Journal of Inequalities in Pure and
Applied Mathematics
NONLINEAR DELAY INTEGRAL INEQUALITIES FOR PIECEWISE
CONTINUOUS FUNCTIONS AND APPLICATIONS
volume 5, issue 4, article 88,
2004.
SNEZHANA HRISTOVA
Department of Mathematics
Denison University
Granville, Ohio
OH 43023,USA
EMail: snehri13@yahoo.com
Received 03 March, 2004;
accepted 08 August, 2004.
Communicated by: S.S. Dragomir
Abstract
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c 2000 Victoria University
ISSN (electronic): 1443-5756
046-04
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Abstract
Nonlinear integral inequalities of Gronwall-Bihari type for piecewise continuous
functions are solved. Inequalities for functions with delay as well as functions
without delays are considered. Some of the obtained results are applied in the
deriving of estimates for the solutions of impulsive integral, impulsive integrodifferential and impulsive differential-difference equations.
2000 Mathematics Subject Classification: 26D10, 34A37, 35K12
Key words: Integral inequalities, Piecewise continuous functions, Impulsive differential equations.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Preliminary Notes and Definitions . . . . . . . . . . . . . . . . . . . . . . 4
3
Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
4
Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
References
Nonlinear Delay Integral
Inequalities For Piecewise
Continuous Functions And
Applications
Snezhana Hristova
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1.
Introduction
Integral inequalities are a powerful mathematical apparatus, by the aid of which,
various properties of the solutions of differential and integral equations can be
studied, such as uniqueness of the solutions, boundedness, stability, etc. This
leads to the necessity of solving various types of linear and nonlinear inequalities, which generalize the classical inequalities of Gronwall and Bihari. In
recent years, many authors, such as S.S. Dragomir ([2] – [4]) and B.G. Pachpatte ([5] – [10]) have discovered and applied several new integral inequalities
for continuous functions.
The development of the qualitative theory of impulsive differential equations, whose solutions are piecewise continuous functions, is connected with
the preliminary deriving of results on integral inequalities for such types of
functions (see [1] and references there).
In the present paper we prove some generalizations of the classical Bihari
inequality for piecewise continuous functions. Two main types of nonlinear inequalities are considered – inequalities with constant delay of the argument as
well as inequalities without delays. The obtained inequalities are used to investigate some properties of the solutions of impulsive integral equations, impulsive
integro-differential and impulsive differential-difference equations.
Nonlinear Delay Integral
Inequalities For Piecewise
Continuous Functions And
Applications
Snezhana Hristova
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2.
Preliminary Notes and Definitions
Let the points tk ∈ (0, ∞), k = 1, 2, . . . are fixed such that tk+1 > tk and
limk↓∞ tk = ∞.
We consider the set P C(X, Y ) of all functions u : X → Y , (X ⊂ R, Y ⊂
Rn ) which are piecewise continuous in X with points of discontinuity of the first
kind at the points tk ∈ X, i.e. there exist the limits limt↓tk u(t) = u(tk +) < ∞
and limt↑tk u(t) = u(tk −) = u(tk ) < ∞.
Definition 2.1. We will say that the function G(u) belongs to the class W1 if
1. G ∈ C([0, ∞), [0, ∞)).
2. G(u) is a nondecreasing function.
Definition 2.2. We will say that the function G(u) belongs to the class W2 (ϕ) if
1. G ∈ W1 .
2. There exists a function ϕ ∈ C([0, ∞), [0, ∞)) such that G(uv) ≤ ϕ(u)G(v)
for u, v ≥ 0.
We note that if the function G ∈ W1 and satisfies the inequality G(uv) ≤
G(u)G(v) for u, v ≥ 0 then G ∈ W2 (G).
P
Q
Further we will use the following notations ki=1 αk = 0 and ki=1 αk = 1
for k ≤ 0.
Nonlinear Delay Integral
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3.
Main Results
As a first result we will consider integral inequalities with delay for piecewise
continuous functions.
Theorem 3.1. Let the following conditions be fulfilled:
1. The functions f1 , f2 , f3 , p, g ∈ C([0, ∞), [0, ∞)).
2. The function ψ ∈ C([−h, 0], [0, ∞)).
3. The function Q ∈ W2 (ϕ) and Q(u) > 0 for u > 0.
4. The function G ∈ W1 .
Nonlinear Delay Integral
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Continuous Functions And
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5. The function u ∈ P C([−h, ∞), [0, ∞)) and it satisfies the following inequalities
(3.1) u(t) ≤ f1 (t)
Z t
Z t
+ f2 (t)G c +
p(s)Q(u(s))ds +
g(t)Q(u(s − h))ds
0
0
X
+ f3 (t)
βk u(tk ) for t ≥ 0,
0<tk <t
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(3.2)
u(t) ≤ ψ(t)
for t ∈ [−h, 0],
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where c ≥ 0, βk ≥ 0, (k = 1, 2, . . . ).
J. Ineq. Pure and Appl. Math. 5(4) Art. 88, 2004
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Then for t ∈ (tk , tk+1 ] ∩ [0, γ), k = 0, 1, 2, . . . we have the inequality
k
Y
(3.3) u(t) ≤ ρ(t) (1 + βi ρ(ti ))
i=1
(
1 + G H −1
×
H(A) +
k Z
X
i=1
Z
"
t
+
p(s)ϕ ρ(s)
#
i−1
Y
p(s)ϕ ρ(s) (1 + βj ρ(tj )) ds
"
ti
ti−1
j=1
#
k
Y
(1 + βj ρ(tj )) ds
tk
j=1
+ Λ(t)
k Z
X

