2012 International Conference on Image, Vision and Computing (ICIVC 2012)
IPCSIT vol. 50 (2012) © (2012) IACSIT Press, Singapore
DOI: 10.7763/IPCSIT.2012.V50.45
Analysis of Boundedness for Unknown Functions by a Delay Integral
Inequality on Time Scales
Qinghua Feng+
School of Science, Shandong University of Technology, Zhangzhou Road 12, Zibo, Shandong, China, 255049
Abstract. In this paper, new explicit bound for unknown functions is derived by a new Volterra-Fredholm
type delay integral inequality on time scales, which can be used as a hand tool in the investigation of qualitative
properties as well as quantitative properties of delay dynamic equations.
Keywords: Delay integral inequality; Time scales; Dynamic equation; Bounded
1. Introduction
Integral inequalities play an important role in the research of qualitative properties of solutions of dynamic
equations, and many integral inequalities as well as difference inequalities have been established since then, for
example [1-10], and the references therein.
Our aim in this paper is to establish a new Volterra-Fredholm type delay integral inequality on time scales,
which provides new bound for unknown functions.
In the rest of the paper, R denotes the set of real numbers and R+ = [0, ∞) . T denotes an arbitrary time scale
and T0 = [ x0 , ∞) ∩ T , T 0 = [ y0 , ∞) ∩ T , where x0 , y0 ∈ T . The set T κ is defined to be T if T does not have a
left-scattered maximum, otherwise it is T without the left-scattered maximum. On T we define the forward
and backward jump operators σ ∈ (T , T ) and ρ ∈ (T , T ) such that σ (t ) = inf{s ∈T , s > t}, ρ (t ) = sup{s ∈ T , s < t} .
Definition 1: A point t ∈ T with t > infT is said to be left-dense if ρ (t ) = t and right-dense if σ (t ) = t ,
left-scattered if ρ (t ) < t and right-scattered if σ (t ) > t .
Definition 2: A function f ∈ (T , R ) is called rd-continuous if it is continuous in right-dense points and if the
left-sided limits exist in left-dense points, while f is called regressive if 1 + μ (t ) f (t ) ≠ 0 , where μ (t ) = σ (t ) − t .
Crd denotes the set of rd-continuous functions, while R denotes the set of all regressive and rd-continuous
+
functions, and R = { f | f ∈ R,1 + μ (t ) f (t ) > 0, ∀t ∈ T } .
Definition 3: For some t ∈ T κ , and a function f ∈ (T , R ) , the delta derivative of f is denoted by f Δ (t ) ,
and satisfies | f (σ (t )) − f ( s) − f Δ (t )(σ (t ) − s) |≤ ε | σ (t ) − s | for ∀ε > 0 , where s ∈ U , and U is a neighborhood
of t . The function f is called delta differential on T κ
Similarly, for some y ∈ T κ , and a function f ∈ (T × T , R ) , the partial delta derivative of f with respect to
y is denoted by ( f ( x, y )) Δy , and satisfies
| f ( x, σ ( y )) − f ( x, s ) − ( f ( x, y )) Δy (σ ( y ) − s ) |≤ ε | σ ( y ) − s | for ∀ε > 0 , where s ∈ U , and U is a neighborho-
od of y .
Definition 4: For some a, b ∈ T and a function f ∈ (T , R ) , the Cauchy integral of f is defined by
+
Corresponding author.
E-mail address: fqhua@sina.com
b
∫ f (t )Δt = F (b) − F (a ) , where
F Δ (t ) = f (t ), t ∈ T κ .
a
Similarly, for some a, b ∈ T and a function f ∈ (T × T , R ) , the Cauchy partial integral of f with respect to
y is defined by
b
∫ f ( x, y )Δy = F ( x, b) − F ( x, a ) , where ( F ( x, y ))
Δ
y
= f ( x, y ), y ∈ T κ .
a
2. Main Results
We will give some lemmas for further use.
