Journal of Inequalities in Pure and
Applied Mathematics
ON WEIGHTED OSTROWSKI TYPE INEQUALITIES FOR OPERATORS
AND VECTOR-VALUED FUNCTIONS
volume 3, issue 1, article 12,
2002.
1
N.S. BARNETT, 2 C. BUSE, 1 P. CERONE AND 1 S.S. DRAGOMIR
1 School
of Communications and Informatics
Victoria University of Technology
PO Box 14428
Melbourne City MC
8001 Victoria, Australia
EMail: neil@matilda.vu.edu.au
URL: http://sci.vu.edu.au/staff/neilb.html
2 Department
of Mathematics
West University of Timişoara
Bd. V. Pârvan 4
1900 Timişoara, România.
EMail:buse@hilbert.math.uvt.ro
URL: http://rgmia.vu.edu.au/BuseCVhtml/
EMail:pc@matilda.vu.edu.au
URL: http://rgmia.vu.edu.au/cerone
EMail:sever@matilda.vu.edu.au
URL: http://rgmia.vu.edu.au/SSDragomirWeb.html
Received 22 June, 2001;
accepted 25 October, 2001.
Communicated by: Th. M. Rassias
Abstract
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2000
School of Communications and Informatics,Victoria University of Technology
ISSN (electronic): 1443-5756
050-01
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Abstract
Some weighted Ostrowski type integral inequalities for operators and vectorvalued functions in Banach spaces are given. Applications for linear operators
in Banach spaces and differential equations are also provided.
On Weighted Ostrowski Type
Inequalities for Operators and
Vector-Valued Functions
2000 Mathematics Subject Classification: Primary 26D15; Secondary 46B99
Key words: Ostrowski’s Inequality, Vector-Valued Functions, Bochner Integral.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Weighted Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Inequalities for Linear Operators . . . . . . . . . . . . . . . . . . . . . . . .
4
Quadrature Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
Application for Differential Equations in Banach Spaces . . . .
6
Some Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
N.S. Barnett, C. Buşe, P. Cerone
and S.S. Dragomir
3
5
14
18
29
34
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1.
Introduction
In [12], Pečarić and Savić obtained the following Ostrowski type inequality for
weighted integrals (see also [7, Theorem 3]):
Theorem 1.1. Let w : [a, b] → [0, ∞) be a weight function on [a, b] . Suppose
that f : [a, b] → R satisfies
(1.1)
|f (t) − f (s)| ≤ N |t − s|α , for all t, s ∈ [a, b] ,
where N > 0 and 0 < α ≤ 1 are some constants. Then for any x ∈ [a, b]
Rb
Rb
|t − x|α w (t) dt
w
(t)
f
(t)
dt
a
.
(1.2)
≤ N · a Rb
f (x) − R b
w
(t)
dt
w
(t)
dt
a
a
Further, if for some constants c and λ
N.S. Barnett, C. Buşe, P. Cerone
and S.S. Dragomir
Title Page
0 < c ≤ w (t) ≤ λc, for all t ∈ [a, b] ,
then for any x ∈ [a, b], we have
Rb
w
(t)
f
(t)
dt
λL (x) J (x)
a
(1.3)
f
(x)
−
,
≤N·
Rb
L (x) − J (x) + λJ (x)
w
(t)
dt
a
where
On Weighted Ostrowski Type
Inequalities for Operators and
Vector-Valued Functions
α
1
a + b L (x) :=
(b − a) + x −
2
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and
1+α
J (x) :=
1+α
(x − a)
+ (b − x)
(1 + α) (b − a)
.
J. Ineq. Pure and Appl. Math. 3(1) Art. 12, 2002
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The inequality (1.2) was rediscovered in [4] where further applications for
different weights and in Numerical Analysis were given.
For other results in connection to weighted Ostrowski inequalities, see [3],
[8] and [10].
In the present paper we extend the weighted Ostrowski’s inequality for vectorvalued functions and Bochner integrals and apply the obtained results to operatorial inequalities and linear differential equations in Banach spaces. Some
numerical experiments are also conducted.
On Weighted Ostrowski Type
Inequalities for Operators and
Vector-Valued Functions
N.S. Barnett, C. Buşe, P. Cerone
and S.S. Dragomir
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2.
Weighted Inequalities
Let X be a Banach space and −∞ < a < b < ∞. We denote by L (X)
the Banach algebra of all bounded linear operators acting on X. The norms of
vectors or operators acting on X will be denoted by k·k .
A function f : [a, b] → X is called measurable if there exists a sequence
of simple functions fn : [a, b] → X which converges punctually almost everywhere on [a, b] at f. We recall also that a measurable function f : [a, b] → X
is Bochner integrable if and only if its norm function (i.e. the function t 7→
kf (t)k : [a, b] → R+ ) is Lebesgue integrable on [a, b].
The following theorem holds.
Theorem 2.1. Assume that B : [a, b] → L (X) is Hölder continuous on [a, b],
i.e.,
On Weighted Ostrowski Type
Inequalities for Operators and
Vector-Valued Functions
N.S. Barnett, C. Buşe, P. Cerone
and S.S. Dragomir
kB (t) − B (s)k ≤ H |t − s|α for all t, s ∈ [a, b] ,
Title Page
where H > 0 and α ∈ (0, 1].
If f : [a, b] → X is Bochner integrable on [a, b], then we have the inequality:
Contents
(2.1)
(2.2)
Z b
Z b
Z b
B (t)
f (s) ds −
B (s) f (s) ds
|t − s|α kf (s)k ds
≤H
a
a
a

(b−t)α+1 +(t−a)α+1

k|f |k[a,b],∞
if
f ∈ L∞ ([a, b] ; X) ;

α+1




 h
i1
(b−t)qα+1 +(t−a)qα+1 q
≤H×
k|f |k[a,b],p
if
p > 1, p1 + 1q = 1
qα+1




and f ∈ Lp ([a, b] ; X) ;


