GENERALIZATION OF AN IMPULSIVE
NONLINEAR SINGULAR GRONWALL-BIHARI
INEQUALITY WITH DELAY
Gronwall-Bihari Inequality
Shengfu Deng and Carl Prather
vol. 9, iss. 2, art. 34, 2008
SHENGFU DENG AND CARL PRATHER
Department of Mathematics
Virginia Polytechnic Institute and State University
Blacksburg, VA 24061, USA
EMail: sfdeng@vt.edu
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prather@math.vt.edu
Received:
25 August, 2007
Accepted:
23 May, 2008
Communicated by:
S.S. Dragomir
2000 AMS Sub. Class.:
26D15, 26D20.
Key words:
Gronwall-Bihari inequality, Nonlinear, Impulsive.
Abstract:
This paper generalizes a Tatar’s result of an impulsive nonlinear singular
Gronwall-Bihari inequality with delay [J. Inequal. Appl., 2006(2006), 1-12] to a
new type of inequalities which includes n distinct nonlinear terms.
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Contents
1
Introduction
3
2
Main Results
5
Gronwall-Bihari Inequality
Shengfu Deng and Carl Prather
vol. 9, iss. 2, art. 34, 2008
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1.
Introduction
In order to investigate problems of the form
x0 = f (t, x),
∆x = Ik (x),
t 6= tk ,
t = tk ,
Samoilenko and Perestyuk [6] first used the following impulsive integral inequality
Z t
X
u(t) ≤ a +
b(s)u(s)ds +
ηk u(tk ), t ≥ 0.
c
Gronwall-Bihari Inequality
Shengfu Deng and Carl Prather
vol. 9, iss. 2, art. 34, 2008
0<tk <t
Then Bainov and Hristova [2] studied a similar inequality with constant delay. In
2004, Hristova [3] considered a more general inequality with nonlinear functions in
u. All of these papers treated the functions (kernels) involved in the integrals which
are regular. Recently, Tatar [7] investigated the following singular inequality
Z t
Z t
m
u(t) ≤ a(t) + b(t)
k1 (t, s)u (s)ds + c(t)
k2 (t, s)un (s − τ )ds
0
0
X
+d(t)
ηk u(tk ), t ≥ 0,
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0<tk <t
(1.1)
u(t) ≤ ϕ(t), t ∈ [−τ, 0], τ > 0
where the kernels ki (t, s) are defined by ki (t, s) = (t − s)βi −1 sγi Fi (s) for βi > 0 and
γi > −1, i = 1, 2, the points tk (called "instants of impulse effect") are in increasing
order and limk→∞ tk = +∞. This inequality was called the impulsive nonlinear
singular version of the Gronwall inequality with delay by Tatar [7]. In this paper, we
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will consider an inequality
n Z
X
u(t) ≤ a(t) +
+
j=n+1
(1.2)
+ d(t)
Z
(t − s)βi −1 sri fi (t, s)wi (u(s))ds
0
i=1
m+n
X
bi (t)
bj (t)
(t − s)βj −1 srj fj (t, s)wj (u(s − τ ))ds
0
X
Gronwall-Bihari Inequality
ηL u(tL ),
t ≥ 0,
0<tL <t
u(t) ≤ ϕ(t),
Shengfu Deng and Carl Prather
vol. 9, iss. 2, art. 34, 2008
t ∈ [−τ, 0], τ > 0,
where n, m are positive integers, βl > 0, rl > −1 for l = 1, . . . , n + m and ηL ≥ 0
and other assumptions are given in Section 2. This inequality is more general than
(1.1) since (1.2) has n nonlinear terms.
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2.