Lemma 2.1 ([11], Gronwall’s inequality): Suppose X ∈ T0 is an arbitrarily fixed number, and u ( X , y ),
+
b( X , y ) ∈ Crd , m( X , y ) ∈ R with respect to y , m( X , y) ≥ 0 , then
y
u ( X , y ) ≤ b( X , y ) + ∫ m( X , t )u ( X , t )Δt , y ∈ T 0
y0
implies
y
u ( X , y ) ≤ b( X , y ) + ∫ em ( y , σ (t ))b( X , t ) m( X , t )Δt , y ∈ T 0 ,
y0
where em ( y , y0 ) is the unique solution of the following equation
( z ( X , y )) Δy = m( X , y ) z ( X , y ), z ( X , y0 ) = 1 .
Lemma 2.2 [12]: Assume that a ≥ 0, p ≥ q ≥ 0 , and p ≠ 0 , then for any K > 0 ,
q
ap ≤
q q−p p
p − q qp .
K a+
K
p
p
Consider the following inequality
y x
y x t s
y0 x0
y0 x0 y0 x0
u p ( x, y) ≤ C + ∫ ∫ L(s, t, u(τ1(s),τ 2 (t)))ΔsΔt + ∫ ∫ ∫ ∫ h1 (ξ ,η )uq (τ1 (ξ ),τ 2 (η ))ΔξΔηΔsΔt
N M
N M t s
y0 x0
y0 x0 y0 x0
+ ∫ ∫ L(s, t, u(τ1 (s),τ 2 (t )))ΔsΔt + ∫ ∫ ∫ ∫ h2 (ξ ,η )uq (τ1 (ξ ),τ 2 (η ))ΔξΔηΔsΔt
(1)
with the initial condition
⎧⎪u( x, y) = φ ( x, y), x ∈[α , x0 ] ∩ T , or, y ∈[β , y0 ] ∩ T
,
1
⎨
p
⎪⎩φ (τ1 ( x),τ 2 ( y)) ≤ C ,τ1 ( x) ≤ x0 , or,τ 2 ( y) ≤ y0
(2)
where u , hi ∈ Crd (T0 × T 0 , R+ ), i = 1, 2 , p , q , C m , C are constants, and p ≥ q ≥ 0, p ≠ 0, C > 0 , τ1 ∈(T0 ,T),τ1(x) ≤ x,
−∞< α = inf{τ1(x), x ∈T0} ≤ x0 , τ 2 ∈ (T 0 , T ),τ 2 ( y) ≤ y, −∞ < β = , inf{τ 2 ( y), y ∈T 0 } ≤ y0 φ ∈ Crd (([α , x0 ] ×[β , y0 ]) ∩ T 2 , R+ ) ,
M ∈T0 , N ∈T0 are two fixed numbers.
Theorem 2.1: If for (x, y) ∈([x0 , M] ∩T ×([ y0 , N] ∩T ) , u( x, y) satisfies (1), and K > 0 is an arbitrary constant, then
the following inequality holds
u( x, y) ≤ {[
1
C + B6
]B3 ( x, y) + B4 ( x, y)}p , ( x, y) ∈ ([ x0 , M ] ∩ T ) × ([ y0 , N ] ∩ T )
1 − B5
provided that B5 < 1 , where
(3)
NM
C = C + ∫ ∫ [L(s, t,
y0 x0
y x
B1 ( x, y) = ∫ ∫ [L(s, t,
y0 x0
x
B2 ( x, y) = ∫ [ A(s, y,
x0
t s
p −1 1p
p − m mp
K ) + ∫ ∫ h2 (ξ ,η )
K ΔξΔη ]ΔsΔt
p
p
y0 x0
t s
p −1 1p
p − q qp
K ) + ∫ ∫ h1 (ξ ,η )
K ΔξΔη ]ΔsΔt
p
p
y0 x0
y s
q q− p
p −1 1p 1 1−pp
K ) K + ∫ ∫ h1 (ξ ,η ) K p ΔξΔη ]Δs .
