Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2008, Article ID 195137, 8 pages
doi:10.1155/2008/195137
Research Article
Euler-Lagrange Type Cubic Operators
and Their Norms on Xλ Space
Abbas Najati1 and Asghar Rahimi2
1
Department of Mathematics, Faculty of Sciences, University of Mohaghegh Ardabil,
P.O. Box 56199-11367 Ardabil, Iran
2
Department of Mathematics, University of Maragheh, P.O. Box 55181-83111, Maragheh,
East Azarbayjan, Iran
Correspondence should be addressed to Abbas Najati, a.najati@uma.ac.ir
Received 17 April 2008; Accepted 1 July 2008
Recommended by Jong Kim
We will introduce linear operators and obtain their exact norms defined on the function spaces Xλ
and Zλ5 . These operators are constructed from the Euler-Lagrange type cubic functional equations
and their Pexider versions.
Copyright q 2008 A. Najati and A. Rahimi. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
1. Introduction
Let X and Y be complex normed spaces. For a fixed nonnegative real number λ, we denote by
Xλ the linear space of all functions f : X→Y with pointwise operations for which there exists
a constant Mf ≥ 0 with
fx ≤ Mf eλx
1.1
for all x ∈ X. It is easy to show that the space Xλ with the norm
f : sup e−λx fx
1.2
x∈X
is a normed space. Let us denote by Xλn the linear space of all functions φ : X × · · · × X →Y with
pointwise operations for which there exists a constant Mφ ≥ 0 with
n
φx1 , . . . , xn ≤ Mφ eλ
i1 xi n times
1.3
2
Journal of Inequalities and Applications
for all x1 , . . . , xn ∈ X. It is not difficult to show that the space Xλn with the norm
φ :
sup
x1 ,...,xn ∈X
n
φx1 , . . . , xn e−λ i1 xi 1.4
is a normed space.
We denote by Zλm the normed space m
i1 Xλ {f1 , . . . , fm : f1 , . . . , fm ∈ Xλ } with
pointwise operations together with the norm
f1 , . . . , fm : max{f1 , . . . , fm }.
1.5
The norms of the Pexiderized Cauchy, quadratic, and Jensen operators on the function
space Xλ have been investigated by Czerwik and Dlutek 1, 2. In 3, Moslehian et al. have
extended the results of 2 to the Pexiderized generalized Jensen and Pexiderized generalized
quadratic operators on the function space Xλ and provided more general results regarding
their norms.
In 4, Jung investigated the norm of the cubic operator on the function space Zλ5 .
A function f : X→Y is called a cubic function if and only if f is a solution function of the
cubic functional equation
1
fx y fx − y 2f
xy
2
1
2f
x−y
2
1
12f
x .
2
1.6
Jun and Kim 5 proved that when both X and Y are real vector spaces, a function f : X→Y
satisfies 1.6 if and only if there exists a function B : X × X × X→Y such that fx Bx, x, x
for all x ∈ X, and B is symmetric for each fixed one variable and is additive for fixed two
variables.
In 6, the authors introduced the following Euler-Lagrange-type cubic functional
equation, which is equivalent to 1.6,
fx y fx − y af
1
xy
a
af
1
x−y
a
2aa2 − 1f
1
x
a
1.7
for fixed integers a with a /
0, ±1. Moreover, Jun and Kim 7 introduced the following EulerLagrange-type cubic functional equation
1
1
1
1
1
1
1
1
2
f
x y f
x y aba−b f
x f
y ababf
x y
a
b
b
a
ab
ab
ab
ab
1.8
for fixed integers a, b with a, b / 0, a ± b / 0, and they proved the following theorem.
Theorem 1.1 see 7, Theorem 2.1. Let X and Y be real vector spaces. If a mapping f : X→Y
satisfies the functional equation 1.6, then f satisfies the functional equation 1.8.
We will introduce linear operators which are constructed from the Euler-Lagrange-type
cubic and the Pexiderization of the Euler-Lagrange-type cubic functional equations 1.7 and
1.8.
A. Najati and A. Rahimi
3
Definition 1.2. The operators C1P , C2P : Zλ5 →Xλ2 are defined by
1
xy
m
1
1
x − y − 2m m2 − 1 f5
x ,
− mf4
m
m
1
1
1
1
1
1
2
P
x y f2
x y − a ba − b f3
x f4
y
C2 f1 , . . . , f5 x, y : f1
a
b
b
a
ab
ab
1
1
− aba bf5
x
y ,
ab
ab
1.9
C1P f1 , . . . , f5 x, y : f1 x y f2 x − y − mf3
where a, b, and m are fixed integers with a, b /
0, a ± b /
0, and m /
0, ±1.