Main Results
Notation: Following [1] and [5], we say w1 ∝ w2 for w1 , w2 : A ⊂ R → R\{0}
2
if w
is nondecreasing on A. This concept helps us to compare the monotonicity of
w1
different functions. Now we make the following assumptions:
(H1) all wi (i = 1, . . . , n + m) are continuous and nondecreasing on [0, ∞) and
positive on (0, ∞), and w1 ∝ w2 ∝ · · · ∝ wn
Gronwall-Bihari Inequality
Shengfu Deng and Carl Prather
(H2) a(t) and d(t) are continuous and nonnegative on [0, ∞);
(H3) all bl : [0, ∞) → [0, ∞) are continuously differentiable and nondecreasing
such that 0 ≤ bl (t) ≤ t on [0, ∞), tL ≤ bl (t) ≤ tL + τ for t ∈ [tL , tL + τ ]
and tL + τ ≤ bl (t) ≤ tL+1 for t ∈ [tL + τ, tL+1 ], l = 1, . . . , n + m and
L = 0, 1, 2, . . . where t0 = 0. The points tL are called instants of impulse
effect which are in increasing order, and limL→∞ tL = ∞;
(H4) all fl (t, s) (l = 1, . . . , n + m) are continuous and nonnegative functions on
[0, ∞) × [0, ∞);
(H5) ϕ(t) is nonnegative and continuous;
(H6) u(t) is a piecewise continuous function from R → R+ = [0, ∞) with points of
discontinuity of the first kind at the points tL ∈ R. It is also left continuous at
the points tL . This space is denoted by P C(R, R+ ).
Without loss of generality, we will suppose that the tL satisfy τ < tL+1 −tL ≤ 2τ ,
L = 0, 1, 2, . . . . As in Remark 3.2 of [7], other cases can be reduced to this one.
vol. 9, iss. 2, art. 34, 2008
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Theorem 2.1. Let the above assumptions hold. Suppose that u satisfies (1.2) and is
in P C([−τ, ∞), [0, ∞)). Then if βα > − p1 + 1 and rα > − p1 , it holds that

uL,0 (t), t ∈ (tL , tL + τ ],






uL,1 (t), t ∈ (tL + τ, tL+1 ],



uk,0 (t), t ∈ (tk , tk + τ ] if tk + τ ≤ T,
u(t) ≤




uk,1 (t), t ∈ (tk + τ, T ] if tk + τ < T,





uk,0 (t), t ∈ (tk , T ] if tk + τ > T,
where tk ≤ T < tk+1 and
"
uL,l (t) = Wn−1
Z
+
bn (t)
(n + m + L + 1)q−1 cqn (t)f˜nq (t, s)ds
Wj−1 (γL,l,j−1 (t))
γL,l,1 (t) = (n + m + L + 1)
q−1
q
ã (t) +
n Z
X
i=1
+
j=n+1
0
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j 6= 1,
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"
bj (t)
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bj−1 (t)
q
q−1 q
˜
(n + m + L + 1) cj−1 (t)fj−1 (t, s)ds ,
Z
Contents
,
tL +lτ
n+m
X
vol. 9, iss. 2, art. 34, 2008
!# 1q
tL +lτ
γL,l,j (t) =
Shengfu Deng and Carl Prather
Title Page
Z
Wn (γL,l,n (t)) +
−1
Wj−1
Gronwall-Bihari Inequality
tL +lτ
Close
cqi (t)f˜iq (t, s)wiq (φ(s))ds
0
cqj (t)f˜jq (t, s)wjq (ψ(s − τ ))ds +
L
X
e=1
#
d˜q (t)ηeq uqe−1,1 (te ) ,

u (t), t ∈ (tL , tL + τ ], t ∈ (tk , tk + τ ] if tk + τ ≤ T,

 L,0
and t ∈ (tk , T ] if tk + τ > T,
φ(t) =


uL,1 (t), t ∈ (tL + τ, tL+1 ] and t ∈ (tk + τ, T ] if tk + τ < T,

ϕ(t),
t ∈ [−τ, 0],




 uL,0 (t), t ∈ (tL , tL + τ ], t ∈ (tk , tk + τ ] if tk + τ ≤ T,
ψ(t) =

and t ∈ (tk , T ] if tk + τ > T,




uL,1 (t), t ∈ (tL + τ, tL+1 ] and t ∈ (tk + τ, T ] if tk + τ < T,
˜ = max d(x),
f˜α (t, s) = max fα (x, s), d(t)
0≤x≤t
0≤x≤t
0≤x≤t
Z u
dv
Wi (u) =
u > 0, ui > 0,
1 ,
q
ui wi (v q )
1
1
Γ(1 + p(βα − 1))Γ(1 + prα ) p
+βα +rα −1
p
cα (t) = t
,
Γ(2 + p(βα + rα − 1))
ã(t) = max a(x),
for L = 0, 1, . . . , k − 1, α = 1, 2, . . . , n + m, l = 0, 1, and i, j = 1, . . . , n where
1
+ 1q = 1 for p > 0 and q > 1, and T is the largest number such that
p
Z
bj (t)
(2.1) Wj (γL,l,j (t)) +
tL +lτ
(n + m + L + 1)q−1 cj (t)f˜jq (t, s)ds ≤
Z
∞
uj
dz
,
wjq (z 1/q )
for all t ∈ (tL , tL + τ ], all t ∈ (tk , tk + τ ] if tk + τ ≤ T and all t ∈ (tL , T ] if
tk + τ > T as l = 0, or all t ∈ [tL + τ, tL+1 ] and all t ∈ [tk + τ, T ) if tk + τ < T as
l = 1 where j = 1, . . . , n, l = 0, 1 and L = 0, 1 . . . , k − 1.