 1 (b − a) + t − a+b α k|f |k
[a,b],1
2
2
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for any t ∈ [a, b], where
|kf k|[a,b],∞ := ess sup kf (t)k
t∈[a,b]
and
Z
|kf k|[a,b],p :=
b
p1
kf (t)kp dt ,
p ≥ 1.
a
Proof. Firstly, we prove that the X−valued function s 7→ B (s) f (s) is Bochner
integrable on [a, b]. Indeed, let (fn ) be a sequence of X−valued, simple functions which converge almost everywhere on [a, b] at the function f . The maps
s 7→ B (s) fn (s) are measurable (because they are continuous with the exception of a finite number of points s in [a, b]). Then
On Weighted Ostrowski Type
Inequalities for Operators and
Vector-Valued Functions
N.S. Barnett, C. Buşe, P. Cerone
and S.S. Dragomir
kB (s) fn (s) − B (s) f (s)k ≤ kB (s)k kfn (s) − f (s)k → 0 a.e. on [a, b]
Title Page
when n → ∞ so that the function s 7→ B (s) f (s) : [a, b] → X is measurable.
Now, using the estimate
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kB (s) f (s)k ≤ sup kB (ξ)k · kf (s)k , for all s ∈ [a, b] ,
ξ∈[a,b]
it is easy to see that the function s 7→ B (s) f (s) is Bochner integrable on [a, b].
We have successively
Z b
Z b
Z b
B (t)
f
(s)
ds
−
B
(s)
f
(s)
ds
=
(B
(t)
−
B
(s))
f
(s)
ds
a
a
a
Z b
≤
k(B (t) − B (s)) f (s)k ds
a
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Z
≤
b
Z
k(B (t) − B (s))k kf (s)k ds ≤ H
a
b
|t − s|α kf (s)k ds =: M (t)
a
for any t ∈ [a, b], proving the first inequality in (2.2).
Now, observe that
Z b
M (t) ≤ H k|f |k[a,b],∞
|t − s|α ds
a
(b − t)α+1 + (t − a)α+1
α+1
and the first part of the second inequality is proved.
Using Hölder’s integral inequality, we may state that
p1
Z b
1q Z b
qα
p
kf (s)k ds
M (t) ≤ H
|t − s| ds
= H k|f |k[a,b],∞ ·
a
a
"
= H
qα+1
(b − t)
qα+1
+ (t − a)
qα + 1
N.S. Barnett, C. Buşe, P. Cerone
and S.S. Dragomir
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# 1q
k|f |k[a,b],p ,
proving the second part of the second inequality.
Finally, we observe that
Z b
α
M (t) ≤ H sup |t − s|
kf (s)k ds
s∈[a,b]
On Weighted Ostrowski Type
Inequalities for Operators and
Vector-Valued Functions
a
= H max {(b − t)α , (t − a)α } k|f |k[a,b],1
α
1
a
+
b
k|f |k
= H
(b − a) + t −
[a,b],1
2
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and the theorem is proved.
The following corollary holds.
Corollary 2.2. Assume that B : [a, b] → L (X) is Lipschitzian with the constant
L > 0. Then we have the inequality
Z b
Z b
B (t)
(2.3)
f
(s)
ds
−
B
(s)
f
(s)
ds
a
Z
a
b
≤L
|t − s| kf (s)k ds
h
i
2
a+b 2
1

(b − a) + t − 2
k|f |k[a,b],∞ if
f ∈ L∞ ([a, b] ; X) ;


4



 h
i1
(b−t)q+1 +(t−a)q+1 q
≤L×
k|f |k[a,b],p
if
p > 1, p1 + 1q = 1
q+1




and f ∈ Lp ([a, b] ; X) ;


 1 (b − a) + t − a+b k|f |k
[a,b],1
2
2
a
for any t ∈ [a, b].
Remark 2.1. If we choose t = a+b
in (2.2) and (2.3), then we get the following
2
midpoint inequalities:
Z b
Z b
a
+
b
(2.4) B
f
(s)
ds
−
B
(s)
f
(s)
ds
2
a
a
α
Z b
a
+
b
s −
kf (s)k ds
≤H
2 a
On Weighted Ostrowski Type
Inequalities for Operators and
Vector-Valued Functions
N.S. Barnett, C. Buşe, P. Cerone
and S.S. Dragomir
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≤H×







1
2α (α+1)
(b − a)α+1 k|f |k[a,b],∞
1
1
2α (qα+1) q






1
2α
1
if
(b − a)α+ q k|f |k[a,b],p if
(b − a)α k|f |k[a,b],1
f ∈ L∞ ([a, b] ; X) ;
p > 1,
1
p
+
1
q
=1
and f ∈ Lp ([a, b] ; X) ;
and
Z b
Z b
a+b
(2.5) B
f (s) ds −
B (s) f (s) ds
2
a
a
Z b
s − a + b kf (s)k ds
≤L
2 a
 1
(b − a)2 k|f |k[a,b],∞
if
f ∈ L∞ ([a, b] ; X) ;


4




1+ 1q
1
k|f |k[a,b],p if
p > 1, p1 + 1q = 1
1 (b − a)
≤L×
q
2(q+1)



and f ∈ Lp ([a, b] ; X) ;


 1 (b − a) k|f |k
[a,b],1
2
respectively.
Rb
Remark 2.2. Consider the function Ψα : [a, b] → R, Ψα (t) := a |t − s|α kf (s)k ds,
α ∈ (0, 1). If f is continuous on [a, b], then Ψα is differentiable and
Z t
Z b
dΨα (t)
d
α
α
=
(t − s) kf (s)k ds +
(s − t) kf (s)k ds
dt
dt a
t
Z t
Z b
kf (s)k
kf (s)k
= α
1−α ds −
1−α ds .
a (t − s)
t (s − t)
On Weighted Ostrowski Type
Inequalities for Operators and
Vector-Valued Functions
N.S. Barnett, C. Buşe, P. Cerone
and S.S. Dragomir
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If t0 ∈ (a, b) is such that
Z
t0
kf (s)k
ds =
(t0 − s)1−α
a
Z
b
t0
kf (s)k
ds
(s − t0 )1−α
and Ψ0 (·) is negative on (a, t0 ) and positive on (t0 , b) , then the best inequality
we can get in the first part of (2.2) is the following one
Z b
Z b
Z b
(2.6) f (s) ds −
B (s) f (s) ds
|t0 − s|α kf (s)k ds.
B (t0 )
≤H
a
a
On Weighted Ostrowski Type
Inequalities for Operators and
Vector-Valued Functions
a
N.S. Barnett, C. Buşe, P. Cerone
and S.S. Dragomir
If α = 1, then, for
Z
b
|t − s| kf (s)k ds,
Ψ (t) :=
a
Title Page
we have
Contents
Z
t
b
Z
dΨ (t)
=
kf (s)k ds −
kf (s)k ds, t ∈ (a, b) ,
dt
a
t
d2 Ψ (t)
= 2 kf (t)k ≥ 0, t ∈ (a, b) ,
dt2
tm
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Z
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b
kf (s)k ds =
a
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which shows that Ψ is convex on (a, b).
If tm ∈ (a, b) is such that
Z
JJ
J
kf (s)k ds,
tm
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then the best inequality we can get from the first part of (2.3) is
Z b
Z b
(2.7) B (tm )
f (s) ds −
B (s) f (s) ds
a
a
Z b
≤L
sgn (s − tm ) s kf (s)k ds.
a
Indeed, as
Z
inf
t∈[a,b]
a
b
|t − s| kf (s)k ds
Z b
=
|tm − s| kf (s)k ds
a
Z tm
Z b
=
(tm − s) kf (s)k ds +
(s − tm ) kf (s)k ds
a
tm
Z tm
Z b
= tm
kf (s)k ds −
kf (s)k ds
a
tm
Z
b
Z
s kf (s)k ds −
+
tm
Z
b
tm
Z b
s kf (s)k ds
a
Z
a
sgn (s − tm ) s kf (s)k ds,
=
N.S. Barnett, C. Buşe, P. Cerone
and S.S. Dragomir
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tm
s kf (s)k ds
s kf (s)k ds −
=
tm
On Weighted Ostrowski Type
Inequalities for Operators and
Vector-Valued Functions
a
then the best inequality we can get from the first part of (2.3) is obtained for
t = tm ∈ (a, b).
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We recall that a function F : [a, b] → L (X) is said to be strongly continuous
if for all x ∈ X, the maps s 7→ F (s) x : [a, b] → X are continuous on [a, b].
In this case the function s 7→ kB (s)k : [a, b] → R+ is (Lebesgue) measurable
Rb
and bounded ([6]). The linear operator L = a F (s) ds (defined by Lx :=
Rb
F (s) xds for all x ∈ X) is bounded, because
a
Z
kLxk ≤
b
kF (s)k ds · kxk for all x ∈ X.
On Weighted Ostrowski Type
Inequalities for Operators and
Vector-Valued Functions
a
In a similar manner to Theorem 2.1, we may prove the following result as
well.
N.S. Barnett, C. Buşe, P. Cerone
and S.S. Dragomir
Theorem 2.3. Assume that f : [a, b] → X is Hölder continuous, i.e.,
Title Page
kf (t) − f (s)k ≤ K |t − s|β for all t, s ∈ [a, b] ,
(2.8)
Contents
where K > 0 and β ∈ (0, 1].
If B : [a, b] → L (X) is strongly continuous on [a, b] , then we have the
inequality:
(2.9)
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Z b
Z b
B
(s)
ds
f
(t)
−
B
(s)
f
(s)
ds
a
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a
Z
≤K
b
β
|t − s| kB (s)k ds
a
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
(b−t)β+1 +(t−a)β+1