ti

g(s)ϕ ρ(s − h))
ti−1
i=1
Z
t
(1 + βj ρ(tk )) ds


(1 + βj ρ(tk )) ds  ,


tk
Y
j:0<tj <s−h
where
(
(3.4)
Snezhana Hristova
j:0<tj <s−h

g(s)ϕ ρ(s − h))
+Λ(t)
Y
Nonlinear Delay Integral
Inequalities For Piecewise
Continuous Functions And
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0 for t ∈ [0, h],
Λ(t) =
1 for t > h,
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ρ(t) = max{fi (t) : i = 1, 2, 3},
B1 = max{g(t) : t ∈ [0, h]},
A = c + hB1 Q(B2 ),
B2 = max{ψ(t) : t ∈ [−h, 0]},
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Z
u
H(u) =
(3.5)
u0
(
γ = sup t ≥ 0 : H(A) +
ds
, A ≥ u0 ≥ 0,
Q(1 + G(s))
k Z
X
i=1
Z
t
+
"
"
#
i−1
Y
p(s)ϕ ρ(s) (1 + βj ρ(tj )) ds
ti
ti−1
j=1
Nonlinear Delay Integral
Inequalities For Piecewise
Continuous Functions And
Applications
#
k
Y
p(s)ϕ ρ(s) (1 + βj ρ(tj )) ds
tk
j=1
+ Λ(t)
k Z
X
i=1
Z
t

ti
g(s)ϕ ρ(s − h))
ti−1
Snezhana Hristova
(1 + βj ρ(tk )) ds
0<tj <s−h

g(s)ϕ ρ(s − h))
+ Λ(t)

Y
tk
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
Y
(1 + βj ρ(tk )) ds ∈ dom(H −1 )
0<tj <s−h
for τ ∈ (tk , tk+1 ] ∩ [0, t], k = 0, 1, . . . ,
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and H
−1
is the inverse function of H(u).
Proof.
Case 1. Let t1 ≥ h.
Let t ∈ (0, h] ∩ [0, γ) 6= ∅.
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It follows from the inequalities (3.1) and (3.2) that for t ∈ (0, h] ∩ [0, γ) the
inequality
Z t
(3.6)
u(t) ≤ ρ(t) 1 + G A +
p(s)Q(u(s))ds
0
holds.
(0)
Define the function v0 : [0, h] ∩ [0, γ) → [0, ∞) by the equality
Z t
(0)
(3.7)
v0 (t) = A +
p(s)Q(u(s))ds.
0
Nonlinear Delay Integral
Inequalities For Piecewise
Continuous Functions And
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(0)
The function v0 (t) is a nondecreasing differentiable function on [0, h] ∩ [0, γ)
and it satisfies the inequality
(0)
(3.8)
u(t) ≤ ρ(t) 1 + G(v0 (t)) .
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The inequality (3.8) and the definition (5) of the function H(u) yield
(0)
(v0 (t))0
d
(0)
≤ p(t)ϕ ρ(t) .
H(v0 (t)) = (0)
dt
Q 1 + G(v0 (t))
(3.9)
We integrate the inequality (3.9) from 0 to t for t ∈ [0, h] ∩ [0, γ), and we use
(0)
v0 (0) = A in order to obtain
Z t
(0)
(3.10)
H(v0 (t)) ≤ H(A) +
p(s)ϕ ρ(s) ds.
0
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The inequalities (3.8) and (3.10) imply the validity of the inequality (3.3) for
t ∈ [0, h] ∩ [0, γ).
Let t ∈ (h, t1 ] ∩ [0, γ) 6= ∅ .
Then
Z t
(0)
u(t) ≤ ρ(t) 1 + G v0 (h) +
p(s)Q(u(s))ds
h
Z t
+
g(s)Q(u(s − h))ds
h
(1)
= ρ(t) 1 + G v0 (t) ,
(3.11)
Nonlinear Delay Integral
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where
(3.12)
(1)
v0
: [h, t1 ] ∩ [0, γ) → [0, ∞) is defined by the equality
Z t
Z t
(1)
(0)
v0 (t) = v0 (h) +
p(s)Q(u(s))ds +
g(s)Q(u(s − h))ds.
h
h
(1)
(0)
Using the fact that the function v0 (t) is nondecreasing continuous and v0 (t −
(1)
(0)
h) ≤ v0 (h) ≤ v0 (t) for h < t ≤ min{2h, t1 }, we can prove as above that
Z t
(0)
(1)
p(s)ϕ ρ(s) ds
H(v0 (t)) ≤ H(v0 (h)) +
h
Z t
+
g(s)ϕ ρ(s − h) ds
h
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t
≤ H(A) +
p(s)ϕ ρ(s) ds
Z t 0 +
g(s)ϕ ρ(s − h) ds.
Z
(3.13)
h
The inequalities (3.11), (3.13) prove the validity of (3.3) on t ∈ (h, t1 ] ∩ [0, γ).
Define the function