Gronwall-Bihari Inequality
Shengfu Deng and Carl Prather
vol. 9, iss. 2, art. 34, 2008
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Before the proof, we introduce a lemma which will play a very important role.
Lemma 2.2 ([1]). Suppose that
1. all wi (i = 1, . . . , n) are continuous and nondecreasing on [0, ∞) and positive
on (0, ∞), and w1 ∝ w2 ∝ · · · ∝ wn .
2. a(t) is continuously differentiable in t and nonnegative on [t0 , t1 ),
3. all bl are continuously differentiable and nondecreasing such that bl (t) ≤ t for
t ∈ [t0 , t1 )
where t0 , t1 are constants and t0 < t1 . If u(t) is a continuous and nonnegative
function on [t0 , t1 ) satisfying
n Z bi (t)
X
u(t) ≤ a(t) +
fi (t, s)wi (u(s))ds, t0 ≤ t < t1 ,
i=1
then
Gronwall-Bihari Inequality
Shengfu Deng and Carl Prather
vol. 9, iss. 2, art. 34, 2008
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bi (t0 )
"
u(t) ≤ W̃n−1 W̃n (γn (t)) +
Z
bn (t)
#
f˜n (t, s)ds ,
t0 ≤ t ≤ T1 ,
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bn (t0 )
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where
"
γi (t) =
−1
W̃i−1
Z
bi−1 (t)
W̃i−1 (γi−1 (t)) +
#
f˜i−1 (t, s)ds ,
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i = 2, 3, . . . , n,
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bi−1 (t0 )
Z
t
γ1 (t) = a(t0 ) +
|a0 (s)|ds,
u
Z
W̃i (u) =
t0
ui
T1 < t1 and T1 is the largest number such that
Z bi (T1 )
Z
˜
W̃i (γi (T1 )) +
fi (T1 , s)ds ≤
∞
bi (t0 )
ui
dz
,
wi (z)
ui > 0,
dz
,
wi (z)
i = 1, . . . , n.
Close
Proof of Theorem 2.1. Since βα > − p1 + 1 and rα > − p1 for α = 1, . . . , n + m, by
Hölder’s inequality we obtain
! 1q
p1 Z bi (t)
n Z t
X
q
q
u(t) ≤ a(t) +
(t − s)p(βi −1) spri ds
fi (t, s)wi (u(s))ds
0
i=1
+
m+n
X
Z
p(βj −1) prj
(t − s)
s ds
X
Z
bj (t)
! 1q
fjq (t, s)wjq (u(s
− τ ))ds
Gronwall-Bihari Inequality
0
0
j=n+1
+
t
0
1
p
Shengfu Deng and Carl Prather
vol. 9, iss. 2, art. 34, 2008
d(t)ηL u(tL )
0<tL <t
≤ a(t) +
n
X
ci (t)
+
! 1q
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fiq (t, s)wiq (u(s))ds
Contents
0
i=1
m+n
X
bi (t)
Z
Z
bj (t)
cj (t)
! 1q
fjq (t, s)wjq (u(s − τ ))ds
+
0
j=n+1
X
d(t)ηL u(tL )
0<tL <t
where we use bα (t)≤t and the definition of cα(t). Now we use the following result [4]:
If A1 , . . . , An are nonnegative for n ∈ N, then for q > 1,
(A1 + · · · + An )q ≤ nq−1 (Aq1 + · · · + Aqn ).