k|B|k[a,b],∞
if
kB (·)k ∈ L∞ ([a, b] ; R+ ) ;

β+1




 h
i1
(b−t)qβ+1 +(t−a)qβ+1 q
≤ K×
k|B|k[a,b],p
if
p > 1, p1 + 1q = 1
qβ+1



and kB (·)k ∈ Lp ([a, b] ; R+ ) ;



 1 (b − a) + t − a+b β k|B|k
[a,b],1
2
2
for any t ∈ [a, b].
The following corollary holds.
Corollary
2.4. Assume that f and B are as in Theorem 2.3. If, in addition,
Rb
B (s) ds is invertible in L (X) , then we have the inequality:
a
(2.10)
Z b
−1 Z b
B (s) ds
B (s) f (s) ds
f (t) −
a
a
Z
−1 Z b
b
≤K
B (s) ds
|t − s|β kB (s)k ds
a
a
for any t ∈ [a, b].
Remark 2.3. It is obvious that the inequality (2.10) contains as a particular case
what is the so called Ostrowski’s inequality for weighted integrals (see (1.2)).
On Weighted Ostrowski Type
Inequalities for Operators and
Vector-Valued Functions
N.S. Barnett, C. Buşe, P. Cerone
and S.S. Dragomir
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3.
Inequalities for Linear Operators
Let 0 ≤ a < b < ∞ and A ∈ L (X). We recall that the operatorial norm of A
is given by
kAk = sup {kAxk : kxk ≤ 1} .
The resolvent set of A (denoted by ρ (A)) is the set of all complex scalars λ for
which λI − A is an invertible operator. Here I is the identity operator in L (X).
The complementary set of ρ (A) in the complex plane, denoted by σ (A), is
the
Pspectrum
of A. It is known that σ (A) is a compact set in C. The series
(tA)n
converges absolutely and locally uniformly for t ∈ R. If we
n≥0 n!
denote by etA its sum, then
tA e ≤ e|t|kAk , t ∈ R.
Proposition 3.1. Let X be a real or complex Banach space, A ∈ L (X) and β
be a non-null real number such that −β ∈ ρ (A). Then for all 0 ≤ a < b < ∞
and each s ∈ [a, b], we have
βb
e − eβa sA
b(βI+A)
−1
a(βI+A)
(3.1) · e − (βI + A)
e
−e
β
#
"
2
1
a+b
2
bkAk
≤ kAk e
·
(b − a) + s −
· max eβb , eβa .
4
2
On Weighted Ostrowski Type
Inequalities for Operators and
Vector-Valued Functions
N.S. Barnett, C. Buşe, P. Cerone
and S.S. Dragomir
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Proof. We apply the second inequality from Corollary 2.2 in the following particular case.
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B (τ ) := eτ A , f (τ ) = eβτ x, τ ∈ [a, b] , x ∈ X.
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For all ξ, η ∈ [a, b] there exists an α between ξ and η such that
∞
X
n
n
(ξ
−
η
)
kB (ξ) − B (η)k = An n=1
n!
∞
n
X
(αA)
= (ξ − η) A
n!
αA n=0
≤ kAk e · |ξ − η|
≤ kAk ebkAk · |ξ − η| .
The function τ 7→ eτ A is thus Lipschitzian on [a, b] with the constant L :=
kAk eb·kAk . On the other hand we have
Z
b
τA
e
βτ
Z
τA
e
βτ
e Ix dτ
a
Z
N.S. Barnett, C. Buşe, P. Cerone
and S.S. Dragomir
Title Page
b
e x dτ =
a
On Weighted Ostrowski Type
Inequalities for Operators and
Vector-Valued Functions
b
eτ (A+βI) xdτ
a
= (A + βI)−1 eb(A+βI) − ea(A+βI) x,
=
and
|kf k|[a,b],∞ = sup eτ β x = max eβb , eβa · kxk .
τ ∈[a,b]
Placing all the above results in the second inequality from (2.3) and taking the
supremum for all x ∈ X, we will obtain the desired inequality (3.1).
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Remark 3.1. Let A ∈ L (X) such that 0 ∈ ρ (A). Taking the limit as β → 0 in
(3.1), we get the inequality
(b − a) esA − A−1 ebA − eaA 2 #
1
a
+
b
≤ kAk ebkAk ·
(b − a)2 + s −
,
4
2
"
On Weighted Ostrowski Type
Inequalities for Operators and
Vector-Valued Functions
where a, b and s are as in Proposition 3.1.
Proposition 3.2. Let A ∈ L (X) be an invertible operator, t ≥ 0 and 0 ≤ s ≤ t.
Then the following inequality holds:
(3.2)
2
t
−2
sin (sA) − A [sin (tA) − tA cos (tA)]
2
≤
Title Page
2s3 + 2t3 − 3st2
kAk .
6
In particular, if X = R, A = 1 and s = 0 it follows the scalar inequality
|sin t − t cos t| ≤
N.S. Barnett, C. Buşe, P. Cerone
and S.S. Dragomir
t3
, for all t ≥ 0.
3
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Proof. We apply the inequality from (2.3) in the following particular case:
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∞
X
Page 16 of 42
B (τ ) = sin (τ A) :=
n=0
(−1)n
(τ A)2n+1
, τ ≥ 0,
(2n + 1)!
J. Ineq. Pure and Appl. Math. 3(1) Art. 12, 2002
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and
f (τ ) = τ · x, for fixed x ∈ X.
(3.3)
For each ξ, η ∈ [0, t] , we have
!
∞
2n
X
(ξ
−
η)
α
kB (ξ) − B (η)k = A
A2n (−1)n
(2n)!
n=0
≤ kAk |ξ − η| · kcos (αA)k
≤ kAk |ξ − η| ,
where α is a real number between ξ and η, i.e., the function τ 7−→ B (τ ) :
R+ → L (X) is kAk −Lipschitzian.
Moreover, it is easy to see that
Z t
B (τ ) f (τ ) dτ = A−2 [sin (tA) − tA cos (tA)] x
0
and
Z
(3.4)
0
t
2s3 + 2t3 − 3st2
|s − τ | |f (τ )| dτ =
kxk .
6
On Weighted Ostrowski Type
Inequalities for Operators and
Vector-Valued Functions
N.S. Barnett, C. Buşe, P. Cerone
and S.S. Dragomir
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Applying the first inequality from (2.3) and taking the supremum for x ∈ X
with kxk ≤ 1, we get (3.2).
Page 17 of 42
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4.
Quadrature Formulae
Consider the division of the interval [a, b] given by
In : a = t0 < t1 < · · · < tn−1 < tn = b
(4.1)
and hi := ti+1 − ti , ν (h) := max hi . For the intermediate points ξ :=
i=0,n−1
(ξ0 , . . . , ξn−1 ) with ξi ∈ [ti , ti+1 ] , i = 0, n − 1, define the sum
Sn(1)
(4.2)
(B, f ; In , ξ) :=
n−1
X
i=0
Z
On Weighted Ostrowski Type
Inequalities for Operators and
Vector-Valued Functions
ti+1
B (ξi )
f (s) ds.
N.S. Barnett, C. Buşe, P. Cerone
and S.S. Dragomir
ti
Then we may state the following result in approximating the integral
Z b
B (s) f (s) ds,
a
Theorem 4.1. Assume that B : [a, b] → L (X) is Hölder continuous on [a, b],
i.e., it satisfies the condition (2.1) and f : [a, b] → X is Bochner integrable on
[a, b]. Then we have the representation
Sn(1)
(B, f ; In , ξ) +
Rn(1)
(B, f ; In , ξ) ,
a
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b
B (s) f (s) ds =
(4.3)
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based on Theorem 2.1.
Z
Title Page
(1)
where Sn (B, f ; In , ξ) is as given by (4.2) and the remainder Rn (B, f ; In , ξ)
Page 18 of 42
J. Ineq. Pure and Appl. Math. 3(1) Art. 12, 2002
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satisfies the estimate
(1)
Rn (B, f ; In , ξ)