 v0(0) (t) for t ∈ [0, h],
v0 (t) =
 (1)
v0 (t) for t ∈ (h, t1 ].
Now let t ∈ (t1 , t2 ] ∩ [0, γ) 6= ∅.
Define the function v1 : [t1 , t2 ] ∩ [0, γ) → [0, ∞) by the equality
Z t
Z t
(3.14)
v1 (t) = v0 (t1 ) +
p(s)Q(u(s))ds +
g(s)Q(u(s − h))ds.
t1
t1
The function v1 (t) is nondecreasing differentiable on t ∈ (t1 , t2 ]∩[0, γ), v1 (t) ≥
v0 (t1 ) and
u(t) ≤ ρ(t) 1 + G v1 (t) + β1 u(t1 )
≤ ρ(t) 1 + G v1 (t) + β1 ρ(t1 ) 1 + G(v0 (t1 ))
(3.15)
≤ ρ(t) 1 + G(v1 (t)) 1 + β1 ρ(t1 ) .
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Consider the following two possible cases:
Case 1.1. Let h ≤ t2 − t1 and t ∈ (t1 + h, t2 ] ∩ [0, γ). Then from (3.15) we
have
u(t − h) ≤ ρ(t − h) 1 + G(v1 (t − h)) 1 + β1 ρ(t1 )
≤ ρ(t − h) 1 + G(v1 (t)) 1 + β1 ρ(t1 )
Y (3.16)
1 + βk ρ(tk ) .
= ρ(t − h) 1 + G(v1 (t))
0<tk <t−h
Case 1.2. Let h > t2 − t1 or t ∈ (t1 , t1 + h] ∩ [0, γ). Then
(3.17)
u(t − h) ≤ ρ(t − h) 1 + G(v0 (t − h)) .
Nonlinear Delay Integral
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Using the inequality (3.17) and v0 (t − h) ≤ v1 (t) we obtain
u(t − h) ≤ ρ(t − h) 1 + G(v1 (t))
Y (3.18)
= ρ(t − h) 1 + G(v1 (t))
1 + βk ρ(tk ) .
0<tk <t−h
The inequalities (3.15), (3.16), (3.18) and the properties of the function Q(u)
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imply
v10 (t) = p(t)Q(u(t)) + g(t)Q(u(t − h))
n
≤ p(t)ϕ ρ(t)(1 + β1 ρ(t1 ))
o
Y 1 + βk ρ(tk )
+ g(t)ϕ ρ(t − h)
0<t <t−h
(3.19)
k
× Q 1 + G(v1 (t)) .
We obtain from the definition (5) and the inequality (3.19) that
d
(v1 (t))0
H(v1 (t)) = dt
Q 1 + G(v1 (t))
(3.20)
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≤ p(t)ϕ ρ(t)(1 + β1 ρ(t1 ))
Y
+ g(t)ϕ ρ(t − h)
!
1 + βk ρ(tk )
.
0<tk <t−h
We integrate the inequality (3.20) from t1 to t, use the inequality (3.13) and
obtain
Z t
H(v1 (t)) ≤ H(v0 (t1 )) +
p(s)ϕ ρ(s)(1 + β1 ρ(t1 )) ds
t1
!
Z t
Y
+
g(s)ϕρ(s − h)
(1 + βk ρ(tk )) ds
t1
Nonlinear Delay Integral
Inequalities For Piecewise
Continuous Functions And
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0<tk <t−h
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t1
≤ H(A) +
p(s)ϕ ρ(s) ds
0
Z t
+
p(s)ϕ ρ(s)(1 + β1 ρ(t1 )) ds
Z
t1
Z
(3.21)
!
t
Y
g(s)ϕρ(s − h)
+
t1
(1 + βk ρ(tk )) ds.
0<tk <t−h
The inequalities (3.15) and (3.21) imply the validity of the inequality (3.3) for
t ∈ (t1 , t2 ] ∩ [0, γ).
We define functions vk : [tk , tk+1 ] ∩ [0, γ) → [0, ∞) by the equalities
Z t
Z t
(3.22)
vk (t) = vk−1 (tk ) +
p(s)Q(u(s))ds +
g(s)Q(u(s − h))ds.
tk
tk
≤ ρ(t) 1 + G(vk (t)) +
i=1
k−1
X
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βi u(ti )
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i=1
+βk ρ(tk ) 1 + G(vk−1 (tk )) +
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The functions vk (t) are nondecreasing functions, vk (t) ≥ vk−1 (tk ) and for t ∈
(tk , tk+1 ] ∩ [0, γ) the inequalities
!
k
X
u(t) ≤ ρ(t) 1 + G vk (t) +
βi u(ti )
(
Nonlinear Delay Integral
Inequalities For Piecewise
Continuous Functions And
Applications
k−1
X
!)
βi u(ti )
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≤ ρ(t) 1 + G(vk (t)) +