Since tk ≤ t ≤ T < tk+1 , we have
"
Z bi (t)
n
X
q
q
q−1
q
u (t) ≤ (1 + n + m + k)
a (t) +
ci (t)
fiq (t, s)wiq (u(s))ds
i=1
+
m+n
X
j=n+1
cqj (t)
Z
0
bj (t)
0
fjq (t, s)wjq (u(s − τ ))ds +
k
X
L=1
#
dq (t)ηLq uq (tL ) .
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˜ ≥ d(t) and f˜α (t, s) ≥ fα (t, s) and they are continuous
We note that ã(t) ≥ a(t), d(t)
and nondecreasing in t. The above inequality becomes
"
Z tL+1
n
k−1
X
X
q
q
q−1
q
u (t) ≤ (1 + n + m + k)
ã (t) +
ci (t)
f˜iq (t, s)wiq (u(s))ds
i=1
+ cqi (t)
Z
bi (t)
tL
L=0
!
f˜iq (t, s)wiq (u(s))ds
Gronwall-Bihari Inequality
tk
+
(2.2)
+
m+n
X
k−1
X
j=n+1
L=0
cqj (t)
Z
bj (t)
Shengfu Deng and Carl Prather
cqj (t)
Z
tL+1
vol. 9, iss. 2, art. 34, 2008
f˜jq (t, s)wjq (u(s − τ ))ds
tL
!
f˜jq (t, s)wjq (u(s − τ ))ds
k
X
+
tk
Title Page
#
d˜q (t)ηLq uq (tL ) .
Contents
L=1
In the following, we apply mathematical induction with respect to k.
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(1) k = 0. We note that t0 = 0 and we have for any fixed t̃ ∈ [0, t1 ]
"
Z bi (t)
n
X
q
q
q−1
q
(2.3) u (t) ≤ (n + m + 1)
ã (t̃) +
c (t̃)
f˜q (t̃, s)wq (u(s))ds
J
I
i
+
i=1
m+n
X
j=n+1
i
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i
Go Back
0
cqj (t̃)
Z
bj (t)
#
f˜jq (t̃, s)wjq (u(s − τ ))ds
0
for t ∈ [0, t̃] since cα (t) are nondecreasing.
Now we consider t̃ ∈ [0, τ ] ⊂ [0, t1 ] and t ∈ [0, t̃]. Note that 0 ≤ bj (t) ≤ t so
−τ ≤ bj (t) − τ ≤ 0 for t ∈ [0, t̃]. Since u(t) ≤ ϕ(t) for t ∈ [−τ, 0], we have
uq (t) ≤ z0,0 (t),
t ∈ [0, t̃],
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where
"
(2.4) z0,0 (t) = (n + m + 1)
q−1
q
ã (t̃) +
n
X
cqi (t̃)
+
f˜iq (t̃, s)wiq (u(s))ds
0
i=1
m+n
X
bi (t)
Z
cqj (t̃)
Z
bj (t̃)
#
f˜jq (t̃, s)wjq (ϕ(s − τ ))ds .
0
j=n+1
Gronwall-Bihari Inequality
It implies that
Shengfu Deng and Carl Prather
1/q
u(t) ≤ z0,0 (t)
(2.5)
,
t ∈ [0, t̃].
n
X
Z
vol. 9, iss. 2, art. 34, 2008
Thus, (2.4) becomes
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"
(2.6) z0,0 (t) ≤ (n + m + 1)q−1 ãq (t̃) +
cqi (t̃)
+
cqj (t̃)
Z
bj (t̃)
#
f˜jq (t̃, s)wjq (ϕ(s
− τ ))ds .
0
j=n+1
By Lemma 2.2, (2.6) and (2.1), we have
"
Z
−1
z0,0 (t) ≤ Wn Wn (γ̃0,0,n (t)) +
1/q
f˜iq (t̃, s)wiq (z0,0 (s))ds
0
i=1
m+n
X
bi (t)
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#
bn (t)
(n + m + 1)
q−1
cn (t̃)f˜nq (t̃, s)ds
0
,
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−1
γ̃0,0,j (t) = Wj−1 Wj−1 (γ̃0,0,j−1 (t))
Z
+
#
bj−1 (t)
(n + m + 1)
0
q−1
q
cj−1 (t̃)f˜j−1
(t̃, s)ds
,
j 6= 1,
"
γ̃0,0,1 (t) = (n + m + 1)q−1 ãq (t̃) +
n+m
X
bj (t̃)
Z
j=n+1
#
cqj (t̃)f˜jq (t̃, s)wjq (ψ(s − τ ))ds
0
since ψ(t) = ϕ(t) for t ∈ [−τ, 0].