Definition 1.3. The operators C1 , C2 : Xλ →Xλ2 are defined by
1
C1 fx, y : fx y fx − y − mf
xy
m
1
1
− mf
x − y − 2mm2 − 1f
x ,
m
m
1
1
1
1
C2 fx, y : f
x y f
x y
a
b
b
a
1
1
1
1
2
x f
y
− aba bf
x
y ,
− a ba − b f
ab
ab
ab
ab
1.10
where a, b, and m are fixed integers with a, b / 0, a ± b / 0, and m / 0, ±1.
In this paper, we will give the exact norms of the operators C1P , C2P on the function space
and norms of the operators C1 , C2 on the function space Xλ . The results extend the results
of 4.
Zλ5 ,
2. Main results
Throughout this section, a, b, and m are fixed integers with a, b / 0, a ± b / 0, and m / 0, ±1.
P
P
The next theorems give us the exact norms of operators C1 , C2 , C1 , and C2 .
Theorem 2.1. The operator C1P : Zλ5 →Xλ2 is a bounded linear operator with
C1P 2|m|3 2.
2.1
Proof. First, we show that C1P ≤ 2|m|3 2. Since
1
1
, x − y , 1 x ≤ x y
max x y, x − y, x
y
m
m
m 2.2
4
Journal of Inequalities and Applications
for all x, y ∈ X, we get
1
f
C1P f1 , . . . , f5 sup e−λxy x
y
x
y
f
x
−
y
−
mf
2
3
1
m
x,y∈X
1
1
2
− mf4
x − y − 2mm − 1f5
x m
m ≤ sup e−λxy f1 x y sup e−λx−y f2 x − y
x,y∈X
x,y∈X
1
f
|m| sup e−λ1/mxy x
y
3
m
x,y∈X
|m| sup e
1
f
x
−
y
4 m
−λ1/mx−y x,y∈X
2|m|m − 1supe
2
x∈X
2.3
1
f5 m x −λ1/mx f1 f2 |m|f3 |m|f4 2|m|m2 − 1f5 ≤ 2|m|3 2 max{f1 , f2 , f3 , f4 , f5 }
2|m|3 2f1 , f2 , f3 , f4 , f5 for each f1 , . . . , f5 ∈ Zλ5 . This implies that
C1P ≤ 2|m|3 2.
2.4
Now, let ν ∈ Y be such that ν 1 and let {ξn }n be a sequence of positive real numbers
decreasing to 0. We define
⎧
⎪
if x 2ξn or x 0,
e2λξn ν,
⎪
⎪
⎪
⎨
fn x − |m| e2λξn ν, if mx |m 1|ξn , mx |m − 1|ξn or mx ξn ,
⎪
⎪
m
⎪
⎪
⎩
0,
otherwise
2.5
for all x ∈ X. Hence we have
e−λx fn x ⎧
⎪
if x 0,
e2λξn ,
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪1,
if x 2ξn ,
⎪
⎪
⎪
⎪
⎪
⎪
⎨e2−|m1/m|λξn , if mx |m 1|ξn ,
⎪
⎪
e2−|m−1/m|λξn , if mx |m − 1|ξn ,
⎪
⎪
⎪
⎪
⎪
⎪
2−1/|m|λξn
⎪
,
if mx ξn ,
⎪
⎪e
⎪
⎪
⎪
⎪
⎩0,
otherwise
2.6
A. Najati and A. Rahimi
5
for all x ∈ X, so that fn ∈ Xλ for all positive integers n, with
fn e2λξn .
2.7
Let u ∈ X be such that u 1 and take x0 , y0 ∈ X as x0 y0 ξn u. Then it follows from
the definition of fn that
1
x
y
x
y
f
x
−
y
−
mf
C1P fn , . . . , fn sup e−λxy f
n
n
n
m
x,y∈X
1
1
x − y − 2mm2 − 1fn
x − mfn
m
m 2.8
≥ e−2λξn e2λξn ν e2λξn ν |m|e2λξn ν |m|e2λξn ν 2|m|m2 − 1e2λξn ν
2|m|3 2.