n−1
P
1
α+1
α+1 

(t
+
(ξ
k|f
|k
i+1 − ξi )
i − ti )

[a,b],∞

α+1

i=0



n−1
1


q
P
1

qα+1
qα+1

(ti+1 − ξi )
+ (ξi − ti )
,
1 k|f |k[a,b],p
i=0
≤H×
(qα + 1) q


p > 1, p1 + 1q = 1





α

 1

t
+
t
i+1
i

k|f |k

ν (h) + max ξi −
[a,b],1
2
2
i=0,n−1

n−1
P α+1
1


k|f
|k
hi

[a,b],∞

α
+
1

i=0


n−1
1

P qα+1 q
1
≤H×
hi
,
p > 1,
1 k|f |k[a,b],p

q

i=0
(qα
+
1)

α



1
t
+
t
i+1
i

k|f |k

ν (h) + max ξi −
[a,b],1
2
2
i=0,n−1
On Weighted Ostrowski Type
Inequalities for Operators and
Vector-Valued Functions
N.S. Barnett, C. Buşe, P. Cerone
and S.S. Dragomir
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1
p
+
1
q
=1
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
1


k|f |k[a,b],∞ [ν (h)]α


α
+
1



1
(b − a) q
≤H×
α

1 k|f |k[a,b],p [ν (h)]



(qα + 1) q


 k|f |k
[ν (h)]α .
[a,b],1
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J. Ineq. Pure and Appl. Math. 3(1) Art. 12, 2002
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Proof. Applying Theorem 4.1 on [xi , xi+1 ] i = 0, n − 1 , we may write that
Z
ti+1
ti
Z
ti+1
f (s) ds
B (s) f (s) ds − B (ξi )
 h ti α+1
i
+(ξi −ti )α+1
(ti+1 −ξi )

k|f |k[ti ,ti+1 ],∞


α+1




 h
i1
(ti+1 −ξi )qα+1 +(ξi −ti )qα+1 q
≤H×
k|f |k[ti ,ti+1 ],p

qα+1






 1 (t − t ) + ξ − ti+1 +ti α k|f |k
i+1
i
i
[ti ,ti+1 ],1 .
2
2
Summing over i from 0 to n − 1 and using the generalised triangle inequality
we get
(1)
Rn (B, f ; In , ξ)
Z ti+1
n−1 Z ti+1
X
≤
B
(s)
f
(s)
ds
−
B
(ξ
)
f
(s)
ds
i
i=0
≤H×
ti










ti
1
α+1
n−1
P
i=0
(ti+1 − ξi )α+1 + (ξi − ti )α+1 k|f |k[ti ,ti+1 ],∞
n−1
P
1
(ti+1 − ξi )qα+1 + (ξi − ti )qα+1
1

q
(qα+1)