Since (2.5) is true for any t ∈ [0, t̃] and γ̃0,0,j (t̃) = γ0,0,j (t̃), we have
u(t̃) ≤ z0,0 (t̃)1/q ≤ u0,0 (t̃).
Gronwall-Bihari Inequality
We know that t̃ ∈ [0, τ ] is arbitrary so we replace t̃ by t and get
u(t) ≤ u0,0 (t),
(2.7)
Shengfu Deng and Carl Prather
vol. 9, iss. 2, art. 34, 2008
for t ∈ [0, τ ].
This implies that the theorem is true for t ∈ [0, τ ] and k = 0.
For t ∈ [τ, t̃] and t̃ ∈ [τ, t1 ], use the assumption (H3) and then we know that
bα (τ ) = τ and τ ≤ bα (t) ≤ t1 for t ∈ [τ, t1 ] and α = 1, . . . , n + m. Thus,
0 ≤ bα (t) − τ ≤ t1 − τ ≤ τ
since τ < t1 − t0 = t1 ≤ 2τ . Using this fact, (2.3) and (2.7), we get
"
Z τ
n
X
q
q
q−1
q
u (t) ≤ (n + m + 1)
ã (t̃) +
c (t̃)
f˜q (t̃, s)wq (u(s))ds
i
i=1
+
+
n
X
cqi (t̃)
i=1
m+n
X
+
j=n+1
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i
0
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f˜iq (t̃, s)wiq (u(s))ds
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τ
Close
τ
cqj (t̃)
Z
cqj (t̃)
Z
f˜jq (t̃, s)wjq (ψ(s − τ ))ds
0
j=n+1
m+n
X
bi (t)
Z
i
Title Page
τ
bj (t̃)
#
f˜jq (t̃, s)wjq (u(s − τ ))ds
"
≤ (n + m + 1)
q
q−1
ã (t̃) +
n
X
cqi (t̃)
+
+
cqi (t̃)
i=1
m+n
X
bi (t)
Z
+
cqj (t̃)
f˜iq (t̃, s)wiq (u(s))ds
τ
Z
f˜jq (t̃, s)wjq (ψ(s − τ ))ds
cqj (t̃)
Z
Shengfu Deng and Carl Prather
bj (t̃)
#
τ
≤ (n + m + 1)q−1 ãq (t̃) +
n
X
cqi (t̃)
+
cqi (t̃)
j=n+1
bi (t)
Z
Z
Title Page
τ
f˜iq (t̃, s)wiq (φ(s))ds
f˜iq (t̃, s)wiq (u(s))ds
τ
cqj (t̃)
Z
bj (t̃)
#
f˜jq (t̃, s)wjq (ψ(s − τ ))ds
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0
:= z0,1 (t),
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where we use the definitions of φ and ψ. Thus,
(2.8)
Contents
0
i=1
i=1
m+n
X
vol. 9, iss. 2, art. 34, 2008
f˜jq (t̃, s)wjq (u0,0 (s − τ ))ds
"
+
Gronwall-Bihari Inequality
0
j=n+1
n
X
f˜iq (t̃, s)wiq (u0,0 (s))ds
τ
j=n+1
m+n
X
τ
0
i=1
n
X
Z
1/q
u(t) ≤ z0,1 (t),
t ∈ [τ, t̃].
Close
Therefore,
"
z0,1 ≤ (n + m + 1)
q−1
+
q
ã (t̃) +
n
X
cqi (t̃)
n
X
cqi (t̃)
i=1
bi (t)
Z
τ
Z
f˜iq (t̃, s)wiq (φ(s))ds
0
1/q
f˜iq (t̃, s)wiq (z0,1 (s))ds
τ
i=1
Gronwall-Bihari Inequality
m+n
X
+
cqj (t̃)
bj (t̃)
Z
#
f˜jq (t̃, s)wjq (ψ(s − τ ))ds .