k−1
X
!
βi u(ti )
1 + βk ρ(tk )
i=1
(3.23)
≤ · · · ≤ ρ(t)
( k Y
) 1 + βi ρ(ti )
1 + G(vk (t))
i=1
hold.
Using mathematical induction we prove that inequality (3.3) is true for t ∈
(tk , tk+1 ] ∩ [0, γ), k = 1, 2, . . . .
Case 2. Let there exist a natural number m such that tm ≤ h < tm+1 . As
inCase 1 we prove the validity
of inequality (3.3) using the functions vk ∈
C [tk , tk+1 ] ∩ [0, γ), [0, ∞) defined by the equalities
Z
t
vk (t) = vk−1 (tk ) +
p(s)Q(u(s))ds for k = 0, 1, . . . , m,
t
vk (t) = vk−1 (tk ) +
p(s)Q(u(s))ds
tk
Z t
+
g(s)Q(u(s − h))ds,
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Z kt
(3.24)
Nonlinear Delay Integral
Inequalities For Piecewise
Continuous Functions And
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k > m.
tk
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In the partial case when the function ϕ(s) in Definition 2.2 is multiplicative,
the following result is true.
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Corollary 3.2. Let the conditions of Theorem 3.1 be satisfied and the function
ϕ satisfy the inequality ϕ(ts) ≤ ϕ(t)ϕ(s) for t, s ≥ 0.
Then for t ∈ (tk , tk+1 ] ∩ [0, γ3 ) the inequality
k
Y
(3.25) u(t) ≤ ρ(t) (1 + βi ρ(ti ))
i=1
× 1 + G H −1
!
k
Y
H(A) + ϕ
(1 + βi ρ(ti )
i=1
Z
×
t
p(s)ϕ(ρ(s)) + Λ(s)g(s)ϕ(ρ(s − h)) ds
,
Nonlinear Delay Integral
Inequalities For Piecewise
Continuous Functions And
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0
Snezhana Hristova
holds, where the functions Λ(t) and H(u) are defined by the equalities (3.4) and
(3.5), respectively, and
(
!
k
Y
(3.26) γ3 = sup t ≥ 0 : H(A) + ϕ
(1 + βi ρ(ti )
i=1
Z
×
t
p(s)ϕ(ρ(s)) + Λ(s)g(s)ϕ(ρ(s − h)) ds ∈ Dom(H −1 )
0
for τ ∈ [0, t]},
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In the case when the function f1 (t) = 0 in inequality (3.3), we can obtain
another bound in which the function H(u) is different and in some cases easier
to be used.
Theorem 3.3. Let the following conditions be fulfilled:
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1.
2.
3.
4.
5.
The functions f1 , f2 , p, g ∈ C([0, ∞), [0, ∞)).
The function ψ ∈ C([−h, 0], [0, ∞)).
The function Q ∈ W2 (ϕ) and Q(u) > 0 for u > 0.
The function G ∈ W1 .
The function u ∈ P C([−h, ∞), [0, ∞)) and it satisfies the inequalities
(3.27) u(t)
Z
≤ f1 (t)G c +
t
t
Z
g(t)Q(u(s − h))ds
X
+ f2 (t)
βk u(tk ) for t ≥ 0,
p(s)Q(u(s))ds +
0
0
0<tk <t
u(t) ≤ ψ(t)
(3.28)
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Then for t ∈ (tk , tk+1 ] ∩ [0, γ), (k = 1, 2, . . . ) we have the inequality
k
Y
(3.29) u(t) ≤ ρ(t) (1 + βi ρ(ti ))
i=1
× G H −1
H(A) +
Snezhana Hristova
for t ∈ [−h, 0],
where c ≥ 0, βk ≥ 0, (k = 1, 2, . . . ).
(
Nonlinear Delay Integral
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k Z
X
i=1
"
#
i−1
Y
p(s)ϕ ρ(s) (1 + βj ρ(tj )) ds
ti
ti−1
j=1
Z
t
+
tk
Close
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"
#
k
Y
p(s)ϕ ρ(s) (1 + βj ρ(tj )) ds
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+ Λ(t)
k Z
X
i=1