p
p
p
y0 x0
y
B3 ( x, y) = 1 + ∫ eB2 ( y,σ (t ))B2 ( x, t )Δt
y0
y
B4 ( x, y) = B1 ( x, y) + ∫ eB2 ( y,σ (t ))B2 ( x, t )B1 ( x, t )Δt
y0
y x
B5 = ∫ ∫ A(s, t,
y0 x0
y x
B6 = ∫ ∫ A(s, t,
y0 x0
NM t s
p −1 1p 1 1−pp
q q− p
K ) K B3 (s, t )ΔsΔt + ∫ ∫ ∫ ∫ h2 (ξ ,η ) K p B3 (ξ ,η )ΔξΔη ]ΔsΔt .
p
p
p
y0 x0 y0 x0
NM t s
p −1 1p 1 1−pp
q q− p
K ) K B4 (s, t )ΔsΔt + ∫ ∫ ∫ ∫ h2 (ξ ,η ) K p B4 (ξ ,η )ΔξΔη ]ΔsΔt .
p
p
p
y0 x0 y0 x0
Proof : Let the right side of (1) be v( x, y) . Then
1
(4)
u( x, y) ≤ v p ( x, y),( x, y) ∈ ([ x0 , M ] ∩ T ) × ([ y0 , N ] ∩ T )
From (2) we have
1
u(τ1 ( x),τ 2 ( y)) ≤ v p ( x, y),( x, y) ∈ ([ x0 , M ] ∩ T ) × ([ y0 , N ] ∩ T )
(5)
Given a fixed X ∈[ x0 , M ] ∩ T , and x∈[x0, X]∩T, y∈[y0, N]∩T , then
(6)
v( x, y) ≤ v( X , y), x ∈[ x0 , X ] ∩ T , y ∈[ y0 , N ] ∩ T
Furthermore, considering
NM
NM t s
y0 x0
y0 x0 y0 x0
v( x0 , y0 ) = C + ∫ ∫ L(s, t, u(τ1(s),τ 2 (t )))ΔsΔt + ∫ ∫ ∫ ∫ h2 (ξ ,η )uq (τ1 (ξ ),τ 2 (η ))ΔξΔηΔsΔt
so we have
y X
y X t s
y0 x0
y0 x0 y0 x0
v( X , y) = C + ∫ ∫ L(s, t, u(τ1 (s),τ 2 (t)))ΔsΔt + ∫ ∫ ∫ ∫ h1 (ξ ,η )uq (τ1 (ξ ),τ 2 (η ))ΔξΔηΔsΔt
NM
NM t s
y0 x0
y0 x0 y0 x0
+ ∫ ∫ L(s, t, u(τ1 (s),τ 2 (t )))ΔsΔt + ∫ ∫ ∫ ∫ h2 (ξ ,η )uq (τ1 (ξ ),τ 2 (η ))ΔξΔηΔsΔt
y X
1
y X t s
q
≤ C + ∫ ∫ L(s, t, v p (s, t))ΔsΔt + ∫ ∫ ∫ ∫ h1 (ξ ,η )v p (ξ ,η )ΔξΔηΔsΔt
y0 x0
y0 x0 y0 x0
(7)
NM
NM t s
1
+ ∫ ∫ L(s, t, v p (s, t ))ΔsΔt + ∫ ∫ ∫ ∫ h2 (ξ ,η )uq (τ1 (ξ ),τ 2 (η ))ΔξΔηΔsΔt
y0 x0
y0 x0 y0 x0
y X
y X t s
1
q
= v( x0 , y0 ) + ∫ ∫ L(s, t, v p (s, t))ΔsΔt + ∫ ∫ ∫ ∫ h1 (ξ ,η )v p (ξ ,η )ΔξΔηΔsΔt
y0 x0
(8)
y0 x0 y0 x0
From Lemma 2.2, we have
⎧ qp
⎪ v ( x, y ) ≤
⎪
⎨ 1
⎪ v p ( x, y ) ≤
⎪⎩
q q−p p
p − q qp
K v ( x, y ) +
K
p
p
p − 1 1p
1 1−pp
K v ( x, y ) +
K
p
p
(9)
Combining (8), (9) we have
y X
y X t s
1 1− p
p −1 1p
q q− p
p − q qp
v( X , y) = v(x0 , y0 ) + ∫ ∫ L(s, t,( K p v(s, t ) +
K ))ΔsΔt + ∫ ∫ ∫ ∫ h1 (ξ ,η )( K p v(ξ ,η ) +