0
j=n+1
Using Lemma 2.2, (2.1) and bα (τ ) = τ , we obtain for t ∈ [τ, t̃]
"
#
Z bn (t)
z0,1 (t) ≤ Wn−1 Wn (γ̃0,1,n (t)) +
(n + m + 1)q−1 cqn (t̃)f˜nq (t̃, s)ds ,
γ̃0,1,j (t) =
Contents
Wj−1 (γ̃0,1,j−1 (t))
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Z
bj−1 (t)
+
#
q
(n + m + 1)q−1 cqj−1 (t̃)f˜j−1
(t̃, s)ds , j 6= 1,
τ
q−1
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"
γ̃0,1,1 (t) = (n + m + 1)
vol. 9, iss. 2, art. 34, 2008
Title Page
τ
−1
Wj−1
Shengfu Deng and Carl Prather
q
ã (t̃) +
+
n Z
X
Close
τ
cqi (t̃)f˜iq (t̃, s)wiq (φ(s))ds
i=1 0
n+m
X Z bj (t̃)
j=n+1
0
#
cqj (t̃)f˜jq (t̃, s)wjq (ψ(s − τ ))ds .
Since (2.8) is true for any t ∈ [τ, t1 ] and γ̃0,1,1 (t̃) = γ0,1,1 (t̃), we have
1/q
u(t̃) ≤ z0,1 (t̃) ≤ u0,1 (t̃).
We know that t̃ ∈ [τ, t1 ] is arbitrary so we replace t̃ by t and get
u(t) ≤ u0,1 (t),
(2.9)
t ∈ [τ, t1 ].
This implies that the theorem is valid for t ∈ [τ, t1 ] and L = 0.
(2) L = 1. First we consider t ∈ (t1 , t̃], where t̃ ∈ (t1 , t1 + τ ] is arbitrary. Note that
τ < t2 − t1 ≤ 2τ . (H3) gives bα (t1 ) = t1 and t1 ≤ bα (t) ≤ t1 + τ for t ∈ (t1 , t1 + τ ]
so t1 − τ ≤ bα (t) − τ ≤ t1 for t ∈ (t1 , t1 + τ ]. By (2.7) and (2.9), (2.2) can be
written as
"
Z τ Z t1 n
X
q
q
q−1
q
u (t) ≤ (n + m + 2)
ã (t̃) +
ci (t̃)
+
f˜iq (t̃, s)wiq (u(s))ds
0
i=1
+
+
n
X
cqi (t̃)
i=1
m+n
X
bi (t)
Z
cqj (t̃)
+
cqj (t̃)
τ
t1 +
0
j=n+1
n
X
f˜iq (t̃, s)wiq (u(s))ds
Z
Z
bj (t̃)
τ
i=1
cqi (t̃)
bi (t)
t1
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n
X
i=1
+
Contents
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f˜jq (t̃, s)wjq (u(s − τ ))ds + d˜q (t̃)η1q uq (t1 )
≤ (n + m + 2)q−1 ãq (t̃) +
Z
Title Page
#
"
n
X
vol. 9, iss. 2, art. 34, 2008
Page 15 of 20
f˜jq (t̃, s)wjq (u(s − τ ))ds
t1
j=1
Shengfu Deng and Carl Prather
τ
t1
Z
Gronwall-Bihari Inequality
cqi (t̃)
Z
0
f˜iq (t̃, s)wiq (u(s))ds
t1
f˜iq (t̃, s)wiq (φ(s))ds
+
m+n
X
cqj (t̃)
Z
bj (t̃)
#
f˜jq (t̃, s)wjq (ψ(s − τ ))ds + d˜q (t̃)η1q uq0,1 (t1 )
0
j=n+1
:=z1,0 (t),
where we use the definitions of φ and ψ so
1/q
u(t) ≤ z1,0 (t),
(2.10)
t ∈ (t1 , t̃].
Gronwall-Bihari Inequality
Thus,
Shengfu Deng and Carl Prather
"
z1,0 (t) ≤ (n + m + 2)
q
q−1
ã (t̃) +
n
X
cqi (t̃)
Z
i=1
+
n
X
cqi (t̃)
bi (t)
vol. 9, iss. 2, art. 34, 2008
f˜iq (t̃, s)wiq (φ(s))ds
0
Title Page
1/q
f˜iq (t̃, s)wiq (z1,0 (s))ds
Contents
t1
i=1
+
Z
t1
m+n
X
cqj (t̃)
Z
bj (t̃)
#
f˜jq (t̃, s)wjq (ψ(s − τ ))ds + d˜q (t̃)η1q uq0,1 (t1 ) .
