inequalities for discontinuous functions, J. Inequal. Appl. 2013 (2013) 297.
ScienceDirect Indagationes Mathematicae xx (xxxx) xxx–xxx www.elsevier.com/locate/indag
On some new integral inequalities of Gronwall–Bellman–Bihari type with delay for discontinuous functions and their applications✩ Q1
Xiaohong Liu a,∗ , Lihong Zhang b , Praveen Agarwal c , Guotao Wang b a School of Mathematical Sciences, Xiamen University, Xiamen, Fujian 361005, People’s Republic of China b School of Mathematics and Computer Science, Shanxi Normal University, Linfen, Shanxi 041004,
People’s Republic of China c Department of Mathematics, Anand International College of Engineering, Rajasthan, Jaipur 303012, India
Received 31 October 2014; received in revised form 26 June 2015; accepted 5 July 2015 Communicated by H. Woerdeman
1 2
Abstract
3
In this article, some new explicit bounds on solutions to a class of new nonlinear integral inequalities of Gronwall–Bellman–Bihari type with delay for discontinuous functions are established. These inequalities generalize and improve some former famous results about inequalities, and which provide an excellent tool to discuss the qualitative and quantitative properties for solutions to some nonlinear differential and integral equations. To illustrate our results, we present an example to show estimated solutions for an impulsive differential system. c 2015 Published by Elsevier B.V. on behalf of Royal Dutch Mathematical Society (KWG). ⃝
10
Keywords: Integral inequalities; Discontinuous functions; Gronwall–Bellman–Bihari type inequalities; Impulsive differential systems
11
4 5 6 7 8 9
12
✩ This work was supported by the Natural Science Foundation for Young Scientists of Shanxi Province, China (2012021002-3). ∗ Corresponding author. Tel.: +86 15959262015. E-mail addresses: [email protected] (X. Liu), [email protected] (L. Zhang), [email protected] (P. Agarwal), [email protected] (G. Wang).
http://dx.doi.org/10.1016/j.indag.2015.07.001 c 2015 Published by Elsevier B.V. on behalf of Royal Dutch Mathematical Society (KWG). 0019-3577/⃝
2
1. Introduction
1
2 3 4 5 6 7
8
X. Liu et al. / Indagationes Mathematicae xx (xxxx) xxx–xxx
Q2
During the past years, a number of efforts have been devoted to the integral inequalities. The celebrated results given by Gronwall, Bellman, Bihari and their linear and nonlinear generalizations in the case of continuous and discontinuous functions provide a fundamental role in the study of many qualitative properties of differential and integral equations, which we can find in [3,6,4,11,12,15–17,2,1,10,9]. For some new development on this topic, see [5,8,13,18,14,19,7]. In 2007, Jiang and Meng [8] discussed the following delay integral inequalities: t p x (t) ≤ ρ(t) + π(t) f (s)x q (s) + h(s)x r (σ (s)) ds t0
9
with the initial condition:
10
s(t) = φ(t) t ∈ R+
11
φ(σ (t)) ≤ ρ(t) p
1
12
t ∈ R+ σ (t) ≤ 0
and x (t) ≤ ρ(t) + π(t) p
13
t
f (s)x q (s) + L s, x(σ (s))
t0 14 15 16 17 18 19
20
with the same initial condition. Under the assumption of functions x(t), ρ(t), π(t), f (t), h(t) ∈ C(R + , R + ), authors obtained explicit bounds on unknown function x(t) of the above inequalities. The purpose of this paper is to give explicit bounds to some new nonlinear integral inequalities of Gronwall–Bellman–Bihari type with delay for discontinuous functions. Precisely, we consider the following nonlinear integral inequalities: t x p (t) ≤ ρ(t) + π(t) f (s)x q (s) + h(s)x r (σ (s)) ds + ai x m (ti − 0) to
t0 21
22
and x p (t) ≤ ρ(t) + π(t)
t
f (s)x q (s) + L s, x(σ (s)) + ai x m (ti − 0).
t0
to
25
Our results develop and improve certain results which were proved by Jiang and Meng [8]. At the end of the article, an example of application is presented to show estimated solutions for an impulsive differential system.
26
2. Main results
27
Lemma 2.1 ([8]). Assume that a ≥ 0, p ≥ q ≥ 0 and p ̸= 0, then
23 24
q 28
29
ap ≤
q q−p p p − q qp K a+ K p p
for any K > 0.
