2012 International Conference on Image, Vision and Computing (ICIVC 2012)
IPCSIT vol. 50 (2012) © (2012) IACSIT Press, Singapore
DOI: 10.7763/IPCSIT.2012.V50.44
Analysis of a Nonlinear 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, a new Volterra-Fredholm type delay integral inequality on time scales is established,
which can be used as a hand tool in the investigation of qualitative properties of delay dynamic equations.
Keywords: Delay integral inequality; Time scales; Dynamic equation; Bounded
1. Introduction
During the past decades, many integral inequalities have been established since then, for example [1-10],
which have played an important role in the research of qualitative properties of solutions of dynamic equations.
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
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
+
Corresponding author.
E-mail address: fqhua@sina.com
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
More details on time scales can be referred to [11].
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 [15]: 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
Theorem 2.1: Suppose u , f i , g i , hi ∈ C rd (T0 × T 0 , R+ ), i = 1, 2 , p , q , r , m , C m, C are constants, and
p ≥ q ≥ 0, p ≥ r ≥ 0, p ≥ m ≥ 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, K > 0 is an arbitrary
constant. If for ( x, y) ∈ ([ x0 , M ] ∩ T × ([ y0 , N ] ∩ T ) , u( x, y) satisfies the following inequality
y x
y x t s
y0 x0
y0 x0 y0 x0
u p ( x, y) ≤ C + ∫ ∫ [ f1(s, t)uq (τ1(s),τ 2 (t )) + g1(s, t)ur (τ1 (s),τ 2 (t))]ΔsΔt + ∫ ∫ ∫ ∫ h1 (ξ ,η )u m (τ1 (ξ ),τ 2 (η ))ΔξΔηΔsΔt
NM
NM t s
y0 x0
y0 x0 y0 x0
+ ∫ ∫ [ f2 (s, t )uq (τ1 (s),τ 2 (t )) + g2 (s, t )u r (τ1 (s),τ 2 (t ))]ΔsΔt + ∫ ∫ ∫ ∫ h2 (ξ ,η )u m (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)
then
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 + ∫ ∫ [ f2 (s, t )
y0 x0
t s
p − q qp
p − r rp
p − m mp
K + g2 (s, t )
K + ∫ ∫ h2 (ξ ,η )
K ΔξΔη ]ΔsΔt
p
p
p
y0 x0
y x
B1 ( x, y) = ∫ ∫ [ f1 (s, t )
y0 x0
x
B2 ( x, y) = ∫ [ f1 (s, y)
x0
t s
p − q qp
p − r rp
p − m mp
K + g1 (s, t )
K + ∫ ∫ h1 (ξ ,η )
K ΔξΔη ]ΔsΔt
p
p
p
y0 x0
y s
m m− p
q q−pp
r r− p
K + g1 (s, y) K p + ∫ ∫ 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
NM t s
y x
NM t s
q q− p
r r− p
m m− p
B5 = ∫ ∫ [ f2 (s, t ) K p B3 (s, t ) + g2 (s, t ) K p B3 (s, t )]ΔsΔt + ∫ ∫ ∫ ∫ h2 (ξ ,η ) K p B3 (ξ ,η )ΔξΔη ]ΔsΔt .
p
p
p
y0 x0
y0 x0 y0 x0
q q− p
r r− p
m m− p
B6 = ∫ ∫ [ f2 (s, t ) K p B4 (s, t ) + g2 (s, t ) K p B4 (s, t )]ΔsΔt + ∫ ∫ ∫ ∫ h2 (ξ ,η ) K p B4 (ξ ,η )ΔξΔη ]ΔsΔt .