0
j=n+1
By Lemma 2.2, (2.1) and bα (t1 ) = t1 , we obtain for t ∈ (t1 , t̃]
"
#
Z bn (t)
z1,0 (t) ≤ Wn−1 Wn (γ̃1,0,n (t)) +
(n + m + 2)q−1 cqn (t̃)f˜nq (t̃, s)ds ,
t1
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−1
γ̃1,0,j (t) = Wj−1 Wj−1 (γ̃1,0,j−1 (t))
Z
#
bj−1 (t)
+
(n + m +
t1
q
2)q−1 cqj−1 (t̃)f˜j−1
(t̃, s)ds
, j 6= 1,
"
γ̃1,0,1 (t) = (n + m + 2)
q
q−1
ã (t̃) +
n Z
X
i=1
n+m
X Z
+
bj (t̃)
t1
cqi (t̃)f˜iq (t̃, s)wiq (φ(s))ds
0
#
cqj (t̃)f˜jq (t̃, s)wjq (ψ(s − τ ))ds + d˜q (t̃)η1q uq0,1 (t1 ) .
0
j=n+1
Since (2.10) is true for any t ∈ (t1 , t̃] and γ̃1,0,1 (t̃) = γ1,0,1 (t̃), we have
Gronwall-Bihari Inequality
Shengfu Deng and Carl Prather
1/q
u(t̃) ≤ z1,0 (t̃) ≤ u1,0 (t̃).
vol. 9, iss. 2, art. 34, 2008
We know that t̃ ∈ (t1 , t1 + τ ] is arbitrary so we replace t̃ by t and get
u(t) ≤ u1,0 (t),
(2.11)
t ∈ (t1 , t1 + τ ].
Title Page
Contents
This implies that the theorem is valid for t ∈ (t1 , t1 + τ ] and L = 1.
We now consider t ∈ [t1 + τ, t̃], where t̃ ∈ [t1 + τ, t2 ] is arbitrary. Again, by
(H3) we have t1 + τ ≤ bα (t) ≤ t2 for t ∈ [t1 + τ, t2 ] and bα (t1 + τ ) = t1 + τ so
t1 ≤ bα (t) − τ ≤ t2 − τ ≤ t1 + τ since τ < t2 − t1 ≤ 2τ . Obviously, by (2.7), (2.9)
and (2.11), (2.2) becomes
"
Z t1 +τ
n
X
q
q
q−1
q
u (t) ≤ (n + m + 2)
ã (t̃) +
ci (t̃)
f˜iq (t̃, s)wiq (u(s))ds
i=1
+
+
n
X
cqi (t̃)
i=1
m+n
X
j=n+1
Z
bi (t)
0
f˜iq (t̃, s)wiq (u(s))ds
t1 +τ
cqj (t̃)
Z
0
t1 +τ
f˜jq (t̃, s)wjq (u(s − τ ))ds
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+
m+n
X
cqj (t̃)
bj (t̃)
Z
#
f˜jq (t̃, s)wjq (u(s − τ ))ds + d˜q (t̃)η1q uq1,0 (t1 )
t1 +τ
j=n+1
"
≤ (n + m + 2)
q−1
q
ã (t̃) +
n
X
cqi (t̃)
+
+
n
X
cqi (t̃)
i=1
m+n
X
bi (t)
t1 +τ
f˜iq (t̃, s)wiq (φ(s))ds
0
i=1
Z
Z
f˜iq (t̃, s)wiq (u(s))ds
Gronwall-Bihari Inequality
t1 +τ
cqj (t̃)
Z
Shengfu Deng and Carl Prather
bj (t̃)
#
vol. 9, iss. 2, art. 34, 2008
f˜jq (t̃, s)wjq (ψ(s − τ ))ds + d˜q (t̃)η1q uq0,1 (t1 )
0
j=n+1
Title Page
:= z1,1 (t),
Contents
that is,
1/q
u(t) ≤ z1,1 (t),
(2.12)
t ∈ [t1 + τ, t̃].