ti
g(s)ϕ ρ(s − h))
ti−1
Y
(1 + βj ρ(tk )) ds
j:0<tj <s−h
Z

t
g(s)ϕ ρ(s − h))
+Λ(t)
tk


(1 + βj ρ(tk )) ds  ,


Y
j:0<tj <s−h
where Λ(t) is defined by equality (3.4), the constants A, B1 , B2 , γ are the same
as in Theorem 3.1, ρ(t) = max{fi (t) : i = 1, 2},
Z u
ds
, A ≥ u0 > 0.
(3.30)
H(u) =
u0 Q(G(s))
The proof of Theorem 3.3 is similar to the proof of Theorem 3.1.
As a partial case of Theorem 3.1 we can obtain the following result about
integral inequalities for piecewise continuous functions without delay.
Theorem 3.4. Let the following conditions be satisfied:
1. The functions f1 , f2 , f3 , p ∈ C([0, ∞), [0, ∞)).
2. The function Q ∈ W2 (ϕ) and Q(u) > 0 for u > 0.
3. The function G ∈ W1 .
4. The function u ∈ P C([0, ∞), [0, ∞)) and it satisfies the inequalities
Z t
(3.31) u(t) ≤ f1 (t) + f2 (t)G c +
p(s)Q(u(s))ds
0
X
+ f3 (t)
βk u(tk ), for t ≥ 0.
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where c ≥ 0, βk ≥ 0, (k = 1, 2, . . . ).
Then for t ∈ (tk , tk+1 ] ∩ [0, γ1 ), k = 0, 1, 2, . . . we have the inequality
(3.32) u(t) ≤ ρ(t)
k
Y
(1 + βi ρ(ti ))
i=1
(
×
1 + G H −1
H(c) +
k Z
X
t
+
"
#
i−1
Y
p(s)ϕ ρ(s) (1 + βj ρ(tj )) ds
ti−1
i=1
Z
ti
j=1
"
#
k
Y
p(s)ϕ ρ(s) (1 + βj ρ(tj )) ds
tk
)!!
,
j=1
Nonlinear Delay Integral
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where the function H(u) is defined by equality (3.5), ρ(t) = max{fi (t); i =
1, 2},
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(
γ1 = sup t ≥ 0 : H(c) +
k Z
X
i=1
Z
t
+
"
p(s)ϕ ρ(s)
tk
k
Y
ti
"
#
i−1
Y
p(s)ϕ ρ(s) (1 + βj ρ(tj )) ds
ti−1
j=1
#
)
(1 + βj ρ(tj )) ds ∈ dom(H −1 ) for τ ∈ [0, t] .
j=1
Remark 1. We note that the obtained inequalities are generalizations of many
known results. For example, in the case when f1 (t) = 0, f2 (t) = 0, βk = 0,
G(u) = u, Q(u) = u, h = 0, g(t) = 0 the result in Theorem 3.3 reduces to the
classical Gronwall inequality.
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Now we will consider different types of nonlinear integral inequalities in
which the unknown function is powered.
Theorem 3.5. Let the following conditions be fulfilled:
1. The functions f, g, h, r ∈ C([0, ∞), [0, ∞)).
2. The function ψ ∈ C([−h, 0], [0, ∞)) and ψ(t) ≤ c for t ∈ [−h, 0] where
c ≥ 0. The constants p > 1, 0 ≤ q ≤ p.
3. The function u ∈ P C([0, ∞), [0, ∞)) and satisfies the inequalities
Z t
p
u (t) ≤ c +
[f (s)up (t) + g(s)uq (s)up−q (s − h)
Nonlinear Delay Integral
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0
(3.33)
+ h(s)u(s) + r(s)u(s − h)]ds
X
+
βk up (tk ), for t ≥ 0,
0<tk <t
(3.34)
u(t) ≤ ψ(t) for t ∈ [−h, 0].
Then for t ∈ (tk , tk+1 ], k = 0, 1, 2, . . . the inequality
v
s
u k
Z
uY
p−1 t
p
p
t
(3.35) u(t) ≤
(h(s) + r(s))ds
(1 + βi ) ×
c+
p
0
i=1
s
Z t
h(s) + r(s)
p
)ds
(f (s) + g(s) +
× exp
p
0
holds.
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Proof.
Case 1. Let t1 ≥ h.
(0)
Let t ∈ (0, h]. We define the function v0 : [−h, h] → [0, ∞) by the
equalities

Rt
c + 0 [f (s)up (t) + g(s)uq (s)up−q (s − h) + h(s)u(s)