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X. Liu et al. / Indagationes Mathematicae xx (xxxx) xxx–xxx
Theorem 2.1. If x(t)(t ≥ t0 ≥ 0) is a nonnegative piecewise continuous function with discontinuities of the 1st kind in the points ti (t0 < t1 < t2 < · · · , limi→∞ ti = ∞) and satisfies the following form of inequality: x p (t) ≤ ρ(t) + π(t)
t
ai x m (ti − 0) f (s)x q (s) + h(s)x r (σ (s)) ds +
(2.1)
1 2 3
4
to
t0
where ρ(t), π(t) ≥ 1 are both non-decreasing functions at t ≥ t0 , h(s), f (s) ∈ C(R+ , R+ ), σ (s) ≤ s, lim|s|→∞ σ (s) = ∞, ai ≥ 0, m > 0, p ≥ q ≥ 0, p ≥ r ≥ 0. Then, for t ≥ t0 , the following estimates hold: t 1 t0 Q(s)ds p ρ(t)π(t)A(t)e , t ∈ I0 = [t0 , t1 ] t 1 Q(s)ds p , t ∈ I1 = [t1 , t2 ] ρ(t)π(t)[M2 (t) + A1 L 2 (t)]e t1 ti s Q(τ )dτ x(t) ≤ ρ(t)π(t) M (t) + Q(s) M (s) + A L (s) e ti−1 ds i i i−1 i t i−1 1 t m t p ti−1 Q(s)ds p ti Q(s)ds e , +Ai (Mi (t) + Ai−1 L i (t))e t ∈ Ii = [ti , ti−1 ]
(2.2)
where
5 6 7
8
9
A(t) = 1 +
t
f (s)
t0 m p
p − r rp p − q qp K + h(s) K ds, p p
m
Ai = ai ρ (ti − 0)π p (ti − 0), q q− p r r−p Q(t) = K p f (t)π(t) + K p h(t)π(t), p p t1 s Q(τ )dτ M2 (t) = A(t) + Q(s)A(s)e t0 ds,
t m Q(s)ds p , L 2 (t) = A(t)e t0
(2.3)
10
t0
Mi (t) = Mi−1 (t) +
ti
s Q(τ )dτ Q(s) Mi−1 (s) + Ai−2 L i−1 (s) e ti−1 ds
(i > 2),
ti−1 t m Q(s)ds p L i (t) = (Mi−1 (t) + Ai−2 L i−1 (t))e ti−2
(i > 2).
Proof. Being ρ(t) ≥ 1, we have t x p (t) x q (s) x r (σ (s)) ≤ 1 + π(t) f (s) + h(s) ds + ai x m (ti − 0). ρ(t) ρ(t) ρ(t) t0 to
11
12
Considering ρ(t) a non-decreasing function, for s ≤ t, we get ρ(s) ≤ ρ(t), then: x p (t) ≤ 1 + π(t) ρ(t)
t t0
f (s)
x q (s) x r (σ (s)) + h(s) ds + ai x m (ti − 0). ρ(s) ρ(s) to
13
(2.4)
14
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By setting u(t) =
x p (t) ρ(t)
u(t) ≤ π(t) 1 +
2
, we can see u(t0 ) = 1 and there is
t t0
3
ω(t) = 1 +
t
f (s)
t0
8
9
It is obvious that x p (t) ≤ π(t)ω(t). ρ(t) This means that x p (t) ≤ ρ(t)π(t)ω(t). The above inequality implies the estimation for x(t) such as 1 p x(t) ≤ ρ(t)π(t)ω(t)
12
13
14
1 p x(ti − 0) ≤ ρ(ti − 0)π(ti − 0)ω(ti − 0) .
16
17
By Lemma 2.1, for any K > 0, we obtain
18
19
20
21
22
q p−q q q q− p p x q (t) ≤ ρ(t)π(t)ω(t) ≤ K p ρ(t)π(t)ω(t) + Kp p p r p−r r r r−p p x r (t) ≤ ρ(t)π(t)ω(t) ≤ K p ρ(t)π(t)ω(t) + K p. p p It follows from (2.7) and (2.8) that t q q− p p−q q Kp ω(t) ≤ 1 + f (s) K p π(s)ω(s) + p p t0 r r−p p−r r + h(s) K p π(s)ω(s) + K p ds p p m m p p + ai ρ(ti − 0)π(ti − 0) ω(ti − 0) to
23
(2.7)
and
Noting σ (s) ≤ s, then 1 1 p p x σ (s) ≤ ρ σ (s) π σ (s) ω σ (s) ≤ ρ(s)π(s)ω(s) .
15
(2.6)
u(t) ≤ π(t)ω(t).