Thus,
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"
z1,1 (t) ≤ (n + m + 2)q−1 ãq (t̃) +
n
X
i=1
+ cqi (t̃)
Z
bi (t)
cqi (t̃)
Z
t1 +τ
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f˜iq (t̃, s)wiq (φ(s))ds
0
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1/q
f˜iq (t̃, s)wiq (z1,1 (s))ds
Close
t1 +τ
+
m+n
X
j=n+1
cqj (t̃)
Z
0
bj (t̃)
#
f˜jq (t̃, s)wjq (ψ(s
− τ ))ds +
d˜q (t̃)η1q uq0,1 (t1 )
.
Using Lemma 2.2, (2.1) and bα (t1 + τ ) = t1 + τ , we obtain for t ∈ (t1 , t̃]
"
#
Z bn (t)
z1,1 (t) ≤ Wn−1 Wn (γ̃1,1,n (t)) +
(n + m + 2)q−1 cqn (t̃)f˜nq (t̃, s)ds ,
t1 +τ
h
−1
Wj−1 (γ̃1,1,j−1 (t))
γ̃1,1,j (t) = Wj−1
#
Z bj−1 (t)
q
+
(n + m + 2)q−1 cqj−1 (t̃)f˜j−1
(t̃, s)ds , j 6= 0,
"
γ̃1,1,1 (t) = (n + m + 2)
q
ã (t̃) +
n Z
X
i=1
+
n+m
X Z
j=n+1
bj (t̃)
t1 +τ
cqi (t̃)f˜iq (t̃, s)wiq (φ(s))ds
Title Page
0
Contents
#
cqj (t̃)f˜jq (t̃, s)wjq (ψ(s − τ ))ds + d˜q (t̃)η1q uq0,1 (t1 ) .
0
Since (2.12) is true for any t ∈ (t1 , t̃] and γ̃1,1,1 (t̃) = γ1,1,1 (t̃), we have
1/q
u(t̃) ≤ z1,1 (t̃) ≤ u1,1 (t̃).
We know that t̃ ∈ [t1 + τ, t2 ] is arbitrary so we replace t̃ by t and get
u(t) ≤ u1,1 (t),
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t ∈ [t1 + τ, t2 ].
This implies that the theorem is valid for t ∈ [t1 + τ, t2 ] and L = 1.
(3) Finally, suppose that the theorem is valid for k, then for k + 1 we redefine φ and
ψ by replacing k with k + 1. In a similar manner as in steps (1) and (2), we can see
that the theorem holds for t ∈ (tk+1 , T ] ⊂ (tk+1 , tk+2 ].
The proof is now completed.
Shengfu Deng and Carl Prather
vol. 9, iss. 2, art. 34, 2008
t1 +τ
q−1
Gronwall-Bihari Inequality
Close
References
[1] R.P. AGARWAL, S. DENG AND W. ZHANG, Generalization of a retarded
Gronwall-like inequality and its applications, Appl. Math. Comput., 165 (2005),
599–612.
[2] D.D. BAINOV AND S.G. HRISTOVA, Impulsive integral inequalities with a
deviation of the argument, Math. Nachr., 171 (1995), 19–27.
[3] S.G. HRISTOVA, Nonlinear delay integral inequalities for piecewise continuous
functions and applications, J. Inequal. Pure Appl. Math., 5(4) (2004), Art. 88.
[ONLINE: http://jipam.vu.edu.au/article.php?sid=441].
[4] M. MEDVED, A new approach to an analysis of Henry type integral inequalities
and their Bihari type versions, J. Math. Anal. Appl., 214 (1997), 349–366.
[5] M. PINTO, Integral inequalities of Bihari-type and applications, Funkcial. Ekvac., 33 (1990), 387–403.
[6] A.M. SAMOILENKO AND N.A. PERESTJUK, Stability of the solutions of
differential equations with impulsive action, Differential Equations, 13 (1977),
1981–1992 (Russian).
[7] N.E. TATAR, An impulsive nonlinear singular version of the Gronwall-Bihari
inequality, J. Inequal. Appl., 2006 (2006), 1–12.
Gronwall-Bihari Inequality
Shengfu Deng and Carl Prather
vol. 9, iss. 2, art. 34, 2008
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Contents
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