(0)
+r(s)u(s − h)]ds, t ∈ [0, h],
v0 (t) =


 p
ψ (t) for t ∈ [−h, 0).
(0)
The function v0 (t) is a nondecreasing differentiable function on [0, h], up (t) ≤
(0)
x
v0 (t) and using the inequality xm y n ≤ m
+ ny , n + m = 1 we obtain
(3.36)
q
(0)
v0 (t) p − 1
p
(0)
u(t) ≤ v0 (t) ≤
+
, t ∈ [0, h]
p
p
and
(3.37)
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u(t − h) ≤
ψ(t − h) p − 1
+
p
p
(0)
≤
c p−1
v (t) p − 1
+
≤ 0
+
, t ∈ [0, h].
p
p
p
p
Therefore the inequality
0
(0)
v0 (t) = f (t)up (t) + g(t)uq (t)up−q (t − h) + h(t)u(t) + r(t)u(t − h)
(0)
(0)
q/p (0)
v0 (t
≤ f (t)v0 (t) + g(t)v0 (t)
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(p−q)/p
− h)
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(0)
+ h(t)
≤
(3.38)
v0 (t) p − 1
+
p
p
!
h(t) + r(t))
f (t) + g(t) +
p
(0)
+ r(t)
v0 (t) p − 1
+
p
p
(0)
v0 (t) + (h(t) + r(t))
!
p−1
p
holds.
According to Corollary 1.2 ([1]) from the inequality (3.38) we obtain the
validity of the inequality
Z
p−1 t
(0)
(3.39)
v0 (t) ≤ c +
(h(s) + r(s))ds
p
0
Z t h(s) + r(s)
× exp
f (t) + g(t) +
ds .
p
0
From inequality (3.39) follows the validity of (3.35) for t ∈ [0, h].
Let t ∈ (h, t1 ].
(1)
Define the function v0 : [h, t1 ] → [0, ∞) by the equation
(1)
v0 (t)
Z
=
(0)
v0 (h)+
t
[f (s)up (t)+g(s)uq (s)up−q (s−h)+h(s)u(s)+r(s)u(s−h)]ds.
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h
(1)
From the definition of the function v0 (t) and the inequality (3.33), the validity
of the inequality
(3.40)
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(1)
up (t) ≤ v0 (t), t ∈ (h, t1 ].
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follows.
Case 1.1. Let h < t ≤ min{t1 , 2h}. Then t − h ∈ (0, h] and
q
q
q
(1)
v0 (t) p − 1
p
p
p
(0)
(0)
(1)
u(t − h) ≤ v0 (t − h) ≤ v0 (h) ≤ v0 (t) ≤
+
.
p
p
Case 1.2. Let t1 > 2h or t ∈ (2h, t1 ]. Then
q
q
(1)
v0 (t) p − 1
p
p
(1)
(1)
u(t − h) ≤ v0 (t − h) ≤ v0 (t) ≤
+
p
p
Nonlinear Delay Integral
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and
(3.41)
0
(1)
v0 (t)
≤
h(t) + r(t))
f (t) + g(t) +
p
(1)
v0 (t)
+ (h(t) + r(t))
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p−1
.
p
From the inequalities (3.40), (3.41) and applying Corollary 1.2 ([1]) we obtain
Z
p−1 t
(1)
(0)
(v0 (t)) ≤ v0 (h) +
(h(s) + r(s))ds
p
h
Z t h(s) + r(s)
× exp
f (t) + g(t) +
ds
p
h
Z
p−1 t
(h(s) + r(s))ds
≤ c+
p
0
Z t h(s) + r(s)
(3.42)
× exp
f (t) + g(t) +
ds .
p
0
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The inequalities (3.40) and (3.43) prove the validity of the inequality (3.35) on
(h, t1 ].
Define a function