10
11
x q (s) x r (σ (s)) ai x m (ti − 0). + h(s) ds + ρ(s) ρ(s) to
Using (2.6) in (2.5), we obtain
6
7
(2.5)
Denote
4
5
x q (s) x r (σ (s)) m ai x (ti − 0) . f (s) ds + + h(s) ρ(s) ρ(s) to
= 1+
t0
− q qp p − r rp K f (s) + K h(s) ds p p
(2.8)
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X. Liu et al. / Indagationes Mathematicae xx (xxxx) xxx–xxx
t r r−p q q−p p f (s)π(s) + K p h(s)π(s) ω(s)ds K + p t0 p m m p p ai ρ(ti − 0)π(ti − 0) + ω(ti − 0) .
1
(2.9)
2
to
Set
3
t p − q qp p − r rp A(t) = 1 + K f (s) + K h(s) ds p p t0 q q−p p r r −p p Q(t) = K f (t)π(t) + K h(t)π(t) p p m p Ai = ai ρ(ti − 0)π(ti − 0) . Then (2.9) can be restated as t m Q(s)ω(s)ds + Ai [ω(ti − 0)] p . ω(t) ≤ A(t) +
(2.10)
4
5
(2.11)
6
to
t0
Let us consider Ii = [ti , ti+1 ](i = 0, 1 · · ·). For t ∈ I0 , t t Q(s)ds ω(t) ≤ A(t) + Q(s)ω(s)ds ≤ A(t)e t0 .
7 8
(2.12)
9
t0
It is obvious that t 1 Q(s)ds p . x(t) ≤ ρ(t)π(t)A(t)e t0
10
11
Now let us consider the next interval I1 . For t ∈ I1 , we have t m p ω(t) ≤ A(t) + Q(s)ω(s)ds + A1 ω(t1 − 0) t0 t1
s
≤ A(t) +
Q(s)A(s)e
t0
Q(τ )dτ
t
ds +
t0
12
13
t m Q(s)ds p Q(s)ω(s)ds + A1 A(t)e t0
14
t1 t1
≤ A(t) +
s
Q(s)A(s)e
t0
Q(τ )dτ
t m t Q(s)ds p Q(s)ds ds + A1 A(t)e t0 e t1 .
(2.13)
15
t0
By setting M2 (t) = A(t) +
16
t1
s
Q(s)A(s)e
t0
Q(τ )dτ
ds
17
t0 t m Q(s)ds p L 2 (t) = A(t)e t0 .
Therefor for the interval I1 , we see t 1 Q(s)ds p x(t) ≤ ρ(t)π(t)[M2 (t) + A1 L 2 (t)]e t1 .
18
Q3
19
20
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2
X. Liu et al. / Indagationes Mathematicae xx (xxxx) xxx–xxx
Consider the next interval I2 , we have t m p ω(t) ≤ A(t) + Q(s)ω(s)ds + A2 ω(t2 − 0) t0 t1
3
s
≤ A(t) +
Q(s)A(s)e
t0
Q(τ )dτ
ds +
t0
t
+
4
t2
Q(s)[M2 (s) + A1 L 2 (s)]e
s t1
Q(τ )dτ
ds
t1
t m Q(s)ds p Q(s)ω(s)ds + A2 [M2 (t) + A1 L 2 (t)]e t1
t2 5
≤ M2 (t) +
t2
Q(s)[M2 (s) + A1 L 2 (s)]e
s t1
Q(τ )dτ
ds
t1
t
+
6
t m Q(s)ds p . Q(s)ω(s)ds + A2 [M2 (t) + A1 L 2 (t)]e t1
(2.14)
t2 7
8
Thus, we have ω(t) ≤
M2 (t) +
t2
s Q(τ )dτ Q(s) M2 (s) + A1 L 2 (s) e t1 ds
t1
9
10 11 12
13 14
15
+ A2
m t p t1 Q(s)ds M2 (t) + A1 L 2 (t) e
t
e
t2
17
.