K )ΔξΔηΔsΔt
p
p
p
p
y0 x0
y0 x0 y0 x0
y X
p −1 1p
p −1 1p
1 1− p
p −1 1p
K ) + L(s, t,
K )]ΔsΔt
= v( x0 , y0 ) + ∫ ∫ [L(s, t,( K p v(s, t ) +
K )) −L(s, t,
p
p
p
p
y0 x0
y X t s
q q− p
p − q qp
+ ∫ ∫ ∫ ∫ h1 (ξ ,η )( K p v(ξ ,η ) +
K )ΔξΔηΔsΔt
p
p
y0 x0 y0 x0
y X
≤ v(x0 , y0 ) + ∫ ∫ A(s, t,
y0 x0
y
y X
p −1 1p 1 1−pp
p −1 1p
K ) K v(s, t )ΔsΔt + ∫ ∫ L(s, t,
K )ΔsΔt
p
p
p
y0 x0
y X t s
X t s
q q− p
p − q qp
+ ∫ [ ∫ ∫ ∫ h1 (ξ ,η ) K p ΔξΔηΔs]v( X , t )Δt + ∫ ∫ ∫ ∫ h1 (ξ ,η )
K ΔξΔηΔsΔt
p
p
y0 x0 y0 x0
y0 x0 y0 x0
y X
≤ v(x0 , y0 ) + ∫ [ ∫ A(s, t,
y0 x0
y
y X
p −1 1p 1 1−pp
p −1 1p
K ) K Δs]v( X , t)Δt + ∫ ∫ L(s, t,
K )ΔsΔt
p
p
p
y0 x0
y X t s
X t s
q q− p
p − q qp
+ ∫ [ ∫ ∫ ∫ h1 (ξ ,η ) K p ΔξΔηΔs]v( X , t )Δt + ∫ ∫ ∫ ∫ h1 (ξ ,η )
K ΔξΔηΔsΔt
p
p
y0 x0 y0 x0
y0 x0 y0 x0
y
= v(x0 , y0 ) + B1( X , y) + ∫ B2 ( X , t)v( X , t)Δt
y0
By Lemma 2.2 we obtain
y
v( X , y) ≤ v( x0 , y0 ) + B1 ( X , y) + eB ( y,σ (t))B2 ( X , t)(v(x0 , y0 ) + B1( X , t))Δt
∫ 2
y0
y
y
y0
y0
= v(x0 , y0 )[1+ ∫ eB2 ( y,σ (t))B2 ( X , t)Δt] + B1( X , y) + ∫ eB2 ( y,σ (t))B2 ( X , t)(v(x0 , y0 ) + B1( X , t))Δt, y ∈[ y0 , N] ∩T
Combining (6), (10), it follows
y
v(x, y) ≤ v(x0, y0 ) + B1(X, y) + ∫ eB2 ( y,σ(t))B2 (X, t)(v(x0, y0 ) + B1(X, t))Δt
y0
(10)
y
= v(x0 , y0 )[1+ ∫ eB2 ( y,σ (t))B2 ( X , t)Δt] + B1( X , y) +
y0
y
∫e
B2
( y,σ (t))B2 ( X , t)(v(x0 , y0 ) + B1( X , t))Δt , x ∈[ x0 , X ] ∩T , y ∈[ y0 , N] ∩T
(11)
y0
Setting x = X in (11), considering X is selected from [ x0 , M ] ∩ T arbitrarily, substituting X with x , yields
y
v(x, y) ≤ v(x0 , y0 ) + B1(x, y) + ∫ eB2 ( y,σ (t))B2 (x, t)(v(x0 , y0 ) + B1(x, t))Δt
y0
y
= v(x0 , y0 )[1+ ∫ eB2 ( y,σ (t))B2 (x, t)Δt] + B1( X , y) +
y0
y
∫e
B2
( y,σ (t))B2 (x, t)(v(x0 , y0 ) + B1(x, t))Δt , x ∈[ x0 , X ] ∩T , y ∈[ y0 , N] ∩T
(12)
y0
that is,
(13)
v( x, y) ≤ v(x0 , y0 )B3 ( x, y) + B4 ( x, y) x ∈[ x0 , X ] ∩T , y ∈[ y0 , N] ∩T