 v0(0) (t) for t ∈ [0, h],
v0 (t) =
 (1)
v0 (t) for t ∈ [h, t1 ].
Now let t ∈ (t1 , t2 ].
Define the function v1 : [t1 , t2 ] → [0, ∞) by the equation
Z t
(3.43) v1 (t) = v0 (t1 ) +
[f (s)up (t) + g(s)uq (s)up−q (s − h) + h(s)u(s)
Nonlinear Delay Integral
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h
+ r(s)u(s − h)]ds + β1 up (t1 ).
p
We note that v0 (t) ≤ v1 (t), up (t) ≤ v1 (t), up (t1 ) ≤ v0 (t1 ), u(t − h) ≤ p v1 (t),
p
p
u(t − h) ≤ p v1 (t) and p v1 (t) ≤ v1p(t) + p−1
for t ∈ (t1 , t2 ].
p
The function v1 (t) satisfies the inequality
Z
p−1 t
v1 (t) ≤ (1 + β1 )v0 (t1 ) +
(h(s) + r(s))ds
p
t1
Z t h(s) + r(s)
× exp
f (t) + g(t) +
ds
p
t1
Z
p−1 t
(h(s) + r(s))ds
≤ (1 + β1 ) c +
p
0
Z t h(s) + r(s)
(3.44)
× exp
f (t) + g(t) +
ds .
p
0
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p
From the inequalities u(t) ≤ p v1 (t) and (3.44) it follows (3.35) for t ∈
(t1 , t2 ].
Using mathematical induction we can prove the validity of (3.35) for t ≥ 0.
Case 2. Let there exist a natural number m such that tm ≤ h < tm+1 . As in
Case 1 we can prove the validity of the inequality (3.35), using functions vk (t),
k = 1, 2, . . . defined by the inequality
(3.45) vk (t) = vk−1 (tk )
Z t
+
[f (s)up (t) + g(s)uq (s)up−q (s − h) + h(s)u(s)
tk
+ r(s)u(s − h)]ds + βk up (tk ).
Remark 2. Some of the inequalities proved by B.G. Pachpatte in [5], [6], [7]
are partial cases of Theorem 3.1 and Theorem 3.3.
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4.
Applications
We will use above results to obtain bounds for the solutions of different type of
equations.
Example 4.1. Consider the nonlinear impulsive integral equation with delay
Z
(4.1)
t
2
Z t
p
p
p(s) u(s)ds +
g(s) u(s − h)ds
u(t) = f (t) +
X0
+
βk u(tk ),
0
for t ≥ 0,
0<tk <t
(4.2)
u(t) = 0
t ∈ [−h, 0],
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where βk ≥ 0, k = 1, 2, . . . , p, g ∈ C([0, ∞), [0, ∞)), f ∈ C([0, ∞), [0, 1]),
h > 0.
We note the solutions of the problem (4.1), (4.2) are nonnegative.
√
2
Define
√ the functions G(u) = u , Q(u) = u. Then Q ∈ W2 (φ), where
φ(u) = u.
Consider the function
Z u
Z u
√
ds
ds
√
(4.3)
H(u) =
=
= ln(u + 1 + u2 ).
1 + s2
0 Q(1 + G(s))
0
Then the inverse function of H(u) will be defined by
(4.4)
1
H −1 (u) = sinh(u) = (eu − e−u ).
2
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According to Corollary 3.2 we obtain the upper bound for the solution


2 
v
u
Z


t
Y
u Y




t
u(t) ≤
(1+βk ) 1 + sinh
(1 + βk ) (p(s) + Λg(s))ds
.


0
0<t <t
0<t <t
k
k
Example 4.2. Consider the initial value problem for the nonlinear impulsive
integro-differential equation
Z t
p
p
0
(4.5)
u (t) = 2f (t) u(t)
f (s) u(s)ds, t > 0, t 6= tk ,
0
(4.6)
(4.7)
u(tk + 0) = βk u(tk ),
u(0) = c,
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where c ≥ 0, βk ≥ 0, k = 1, 2, . . . , f ∈ C([0, ∞), [0, ∞)).
The solutions of the given problem satisfy the inequality
Z
(4.8)
u(t) ≤ c +
0
t
p
f (s) u(s)ds
Nonlinear Delay Integral
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Continuous Functions And
Applications
2
+
X
βk u(tk ), tk ≥ 0.
0<tk <t
√
Define the functions G(u) = u2 , Q(u) = u. Then the functions H(u) and
H −1 (u) are defined by the equations (4.3) and (4.4).
We note that the solutions of the problem (4.5) – (4.7) are nonnegative. Ac-
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cording to Theorem 3.4 the solutions satisfy the estimate
Y
(4.9) u(t) ≤ A
(1 + Aβk )
0<tk <t