i=0



n−1

P 1


ξi − ti+1 +ti α k|f |k
h
+

i
[ti ,ti+1 ],1 .
2
2
i=0
1q
k|f |k[ti ,ti+1 ],p
On Weighted Ostrowski Type
Inequalities for Operators and
Vector-Valued Functions
N.S. Barnett, C. Buşe, P. Cerone
and S.S. Dragomir
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Now, observe that
n−1
X
(ti+1 − ξi )α+1 + (ξi − ti )α+1 k|f |k[ti ,ti+1 ],∞
i=0
n−1
X
≤ k|f |k[a,b],∞
(ti+1 − ξi )α+1 + (ξi − ti )α+1
≤ k|f |k[a,b],∞
i=0
n−1
X
On Weighted Ostrowski Type
Inequalities for Operators and
Vector-Valued Functions
hα+1
i
i=0
≤ k|f |k[a,b],∞ (b − a) [ν (h)]α .
N.S. Barnett, C. Buşe, P. Cerone
and S.S. Dragomir
Using the discrete Hölder inequality, we may write that
" n−1
# 1q
X
(ti+1 − ξi )qα+1 + (ξi − ti )qα+1 k|f |k[ti ,ti+1 ],p
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i=0
1
q
#
" n−1
" n−1
#
X X
1 q
qα+1
qα+1 q
p
≤
(ti+1 − ξi )
+ (ξi − ti )
×
k|f |k[ti ,ti+1 ],p
i=0
i=0
( n−1
) 1q Z
p1
b
X
qα+1
qα+1
p
=
(ti+1 − ξi )
+ (ξi − ti )
kf (t)k ds
a
i=0
≤
n−1
X
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! 1q
hqα+1
i
1
p
k|f |kp[a,b],p
Page 21 of 42
i=0
1
≤ (b − a) q k|f |k[a,b],p [ν (h)]α .
J. Ineq. Pure and Appl. Math. 3(1) Art. 12, 2002
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Finally, we have
n−1 X
1
i=0
α
t
+
t
i+1
i
k|f |k
hi + ξi −
[ti ,ti+1 ],1
2
2
α
1
t
+
t
i+1
i
k|f |k
≤
max hi + max ξi −
[a,b],1
2 i=0,n−1
2
i=0,n−1
≤ [ν (h)]α k|f |k[a,b],1
and the theorem is proved.
The following corollary holds.
Corollary 4.2. If B is Lipschitzian with the constant L, then we have the rep(1)
resentation (4.3) and the remainder Rn (B, f ; In , ξ) satisfies the estimates:
(1)
Rn (B, f ; In , ξ)
(4.4)
n−1

P 2 n−1
P
ti+1 +ti 2

1

k|f |k[a,b],∞ 4
hi +
ξi − 2



i=0
i=0


n−1
1q


P

q+1
q+1 
1
(ti+1 − ξi )
+ (ξi − ti )
,
1 k|f |k[a,b],p
(q+1) q
≤L×
i=0


p > 1, p1 + 1q = 1







t
+t
1

i+1
i
 2 ν (h) + max ξi − 2 k|f |k[a,b],1
On Weighted Ostrowski Type
Inequalities for Operators and
Vector-Valued Functions
N.S. Barnett, C. Buşe, P. Cerone
and S.S. Dragomir
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i=0,n−1
J. Ineq. Pure and Appl. Math. 3(1) Art. 12, 2002
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≤L×









1
2
k|f |k[a,b],∞
n−1
P
h2i
i=0
n−1
P
1
hq+1
i
1q
1 k|f |k[a,b],p

(q+1) q

i=0




ti+1 +ti 1

 2 ν (h) + max ξi − 2
k|f |k[a,b],1
i=0,n−1
 1
k|f |k[a,b],∞ (b − a) ν (h)

2




1
(b−a) q
≤L×
1 k|f |k[a,b],p ν (h)

(q+1) q




k|f |k[a,b],1 ν (h) .
The second possibility we have for approximating the integral
is embodied in the following theorem based on Theorem 2.3.
Rb
a
On Weighted Ostrowski Type
Inequalities for Operators and
Vector-Valued Functions
N.S. Barnett, C. Buşe, P. Cerone
and S.S. Dragomir
B (s) f (s) ds
Theorem 4.3. Assume that f : [a, b] → X is Hölder continuous, i.e., the condition (2.8) holds. If B : [a, b] → L (X) is strongly continuous on [a, b], then we
have the representation:
Z b
(4.5)
B (s) f (s) ds = Sn(2) (B, f ; In , ξ) + Rn(2) (B, f ; In , ξ) ,
a
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where
(4.6)
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Page 23 of 42
Sn(2) (B, f ; In , ξ) :=
n−1 Z ti+1
X
i=0
ti
B (s) ds f (ξi )
J. Ineq. Pure and Appl. Math. 3(1) Art. 12, 2002
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(2)
and the remainder Rn (B, f ; In , ξ) satisfies the estimate:
(2)
Rn (B, f ; In , ξ)
(4.7)
≤K×













1
β+1
k|B|k[a,b],∞
1
1
(qβ+1) q
n−1
Ph
(ti+1 − ξi )β+1 + (ξi − ti )β+1
i=0
k|B|k[a,b],p
n−1 h
P
i=0











 12 ν (h) + max ξi −
i=0,n−1
≤K×










i
qβ+1
(ti+1 − ξi )
qβ+1
+ (ξi − ti )
i 1q
p > 1, p1 + 1q = 1
β
ti+1 +ti k|B|k[a,b],1
2
,
On Weighted Ostrowski Type
Inequalities for Operators and
Vector-Valued Functions
N.S. Barnett, C. Buşe, P. Cerone
and S.S. Dragomir
Title Page
1
β+1
1
k|B|k[a,b],∞
n−1
P
hβ+1
i
Contents
i=0
k|B|k[a,b],p
n−1
P
hqβ+1
i
1q
,
p > 1,
1
p
+
1

(qβ+1) q
i=0




β

ti+1 +ti 
1

k|B|k[a,b],1
 2 ν (h) + max ξi − 2
i=0,n−1
 1
k|B|k[a,b],∞ (b − a) [ν (h)]β


β+1



1
β
(b−a) q
≤K×
p > 1, p1 + 1q = 1
1 k|B|k[a,b],p [ν (h)] ,

q
(qβ+1)




k|B|k[a,b],1 [ν (h)]β .
1
q
= 1;
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If we consider the quadrature
(4.8)
Mn(1)
(B, f ; In ) :=
n−1
X
i=0
B
ti + ti+1
2
Z
ti+1
f (s) ds,
ti
then we have the representation
Z b
(4.9)
B (s) f (s) ds = Mn(1) (B, f ; In ) + Rn(1) (B, f ; In ) ,
a
On Weighted Ostrowski Type
Inequalities for Operators and
Vector-Valued Functions
(1)
Rn
and the remainder
(B, f ; In ) satisfies the estimate:
(1)
(4.10) Rn (B, f ; In )

n−1
P α+1

1


k|f
|k
hi
α (α+1)

[a,b],∞
2

i=0





n−1
1
P qα+1 q
≤H×
1

, p > 1,
hi
1 k|f |k[a,b],p


 2α (qα+1) q
i=0





 1 [ν (h)]α k|f |k
[a,b],1
2α
N.S. Barnett, C. Buşe, P. Cerone
and S.S. Dragomir
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p
+
1
q
= 1;
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≤H×