(2.15)
1 p Taking into account x(t) is such that x(t) ≤ ρ(t)π(t)ω(t) , ∀t ≥ t0 , from inequalities (2.15), we can see for t ∈ I2 estimates (2.2) are valid for x(t). Consequently, by using a similar procedure for t ∈ Ii , we complete the proof of Theorem 2.1. Theorem 2.2. Assume that x(t), ρ(t), π(t), σ (t) are defined in Theorem 2.1 and x(t) satisfies the following form of integral inequality t f (s)x q (s) + L s, x(σ (s)) + ai x m (ti − 0) (2.16) x p (t) ≤ ρ(t) + π(t) t0
16
Q(s)ds
to
2 , R ) satisfying where p ≥ q ≥ 1, m ≥ 0 and L , M ∈ (R+ +
0 ≤ L(t, x) − L(t, y) ≤ M(t, y)(x − y)
18
for x ≥ y ≥ 0, then the inequality (3.1) with the initial condition in Theorem 2.1 implies
19
t 1 ˜ t0 Q(s)ds p ˜ ρ(t)π(t) A(t)e , t ∈ I0 = [t0 , t1 ] t ˜ 1 Q(s)ds p ρ(t)π(t) M˜ 2 (t) + A˜1 L˜2 (t) e t1 , t ∈ I1 = [t1 , t2 ] ti s ˜ ˜ i (t) + ˜ ˜ i (s) + Ai−1 ˜ L˜i (s) e ti−1 Q(τ )dτ ds x(t) ≤ ρ(t)π(t) M Q(s) M ti−1 1 t m t ˜ p ˜ ti−1 Q(s)ds p t1 Q(s)ds ˜ ˜ ˜ ˜ + A ( M (t) + A L (t))e e , i i i−1 i t ∈ Ii = [ti , ti−1 ]
(2.17)
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X. Liu et al. / Indagationes Mathematicae xx (xxxx) xxx–xxx
where
1
t p − q qp p − r rp ˜ =1+ K f (s) + K h(s) ds, A(t) p p t0 m p A˜i = ai ρ(ti − 0)π(ti − 0) q q− p r r−p ˜ Q(t) = K p f (t)π(t) + K p h(t)π(t) p p t1 s ˜ t0 Q(τ )dτ ˜ + ˜ A(s)e ˜ M˜ 2 (t) = A(t) Q(s) ds
(2.18)
t m ˜ t0 Q(s)ds p ˜ L˜2 (t) = A(t)e
2
t0
˜ (t) + M˜ i (t) = Mi−1
ti
s ˜ ˜ ˜ (s) + Ai−2 ˜ L i−1 ˜ (s) e ti−1 Q(τ )dτ ds Q(s) Mi−1
ti−1
˜ (t) + Ai−2 ˜ L i−1 ˜ (t))e L˜i (t) = ( Mi−1
t ti−2
˜ Q(s)ds
m p
.
Proof. Being ρ(t) ≥ 1 a non-decreasing function, the following inequality holds: t x q (s) x p (t) ≤ 1 + π(t) f (s) + L s, x(σ (s)) ds + ai x m (ti − 0) ρ(t) ρ(s) t0 to
3
4
(2.19)
Denote
5
Q4
ω(t) ˜ =1+
t
f (s)
t0
x q (s) ρ(s)
+ L(s, x(σ (s))) ds + ai x m (ti − 0)
6
7
to
it is obvious that 1 p x(t) ≤ ρ(t)π(t)ω(t) ˜
8
(2.20)
1 p x(ti − 0) ≤ ρ(ti − 0)π(ti − 0)ω(t ˜ i − 0) .
9
By Lemma 2.1, we obtain q p−q q p x q (t) ≤ ρ(t)π(t)ω(t) ˜ ρ(t)π(t)ω(t) ˜ + ≤ p p
10
11
and
12
1 p−1 1 p x(t) ≤ ρ(t)π(t)ω(t) ˜ ≤ ρ(t)π(t)ω(t) ˜ + . p p
13
Then there is
Q5
q p−q ρ(s)π(s)ω(s) ˜ π(s)ω(s) ˜ + ds + L s, + p p p t0 t0 m m p p + ai ρ(ti − 0)π(ti − 0) ω(t ˜ i − 0)
ω(t) ˜ ≤ 1+
t
f (s)
to
t
p − 1 p
ds
14
15
16
8
X. Liu et al. / Indagationes Mathematicae xx (xxxx) xxx–xxx
t q ρ(s)π(s)ω(s) ˜ p−q p − 1 ˜ + ds + L s, + ds f (s) π(s)ω(s) p p p p t0 t0 t t p − 1 p − 1 − ds + ds L s, L s, p p t0 t0 m m p + ai ρ(ti − 0)π(ti − 0) [ω(t ˜ i − 0)] p
t
= 1+
1
2
3
to
t q p−q p − 1 ρ(s)π(s)ω(s) ˜ ≤ 1+ f (s) π(s)ω(s) ˜ + ds ds + M s, p p p p t0 t0 m m p p + ai ρ(ti − 0)π(ti − 0) ω(t ˜ i − 0)
4
5
to
t p − 1 ρ(s)π(s) p−q q = 1+ f (s) ds + f (s)π(s) + M s, ω(s)ds ˜ p p p t0 t0 p m m p + ai ρ(ti − 0)π(ti − 0) [ω(t ˜ i − 0)] p . (2.21)
6
7
to
8
9
10
11
12
13
Setting in the last inequality t p−q ˜ A(t) = 1 + f (s)ds p t0 m p A˜i = ai ρ(ti − 0)π(ti − 0) p − 1 ρ(t)π(t) q ˜ Q(t) = f (t)π(t) + M t, p p p we obtain ˜ + ω(t) ˜ ≤ A(t)
t t0
˜ ω(s)ds Q(s) ˜ +
to
m p A˜i ω(t ˜ i − 0) .