p
p
p
y0 x0
y0 x0 y0 x0
Proof : Let the right side of (1) be v( x, y) . Then
(4)
1
u( x, y) ≤ v p ( x, y),( x, y) ∈ ([ x0 , M ] ∩ T ) × ([ y0 , N ] ∩ T )
From (2) we have
(5)
1
u(τ1 ( x),τ 2 ( y)) ≤ v p ( x, y),( x, y) ∈ ([ x0 , M ] ∩ T ) × ([ y0 , N ] ∩ T )
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 + ∫ ∫ [ f1(s, t)uq (τ1 (s),τ 2 (t)) + g1(s, t)ur (τ1(s),τ 2 (t))]ΔsΔt + ∫ ∫ ∫ ∫ h1 (ξ ,η )u m (τ1 (ξ ),τ 2 (η ))ΔξΔηΔsΔt ,
we have
y X
y X t s
y0 x0
y0 x0 y0 x0
v( X , y) = C + ∫ ∫ [ f1 (s, t)uq (τ1(s),τ 2 (t)) + g1(s, t )ur (τ1(s),τ 2 (t))]ΔsΔt + ∫ ∫ ∫ ∫ h1 (ξ ,η )um (τ1 (ξ ),τ 2 (η ))ΔξΔηΔsΔt
NM
NM t s
+ ∫ ∫ [ f2 (s, t )uq (τ1 (s),τ 2 (t )) + g2 (s, t )u r (τ1 (s),τ 2 (t ))]ΔsΔt + ∫ ∫ ∫ ∫ h2 (ξ ,η )u m (τ1 (ξ ),τ 2 (η ))ΔξΔηΔsΔt
y0 x0
y X
y0 x0 y0 x0
q
p
r
p
y X t s
m
≤ C + ∫ ∫ [ f1(s, t )v (s, t ) + g1(s, t )v (s, t)]ΔsΔt + ∫ ∫ ∫ ∫ h1 (ξ ,η )v p (ξ ,η )ΔξΔηΔsΔt
y0 x0
y0 x0 y0 x0
(7)
NM
NM t s
y0 x0
y0 x0 y0 x0
+ ∫ ∫ [ f2 (s, t )uq (τ1 (s),τ 2 (t )) + g2 (s, t )u r (τ1 (s),τ 2 (t ))]ΔsΔt + ∫ ∫ ∫ ∫ h2 (ξ ,η )u m (τ1 (ξ ),τ 2 (η ))ΔξΔηΔsΔt
y X
q
y X t s
= v( x0 , y0 ) + ∫ ∫ [ f1(s, t)v p (s, t ) + g1 (s, t)v p (s, t )]ΔsΔt + ∫ ∫ ∫ ∫ h1 (ξ ,η )v p (ξ ,η )ΔξΔηΔsΔt .
r
y0 x0
m
(8)
y0 x0 y0 x0
Then a suitable application of Lemma 2.1 and Lemma 2.2 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
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 .
(9)
Combining (6), (9), 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
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 x ∈[ x0 , X ] ∩T , y ∈[ y0 , N] ∩T
(10)
Setting x = X in (10), 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
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 , x ∈[x0 , X ] ∩T , y ∈[ y0 , N] ∩T
(11)
that is,
(12)
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 Lemma 2.2, (5) and (7) we obtain
NM
q
r
NM t s
m
v( x0 , y0 ) ≤ C + ∫ ∫ [ f2 (s, t )v p (s, t ) + g2 (s, t )v p (s, t )]ΔsΔt + ∫ ∫ ∫ ∫ h2 (ξ ,η )v p (ξ ,η )ΔξΔηΔsΔt
y0 x0
y0 x0 y0 x0
N M
q q− p
p − q qp
≤ C + ∫ ∫ [ f2 (s, t )( K p v(s, t ) +
K )
p
p
y0 x0
N M t s
r r− p
p − r rp
m m− p
p − m mp
K )]ΔsΔt + ∫ ∫ ∫ ∫ h2 (ξ ,η )( K p v(ξ ,η ) +
+ g2 (s, t )( K p v(s, t ) +
K )ΔξΔηΔsΔt
p
p
p
p
y0 x0 y0 x0
N M
= C + ∫ ∫ [ f2 (s, t )
y0 x0
N M t s
q q−pp
r r− p
m m− p
K v(s, t ) + g2 (s, t ) K p v(s, t )]ΔsΔt + ∫ ∫ ∫ ∫ h2 (ξ ,η ) K p v(ξ ,η )ΔξΔηΔsΔt
p
p
p
y0 x0 y0 x0
(13)
Then using (12) in (13) yields
N M
v( x0 , y0 ) ≤ C + ∫ ∫ { f2 (s, t )
y0 x0
N M
q q−pp
r r− p
K [v( x0 , y0 ) B3 (s, t ) + B4 (s, t )]}ΔsΔt + ∫ ∫ {g2 (s, t ) K p [v( x0 , y0 )B3 (s, t ) + B4 (s, t )]}ΔsΔt
p
p
y0 x0
N M t s
+ ∫ ∫ ∫ ∫ h2 (ξ ,η )
y0 x0 y0 x0
m m−p p
K v[v( x0 , y0 )B3 (ξ ,η ) + B4 (ξ ,η )]ΔξΔηΔsΔt
p
y x
N M t s
q q−p
r r− p
m m− p
= C + v(x0 , y0 ){∫ ∫ [ f2 (s, t) K p B3 (s, t) + g2 (s, t) K p B3 (s, t)]ΔsΔt + ∫ ∫ ∫ ∫ h2 (ξ ,η ) K p B3 (ξ ,η )ΔξΔη ]ΔsΔt}
p
p
p
y0 x0
y0 x0 y0 x0
y x
+ ∫ ∫ [ f2 (s, t )
y0 x0
N M t s
q q−pp
r r− p
m m− p
K B4 (s, t ) + g2 (s, t ) K p 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
(14)
which is followed by
v(x0 , y0 ) ≤
C + B6
1− B5
(15)
Combining (4), (12) and (15) we can obtain the desired inequality (3).
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