1
2α (α+1)
α
(b − a) k|f |k[a,b],∞ [ν (h)]
Quit
1
(b−a) q
α
1







Close
2α (qα+1) q
1
2α
k|f |k[a,b],p [ν (h)] ,
k|f |k[a,b],1 [ν (h)]α ,
p > 1,
1
p
+
1
q
= 1;
Page 25 of 42
J. Ineq. Pure and Appl. Math. 3(1) Art. 12, 2002
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provided that B and f are as in Theorem 4.1.
Now, if we consider the quadrature
n−1 Z ti+1
X
ti + ti+1
(2)
(4.11)
Mn (B, f ; In ) :=
B (s) ds f
,
2
t
i
i=0
then we also have
Z b
(4.12)
B (s) f (s) ds = Mn(2) (B, f ; In ) + Rn(2) (B, f ; In ) ,
On Weighted Ostrowski Type
Inequalities for Operators and
Vector-Valued Functions
a
and in this case the remainder satisfies the bound
(2)
Rn (B, f ; In )
(4.13)
≤K×








1
2β (β+1)
k|B|k[a,b],∞
1







1
2β (qβ+1) q






1
2β (β+1)
1
2β
n−1
P
N.S. Barnett, C. Buşe, P. Cerone
and S.S. Dragomir
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hβ+1
i
Contents
i=0
k|B|k[a,b],p
n−1
P
hqβ+1
i
1q
,
p > 1,
i=0
1
p
+
β
[ν (h)] k|B|k[a,b],1
1
q
= 1;
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≤K×





(b − a) k|B|k[a,b],∞ [ν (h)]
1
(b−a) q
1
2β (qβ+1) q
1
2β
β
k|B|k[a,b],p [ν (h)]β ,
k|B|k[a,b],1 [ν (h)]β ,
p > 1,
Quit
1
p
+
1
q
= 1;
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provided B and f satisfy the hypothesis of Theorem 4.3.
Now, if we consider the equidistant partitioning of [a, b],
b−a
En : ti := a +
· i, i = 0, n,
n
(1)
then Mn (B, f ; En ) becomes
(4.14) Mn(1) (B, f ) :=
n−1
X
i=0
1
B a+ i+
2
b−a
·
n
Z
a+ b−a
·(i+1)
n
f (s) ds
a+ b−a
·i
n
N.S. Barnett, C. Buşe, P. Cerone
and S.S. Dragomir
and then
Z
On Weighted Ostrowski Type
Inequalities for Operators and
Vector-Valued Functions
b
B (s) f (s) ds = Mn(1) (B, f ) + Rn(1) (B, f ) ,
(4.15)
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a
Contents
where the remainder satisfies the bound
 (b−a)α+1

k|f |k[a,b],∞

2α (α+1)nα



α+ 1
(b−a) q
(4.16) Rn(1) (B, f ) ≤ H ×
k|f |k[a,b],p ,
2α (α+1)nα




 (b−a)α
k|f |k[a,b],1 .
2α nα
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p > 1,
Also, we have
1
p
+
1
q
= 1;
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Z
(4.17)
a
b
B (s) f (s) ds = Mn(2) (B, f ) + Rn(2) (B, f ) ,
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where
Mn(2)
(B, f ) :=
Z
n−1
X
i=0
a+ b−a
·(i+1)
n
a+ b−a
·i
n
! 1
b−a
B (s) ds f a + i +
·
,
2
n
(2)
and the remainder Rn (B, f ) satisfies the estimate