2 
 
s Y
Z t






A
(1 + Aβk )
f (s)ds
,
× 1 + sh


0
0<t <t
k
where A = max{1, c}.
Example 4.3. Consider the initial value problem for the nonlinear impulsive
differential-difference equation
u0 (t) = F (t, u(t), u(t − h)) + r(t), t > 0, t 6= tk ,
u(tk + 0) = βk u(tk ),
u(t) = ψ(t), t ∈ [−h, 0],
(4.10)
(4.11)
(4.12)
where βk =const, k = 1, 2, . . . , r ∈ C([0, ∞), R), ψ ∈ C([−h, 0], R), F ∈
C([0, ∞) × R2 , R)pand there exist
p functions p, g ∈
R tC([0, ∞), [0, ∞)) such that
|F (t, u, v)| ≤ p(t) |u| + g(t) |v|. Let |ψ(0) + 0 r(s)ds| ≤ 1, t ≥ 0.
We assume that the solutions of the initial value problem (4.10) – (4.12) exist
on [0, ∞).
The solution satisfies the inequalities
Z t
(4.13) |u(t)| ≤ ψ(0) +
r(s)ds
0
Z t
X
p
p
|βk |.|u(tk )|, t > 0,
+
(p(s) |u(s)| + g(s) |u(s − h)|)ds +
0
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|u(t)| = |ψ(t)|, t ∈ [−h, 0].
Rt
√
Consider functions f1 (t) = √
|ψ(0) + 0 r(s)ds|, G(u) = u, Q(u) = u, f2 (t) =
1, f3 (t) = 1.
to the equality (3.5) the function
√ Then ϕ(u) = u and according
−1
H(u) = 2( 1 + u − 1) and its inverse is H (u) = ( u2 + 1)2 − 1.
According to Corollary 3.2 the following bound for the solution of (4.10) –
(4.12)
q
Y
p
(4.15) |u(t)| ≤
(1 + |βk |)
1 + hB1 B2
(4.14)
0<tk <t
+
s Y
1
2
0<tk <t
Z
(1 + |βk |)
t
2
(p(s) + Λg(s)ds ,
0
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is satisfied, where B1 = max{|g(s)| : s ∈ [0, h]}, B2 = max{ψ(s) : s ∈
[−h, 0]}.
Example 4.4. Consider the initial value problem for the nonlinear impulsive
differential-difference equation
(4.16)
(4.17)
(4.18)
u(t)u0 (t) = F (t, u(t), u(t − h))
+ q(t)u(t) + r(t)u(t − h), t > 0, t 6= tk ,
u(tk + 0) = βk u(tk ),
u(t) = ψ(t), t ∈ [−h, 0],
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where βk = const, k = 1, 2, . . . , r, q ∈ C([0, ∞), R), ψ ∈ C([−h, 0], R),
F ∈ C([0, ∞) × R2 , R) and there exist functions p,
R tg ∈ C([0, ∞), [0, ∞)) such
that |F (t, u, v)| ≤ p(t)u2 + g(t)v 2 . Let |ψ(0) + 0 r(s)ds| ≤ 1, t ≥ 0. We
assume that the solutions of the initial value problem (4.10) – (4.12) exist on
[0, ∞).
The solution satisfies the inequalities
(4.19) u(t)2 ≤ ψ 2 (0)
Z t
2
2
+
p(s)u (s) + g(s)u (s − h) + q(s)u(s) + r(s)u(s − h) ds
0
X
+
βk2 u2 (tk ), t > 0,
0<tk <t
u2 (t) = ψ 2 (t), t ∈ [−h, 0].
(4.20)
Apply Theorem 3.5 to the inequalities (4.17), (4.18) and obtain the following
upper bound for the solution of the initial value problem (4.14) – (4.16)
2
2
(4.21) u(t) ≤ ψ (0)
Z t
2
2
+
p(s)u (s) + g(s)u (s − h) + q(s)u(s) + r(s)u(s − h) ds
0
X
+
βk2 u2 (tk ), t > 0,
0<tk <t
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(4.22)
2
2
u (t) = ψ (t), t ∈ [−h, 0].
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References
[1] D.D. BAINOV AND P.S. SIMEONOV, Integral Inequalities and Applications, Kluwer Academic Publishers, Dordrecht-Boston-London, 1992.
[2] S.S. DRAGOMIR AND N.M. IONESCU, On nonlinear integral inequalities in two independent variables, Studia Univ. Babes-Bolyai, Math., 34
(1989), 11–17.
[3] S.S. DRAGOMIR AND YOUNG-HO KIM, On certain new integral inequalities and their applications, J. Ineq. Pure Appl. Math.,
3(4) (2002), Article 65 [ONLINE: http://jipam.vu.edu.au/
article.php?sid=217].
Nonlinear Delay Integral
Inequalities For Piecewise
Continuous Functions And
Applications
Snezhana Hristova
[4] S.S. DRAGOMIR AND YOUNG-HO KIM, Some integral inequalities for
functions of two variables,EJDE, 2003(10) (2003) [ONLINE: http://
ejde.math.swt.edu/].
[5] B.G. PACHPATTE, On a certain inequality arrizing in the theory of differential equations, J. Math. Anal. Appl., 182 (1994), 143–157.
[6] B.G. PACHPATTE, On some new inequalities related to certain inequalities in the theory of differential equations, J. Math. Anal. Appl., 189
(1995), 128–144.
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[7] B.G. PACHPATTE, A note on certain integral inqualities with delay, Periodica Math. Hungaria, 31(3) (1995), 229–234.
[8] B.G. PACHPATTE, Inequalities for Differential and Integral Equations,
Academic Press, New York, 1998
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[9] B.G. PACHPATTE, On some new inequalities related to a certain inequality arising in the Theory of Differential Equations, J. Math. Anal.
Appl., 251 (2000), 736–751.
[10] B.G. PACHPATTE, On some retarded integral inequalities and applications, J. Ineq. Pure Appl. Math., 3(2) (2002), Article 18, [ONLINE:
http://jipam.vu.edu.au/article.php?sid=170]
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