On the other hand, from (5), (7) (9) we have
NM
1
NM t s
q
v( x0 , y0 ) = C + ∫ ∫ L(s, t, v p (s, t))ΔsΔt + ∫ ∫ ∫ ∫ h2 (ξ ,η )v p (ξ ,η )ΔξΔηΔsΔt
y0 x0 y0 x0
y0 x0
NM
NM t s
1 1− p
p −1 1p
q q− p
p − q qp
≤ C + ∫ ∫ L(s, t, K p v(s, t ) +
K )ΔsΔt + ∫ ∫ ∫ ∫ h2 (ξ ,η )( K p v(ξ ,η ) +
K )ΔξΔηΔsΔt
p
p
p
p
y0 x0
y0 x0 y0 x0
p −1 1p
p −1 1p
1 1− p
p −1 1p
K ) + L(s, t,
K )]ΔsΔt
≤ C + ∫ ∫ [ L(s, t, K p v(s, t ) +
K ) −L(s, t,
p
p
p
p
y0 x0
NM
NM t s
NM
q q− p
p − q qp
p −1 1p 1 1−pp
+ ∫ ∫ ∫ ∫ h2 (ξ ,η )( K p v(ξ ,η ) +
K )ΔξΔηΔsΔt ≤ C + ∫ ∫ A(s, t,
K ) K v(s, t )ΔsΔt
p
p
p
p
y0 x0 y0 x0
y0 x0
NM
+ ∫ ∫ L(s, t,
y0 x0
NM
= C + ∫ ∫ A(s, t,
y0 x0
NM t s
p −1 1p
q q− p
p − q qp
K )ΔsΔt + ∫ ∫ ∫ ∫ h2 (ξ ,η )( K p v(ξ ,η ) +
K )ΔξΔηΔsΔt
p
p
p
y0 x0 y0 x0
NM t s
p −1 1p 1 1−pp
q q− p
p − q qp
K ) K v(s, t )ΔsΔt + ∫ ∫ ∫ ∫ h2 (ξ ,η )( K p v(ξ ,η ) +
K )ΔξΔηΔsΔt
p
p
p
p
y0 x0 y0 x0
Then using (13) in (14) yields
NM
v(x0 , y0 ) ≤ C + ∫ ∫ A(s, t,
y0 x0
p −1 1p 1 1−pp
K ) K [v(x0 , y0 )B3 (s, t) + B4 (s, t)]ΔsΔt
p
p
NM t s
q q−p
p − q qp
+∫ ∫ ∫ ∫ h2 (ξ,η)( K p [v(x0, y0 )B3(ξ,η) + B4 (ξ,η)] +
K )ΔξΔηΔsΔt
p
p
y0 x0 y0 x0
y x
= C + v(x0 , y0 ){∫ ∫ A(s, t,
y0 x0
NM t s
p −1 1p 1 1−pp
q q− p
K ) K B3 (s, t)ΔsΔt + ∫ ∫ ∫ ∫ h2 (ξ ,η ) K p B3 (ξ ,η )ΔξΔη ]ΔsΔt}
p
p
p
y0 x0 y0 x0
(14)
y x
+ ∫ ∫ A(s, t,
y0 x0
NM t s
p −1 1p 1 1−pp
q q− p
K ) K B4 (s, t )ΔsΔt + ∫ ∫ ∫ ∫ h2 (ξ ,η ) K p B4 (ξ ,η )ΔξΔη ]ΔsΔt
p
p
p
y0 x0 y0 x0
= C + v(x0 , y0 )B5 + B6
(15)
which is followed by
v(x0 , y0 ) ≤
C + B6
1− B5
(16)
Combining (4), (13) and (16) we can obtain the desired inequality (3).
3. Conclusions
In this paper, new explicit bound for unknown functions is derived by use of a new Volterra- Fredholm type
integral inequality on time scales, which provides a handy tool in the qualitative analysis of solutions of
dynamic equations on time scales.
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