(b−a)β+1


β (β+1)nβ kBk[a,b],∞

2



β+ 1
(b−a) q
(4.18) Rn(2) (B, f ) ≤ K ×
k|B|k[a,b],p ,
β
2 (β+1)nβ





 (b−a)β k|B|k
[a,b],1 .
2β nβ
On Weighted Ostrowski Type
Inequalities for Operators and
Vector-Valued Functions
p > 1,
1
p
+
1
q
= 1;
N.S. Barnett, C. Buşe, P. Cerone
and S.S. Dragomir
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5.
Application for Differential Equations in Banach
Spaces
We recall that a family of operators U = {U (t, s) : t ≥ s} ⊂ L (X) with t, s ∈
R or t, s ∈ R+ is called an evolution family if:
(i) U (t, t) = I and U (t, s) U (s, τ ) = U (t, τ ) for all t ≥ s ≥ τ ; and
(ii) for each x ∈ X, the function (t, s) 7−→ U (t, s) x is continuous for t ≥ s.
Here I is the identity operator in L (X) .
An evolution family {U (t, s) : t ≥ s} is said to be exponentially bounded if,
in addition,
On Weighted Ostrowski Type
Inequalities for Operators and
Vector-Valued Functions
N.S. Barnett, C. Buşe, P. Cerone
and S.S. Dragomir
(iii) there exist the constants M ≥ 1 and ω > 0 such that
Title Page
kU (t, s)k ≤ M eω(t−s) , t ≥ s.
Contents
(5.1)
Evolution families appear as solutions for abstract Cauchy problems of the
form
(5.2)
u̇ (t) = A (t) u (t) , u (s) = xs , xs ∈ D (A (s)) , t ≥ s, t, s ∈ R (or R+ ),
where the domain D (A (s)) of the linear operator A (s) is assumed to be dense
in X. An evolution family is said to solve the abstract Cauchy problem (5.2) if
for each s ∈ R there exists a dense subset Ys ⊆ D (A (s)) such that for each
xs ∈ Ys the function
t 7−→ u (t) := U (t, s) xs : [s, ∞) → X,
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is differentiable, u (t) ∈ D (A (t)) for all t ≥ s and
d
u (t) = A (t) u (t) , t ≥ s.
dt
This later definition can be found in [15]. In this definition the operators A (t)
can be unbounded. The Cauchy problem (5.2) is called well-posed if there exists
an evolution family {U (t, s) : t ≥ s} which solves it.
It is known that the well-posedness of (5.2) can be destroyed by a bounded
and continuous perturbation [13]. Let f : R →X be a locally integrable function. Consider the inhomogeneous Cauchy problem:
(5.3) u̇ (t) = A (t) u (t) + f (t) , u (s) = xs ∈ X, t ≥ s, t, s ∈ R (or R+ ).
A continuous function t 7−→ u (t) : [s, ∞) → X is said to a mild solution of
the Cauchy problem (5.3) if u (s) = xs and there exists an evolution family
{U (t, τ ) : t ≥ τ } such that
(5.4)
Z
t
U (t, τ ) f (τ ) dτ, t ≥ s, xs ∈ X, t, s ∈ R (or R+ ).
u (t) = U (t, s) xs +
s
The following theorem holds.
Theorem 5.1. Let U = {U (ν, η) : ν ≥ η} ⊂ L (X) be an evolution family and
f : R →X be a locally Bochner integrable and locally bounded function. We
assume that for all ν ∈ R (or R+ ) the function η 7−→ U (ν, η) : [ν, ∞) → L (X)
is locally Hölder continuous (i.e. for all a, b ≥ ν, a < b, there exist α ∈ (0, 1]
and H > 0 such that
kU (ν, t) − U (ν, s)k ≤ H |t − s|α , for all t, s ∈ [a, b] ).
On Weighted Ostrowski Type
Inequalities for Operators and
Vector-Valued Functions
N.S. Barnett, C. Buşe, P. Cerone
and S.S. Dragomir
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We use the notations in Section 4 for a = 0 and b = t > 0. The map u (·) from
(5.4) can be represented as
(5.5)
u (t) = U (t, 0) x0 +
n−1
X
Z
ti+1
U (t, ξi )
i=0
f (s) ds + Rn(1) (U, f, In , ξ)
ti
(1)
where the remainder Rn (U, f, In , ξ) satisfies the estimate
(1)
Rn (U, f, In , ξ) ≤
H
|kf k|[0,t],∞
α+1
n−1
X
(ti+1 − ξi )α+1 + (ξi − ti )α+1 .
i=0
On Weighted Ostrowski Type
Inequalities for Operators and
Vector-Valued Functions
N.S. Barnett, C. Buşe, P. Cerone
and S.S. Dragomir
Proof. It follows by representation (4.3) and the first estimate after it.
Moreover, if n is a natural number, i ∈ {0, . . . , n}, ti := t·i
and ξi := (2i+1)t
,
n
2n
then
Z t·(i+1)
n−1
X
n
(2i + 1) t
(5.6) u (t) = U (t, 0) x0 +
U t,
f (s) ds + Rn(1)
t·i
2n
i=0
n
(1)
and the remainder Rn satisfies the estimate
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(5.7)
(1) Rn ≤
H
tα+1
· α α |kf k|[0,t],∞ .
α+1 2 ·n
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The following theorem also holds.
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Theorem 5.2. Let U = {U (ν, η) : ν ≥ η} ⊂ L (X) be an exponentially bounded
evolution family of bounded linear operators acting on the Banach space X and
f : R →X be a locally Hölder continuous function, i.e., for all a, b ∈ R, a < b
there exist β ∈ (0, 1] and K > 0 such that (2.8) holds. We use the notations of
Section 4 for a = 0 and b = t > 0. The map u (·) from (5.4) can be represented
as
n−1 Z ti+1
X
(5.8) u (t) = U (t, 0) x0 +
U (t, τ ) dτ f (ξi ) + Rn(2) (U, f, In , ξ)
i=0
ti
(2)
N.S. Barnett, C. Buşe, P. Cerone
and S.S. Dragomir
where the remainder Rn (U, f, In , ξ) satisfies the estimate
n−1 h
i
X
(2)
β+1
β+1
Rn (U, f, In , ξ) ≤ KM eωt
(ti+1 − ξi )
+ (ξi − ti )
.
β+1
i=0
Proof. It follows from the first estimate in (4.7) for B (s) := U (t, s), using the
fact that
|kB (·)k|[0,t],∞ = sup kU (t, τ )k ≤ sup M eω(t−τ ) ≤ M eωt .
τ ∈[0,t]
On Weighted Ostrowski Type
Inequalities for Operators and
Vector-Valued Functions
τ ∈[0,t]
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(2i+1)t
2n
Moreover, if n is a natural number, i ∈ {0, . . . , n} , ti := t·i
and ξi :=
n
then
! Z t·(i+1)
n−1
X
n
(2i + 1) t
(5.9) u (t) = U (t, 0) x0 +
+ Rn(2)
U (t, τ ) dτ f
t·i
2n
i=0
n
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(2)
and the remainder Rn satisfies the estimate
(5.10)
β+1
(2) Rn ≤ KM eωt · t
.
β+1
2β · nβ
On Weighted Ostrowski Type
Inequalities for Operators and
Vector-Valued Functions
N.S. Barnett, C. Buşe, P. Cerone
and S.S. Dragomir
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6.
Some Numerical Examples
p
1. Let X = R2 , x = (ξ, η) ∈ R2 , kxk2 = ξ r + η 2 . We consider the linear
2-dimensional system

u̇1 (t) = −1 − sin2 t u1 (t) + (−1 + sin t cos t) u2 (t) + e−t ;





u̇2 (t) = (1 + sin t cos t) u1 (t) + (−1 − cos2 t) u2 (t) + e−2t ;
(6.1)





u1 (0) = u2 (0) = 0.
If we denote

2
−1 − sin t
−1 + sin t cos t
A (t) := 
2

N.S. Barnett, C. Buşe, P. Cerone
and S.S. Dragomir
,
−1 − cos t
1 + sin t cos t
On Weighted Ostrowski Type
Inequalities for Operators and
Vector-Valued Functions
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−t
−2t
f (t) = e , e
, x = (0, 0)
ξ
and we identify (ξ, η) with
, then the above system is a Cauchy problem.
η
The evolution family associated with A (t) is
U (t, s) = P (t) P −1 (s) , t ≥ s, t, s ∈ R,
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where

(6.2)
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P (t) = 
e−t cos t
−t
−e
sin t
e−2t sin t
e
−2t

 , t ∈ R.
cos t
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The exact solution of the system (6.1) is u = (u1 , u2 ) , where
u1 (t) = e−t cos t E1 (t) + e−2t sin t E2 (t)
u2 (t) = − e−t sin t E1 (t) + e−2t cos t E2 (t) , t ∈ R,
and
1
1
E1 (t) = sin t + e−t (cos t + sin t) − ,
2
2
1
1 t
E2 (t) = sin t + e (sin t − cos t) + ,
2
2
see [2, Section 4] for details. The function t 7→ A (t) is bounded on R and
therefore there exist M ≥ 1 and ω > 0
On Weighted Ostrowski Type
Inequalities for Operators and
Vector-Valued Functions
N.S. Barnett, C. Buşe, P. Cerone
and S.S. Dragomir
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ω|t−s|
kU (t, s)k ≤ M e
, for all t, s ∈ R.
Let ξ ≥ 0 be fixed and t, s ≥ ξ. Then there exists a real number µ between t
and s such that
kU (ξ, t) − U (ξ, s)k = |t − s| kU (ξ, µ) A (µ)k ≤ M eωµ |kA (·)k|∞ · |t − s| ,
that is, the function η 7→ U (ξ, η) is locally Lipschitz continuous on [ξ, ∞).
Using (6.2), it follows