14
Thus, we get the similar form of inequality (2.11) in Theorem 2.1, we complete the proof.
15
3. Example
16
17
18 19
20 21 22
As an application, when p = 1, let us consider the following impulsive system: dy = u(y, t) + v(y, t) t ̸= ti dt 1y| = Ii (y) +t=ti y(t0 ) = y0 when t ≥ t0 ≥ 0, limi→∞ ti = ∞, ti−1 < ti , i = 1, 2, . . . . Suppose that u(y, t), v(y, t) and Ii (y) satisfy the nonlinearity conditions: (a) ∥u(y, t)∥ ≤ f (t)∥y∥q f : R+ → R+ ; (b) ∥v(y, t)∥ ≤ h(t)∥y∥r h : R+ → R+ ; (c) ∥Ii (y)∥ ≤ ai ∥y∥m , ai = const. > 0, m > 0.
(2.22)
(3.1)
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X. Liu et al. / Indagationes Mathematicae xx (xxxx) xxx–xxx
Denote y(t) = x(t, t0 , y0 ) the solution for (3.1), so that y(t) = y(t, t0 , y0 )(y(t0 ) = y0 ), it is obvious that its integro-sum representation is the following: t t y(t, t0 , y0 ) = y0 + u(s, y(s, t0 , y0 ))ds + v(s, y(s, t0 , y0 ))ds t0 t0 Ii (y(ti − 0, t0 , y0 )). +
1 2
3
4
to
Thus, (3.1) is equivalent to the above equation. By using conditions (a), (b) and (c), it is easy to have: t t ∥y(t, t0 , y0 )∥ ≤ ∥y0 ∥ + f (s)∥y(s, t0 , y0 )∥ds + h(s)∥y(s, t0 , y0 )∥ t t0 0 + ai ∥y(ti − 0, t0 , y0 )∥m .
5 6
7
8
to
Set x(t) = ∥y(t, t0 , y0 )∥, we obtain the particular case of integro-sum inequality (2.1): t x(t) ≤ ∥y0 ∥ + f (s)x q (s) + h(s)x r (s) ds + ai x m (ti − 0)
9
10
to
t0
if in (2.1) p = 1, ρ(t) = ∥y0 ∥, π(t) = 1, τ (s) = s. Therefore, we get the estimated solutions for impulsive system:
11 12
t Q(s)ds ∥y0 ∥A(t)e t0 , t ∈ I0 = [t0 , t1 ] t t Q(s)ds ∥y0 ∥[M2 (t) + A , t ∈ I1 = [t1 , t2 ] 1 L 2 (t)]e 1 s Q(τ )dτ ti ∥y(t, t0 , y0 )∥ ≤ ∥y0 ∥ Mi (t) + Q(s) Mi (s) + Ai−1 L i (s) e ti−1 ds ti−1 t t m Q(s)ds Q(s)ds +Ai (Mi (t) + Ai−1 L i (t))e ti−1 e ti , t ∈ Ii = [ti , ti−1 ]
(3.2)
where
13
14
A(t) = 1 +
t
(1 − q)K q f (s) + (1 − r )K r h(s) ds,
f (t) + r K r −1 h(t), t1 s Q(τ )dτ ds, Q(s)A(s)e t0 M2 (t) = A(t) + Q(t) = q K
Ai = ai ∥y(t, t0 , y0 )∥,
t0 q−1
t Q(s)ds m , L 2 (t) = A(t)e t0
t0
Mi (t) = Mi−1 (t) +
ti
s Q(τ )dτ Q(s) Mi−1 (s) + Ai−2 L i−1 (s) e ti−1 ds
(3.3)
15
(i > 2),
ti−1 t Q(s)ds m L i (t) = (Mi−1 (t) + Ai−2 L i−1 (t))e ti−2
(i > 2).
References [1] R.P. Agarwal, Difference Equation and Inequalities: Theory, Methods and Applications, New York, 1992. [2] R. Bellan, K.L. Cooke, Differential-Difference Equations, Academic Press, New York, 1963.
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17 18
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3
4
5
6
7
8
9
10 11
12
13
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