a11 (t, s)
a12 (t, s)
,
U (t, s) = 
a21 (t, s)
a22 (t, s)
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where
a11 (t, s) = e(s−t) cos t cos s + e2(s−t) sin t sin s;
1
a12 (t, s) = −e(s−t) cos t sin s + e2(s−t) sin t cos s;
2
(s−t)
a21 (t, s) = −e
sin t cos s + e2(s−t) cos t sin s;
1
a22 (t, s) = e(s−t) sin t sin s + e2(s−t) cos t cos s.
2
Then from (5.6) we obtain the following approximating formula for u (·) :
u1 (t) = −
n−1 X
t(i+1)
(2i + 1) t
e− n − e− n
2n
2ti
(2i + 1) t − 2t(i+1)
1
(1)
−
e n −e n
+ R1,n
+ a12 t,
2
2n
a11 t,
i=0
N.S. Barnett, C. Buşe, P. Cerone
and S.S. Dragomir
ti
and
n−1 X
ti
(2i + 1) t − t(i+1)
u2 (t) = −
a21 t,
e n − e− n
2n
i=0
2ti
1
(2i + 1) t − 2t(i+1)
(1)
−
+ a22 t,
e n −e n
+ R2,n ,
2
2n
(1)
(1)
(1)
where the remainder Rn = R1,n , R2,n satisfies the estimate (5.7) with α =
1, H = M eωt |kA (·)k|∞ and |kf k|[0,t],∞ ≤ 2.
On Weighted Ostrowski Type
Inequalities for Operators and
Vector-Valued Functions
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On Weighted Ostrowski Type
Inequalities for Operators and
Vector-Valued Functions
N.S. Barnett, C. Buşe, P. Cerone
and S.S. Dragomir
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(1) (1) Figure 1: The behaviour of the error εn (t) := R1,n , R2,n for n = 200.
2
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On Weighted Ostrowski Type
Inequalities for Operators and
Vector-Valued Functions
N.S. Barnett, C. Buşe, P. Cerone
and S.S. Dragomir
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Figure 2: The behaviour of the error εn (t) := |Rn | for n = 400.
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(1) (1) Figure 1 contains the behaviour of the error εn (t) := R1,n , R2,n for
2
n = 200.
t+1
2. Let X = R and U (t, s) := s+1
, t ≥ s ≥ 0. It is clear that the family
{U (t, s) : t ≥ s ≥ 0} ⊂ L (R) is an exponentially bounded evolution family
which solves the Cauchy problem
u̇ (t) =
1
u (t) , u (s) = xs ∈ R, t ≥ s ≥ 0.
t+1
Consider the inhomogeneous Cauchy problem

1
u (t) + cos [ln (t + 1)] , t ≥ 0
 u̇ (t) = t+1
(6.3)

u (0) = 0.
The solution of (6.3) is given by
Z t
t+1
u (t) =
cos (ln (τ + 1)) dτ = (t + 1) sin [ln (t + 1)] , t ≥ 0.
0 τ +1
From (5.9) we obtain the approximating formula for u (·) as,
n−1
X
n + ti + t
(2i + 1) t
u (t) = (t + 1)
ln
cos ln 1 +
+ Rn ,
n
+
ti
2n
i=0
where Rn satisfies the estimate (5.10) with K = M = ω = β = 1. Indeed,
t+1
≤ et , for all t ≥ s ≥ 0
s+1
On Weighted Ostrowski Type
Inequalities for Operators and
Vector-Valued Functions
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and
1
|cos [ln (t + 1)] − cos [ln (s + 1)]| = |t − s| sin [ln (c + 1)] ≤ |t − s|
c+1
for all t ≥ s ≥ 0, where c is some real number between s and t.
The following Figure 2 contains the behaviour of the error εn (t) := |Rn | for
n = 400.
On Weighted Ostrowski Type
Inequalities for Operators and
Vector-Valued Functions
N.S. Barnett, C. Buşe, P. Cerone
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References
[1] G.A. ANASTASSIOU, Ostrowski type inequalities, Proc. of Amer. Math.
Soc., 123(12) (1999), 3775–3781.
[2] C. BUŞE, S.S. DRAGOMIR AND A. SOFO, Ostrowski’s inequality for
vector-valued functions of bounded semivariation and applications, New
Zealand J. Math., in press.
[3] S.S. DRAGOMIR, N.S. BARNETT AND P. CERONE, An n−dimensional
version of Ostrowski’s inequality for mappings of Hölder type, Kyungpook
Math. J., 40(1) (2000), 65–75.
[4] S.S. DRAGOMIR, P. CERONE, J. ROUMELIOTIS AND S. WANG, A
weighted version of Ostrowski inequality for mappings of Hölder type and
applications in numerical analysis, Bull. Math. Soc. Sc. Math. Roumanie,
42(90)(4) (1992), 301–304.
[5] A.M. FINK, Bounds on the derivation of a function from its averages,
Czech. Math. J., 42 (1992), 289–310.
[6] E. HILLE AND R.S. PHILIPS, Functional Analysis and Semi-Groups,
Amer. Math. Soc. Colloq. Publ., Vol. 31, Providence, R.I., (1957).
On Weighted Ostrowski Type
Inequalities for Operators and
Vector-Valued Functions
N.S. Barnett, C. Buşe, P. Cerone
and S.S. Dragomir
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[7] M. MATIĆ, J.E. PEČARIĆ AND N. UJEVIĆ, Generalisation of weighted
version of Ostrowski’s inequality and some related results, J. of Ineq. &
App., 5 (2000), 639–666.
[8] G.V. MILOVANOVIĆ, On some integral inequalities, Univ. Beograd Publ.
Elektrotehn Fak. Ser. Mat. Fiz., No. 498-541 (1975), 119–124.
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[9] G.V. MILOVANOVIĆ, O nekim funkcionalnim nejednakostima, Univ.
Beograd Publ. Elektrotehn Fak. Ser. Mat. Fiz., No. 599 (1977), 1–59.
[10] D.S. MITRINOVIĆ, J.E. PEČARIĆ AND A.M. FINK, Inequalities Involving Functions and Their Integrals and Derivatives, Kluwer Academic Publishers, Dordrecht, 1991.
[11] A. OSTROWSKI, Uber die Absolutabweichung einer differentienbaren
Funktionen von ihren Integralmittelwert, Comment. Math. Hel, 10 (1938),
226–227.
[12] J.E. PEČARIĆ AND B. SAVIĆ, O novom postupku razvijanja funkcija u
red i nekim primjenama, Zbornik radova V A KoV (Beograd), 9 (1983),
171–202.
[13] R.S. PHILIPS, Perturbation theory from semi-groups of linear operators,
Trans. Amer. Math. Soc., 74 (1953), 199–221.
[14] J. ROUMELIOTIS, P. CERONE AND S.S. DRAGOMIR, An Ostrowski
type inequality for weighted mappings with bounded second derivatives,
J. KSIAM, 3(2) (1999), 107–119. RGMIA Res. Rep. Coll., 1(1) (1998),
http://rgmia.vu.edu.au/v1n1/index.html.
[15] R. SCHNAUBELT, Exponential bounds and hyperbolicity of evolution
families, Dissertation, Tubingen, 1996.
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Inequalities for Operators and
Vector-Valued Functions
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and S.S. Dragomir
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