If on the contrary C1P < 2|m|3 2, then there exists a δ > 0 such that
C1P fn , . . . , fn ≤ 2|m|3 2 − δfn , . . . , fn 2.9
for all positive integers n. So it follows from 2.7, 2.8, and 2.9 that
2|m|3 2 ≤ C1P fn , . . . , fn ≤ 2|m|3 2 − δe2λξn
2.10
for all positive integers n. Since limn→∞ e2λξn 1, the right-hand side of 2.10 tends to 2|m|3 2 − δ as n→∞, whence 2|m|3 2 ≤ 2|m|3 2 − δ, which is a contradiction. Hence we have
C1P 2|m|3 2.
Theorem 2.1 of 4 is a result of Theorem 2.1 for m 2.
Corollary 2.2. The operator C1 : Xλ →Xλ2 is a bounded linear operator with
C1 2|m|3 2.
2.11
Proof. The result follows from the proof of Theorem 2.1.
Theorem 2.3. The operator C2P : Zλ5 →Xλ2 is a bounded linear operator with
C2P 2|a b|a − b2 |aba b| 2.
2.12
Proof. Since
1
1 , 1 x 1 y , 1 x, 1 y , 1 x 1 y ≤ x y
max x
y
a
b
b
a
ab
ab
ab
ab 2.13
6
Journal of Inequalities and Applications
for all x, y ∈ X, we get
1
1
1
1
C2P f1 , . . . , f5 sup e−λxy f
x
y
f
x
y
1
2
a
b
b
a
x,y∈X
1
1
− a ba − b2 f3
x f4
y
ab
ab
1
1
− aba bf5
x
y ab
ab
1
1 ≤ sup e−λ1/ax1/by f
x
y
1
a
b x,y∈X
1
1
f
sup e−λ1/bx1/ay x
y
2 b
a x,y∈X
1
f
|a b|a − b2 supe−λ1/abx x
3 ab x∈X
1
|a b|a − b2 supe−λ1/aby f
y
4 ab
y∈X
1
1
f
|aba b| sup e−λ1/abx1/aby x
y
5 ab
ab
x,y∈X
2.14
≤ f1 f2 |a b|a − b2 f3 f4 |aba b|f5 ≤ 2|a b|a − b2 |aba b| 2 max{f1 , f2 , f3 , f4 , f5 }
2|a b|a − b2 |aba b| 2f1 , f2 , f3 , f4 , f5 for each f1 , . . . , f5 ∈ Zλ5 . This implies that
C2P ≤ 2|a b|a − b2 |aba b| 2.
2.15
Let η be a real number such that
1−a 1−b a−1 b−1 a
b
.
η/
∈ 0, 1,
,
,
,
,
,
b
a 1−b 1−a 1−b 1−a
2.16
Now, let u ∈ X, ν ∈ Y be such that u ν 1 and let {ξn }n be a sequence of positive
real numbers decreasing to 0. We define
⎧
⎪
⎪
eλ1|η|ξn ν,
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨− |a b| eλ1|η|ξn ν,
ab
fn x ⎪
⎪
⎪
|aba b| λ1|η|ξn
⎪
⎪
e
ν,
−
⎪
⎪
⎪
aba b
⎪
⎪
⎩
0,
1 η
1 η
ξn u, or x ξn u,
if x a b
b a
η
1
ξn u, or x ξn u,
if x ab
ab
1η
ξn u,
if x ab
otherwise
2.17
A. Najati and A. Rahimi
7
for all x ∈ X. Hence we have
⎧
⎪
⎪
e1|η|−|1/aη/b|λξn ,
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
e1|η|−|1/bη/a|λξn ,
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨e1|η|−|1/ab|λξn ,
−λx
e
fn x ⎪
⎪
⎪
⎪
⎪
⎪e1|η|−|η/ab|λξn ,
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
1|η|−|1η/ab|λξn
⎪
,
⎪e
⎪
⎪
⎪
⎩
0,
1 η
ξn u,
if x a b
1 η
ξn u,
if x b a
1
ξn u,
if x ab
η
if x ξn u,
ab
1η
if x ξn u,
ab
otherwise
2.18
for all x ∈ X, so that fn ∈ Xλ for all positive integers n, with
fn max{e1|η|−|1/aη/b|λξn , e1|η|−|1/bη/a|λξn ,
e1|η|−|1/ab|λξn , e1|η|−|η/ab|λξn , e1|η|−|1η/ab|λξn }.
2.19
Let x0 , y0 ∈ X be such that x0 ξn u and y0 ηξn u. Then it follows from the definition of fn that
1
1
1
1
f
C2P fn , . . . , fn sup e−λxy x
y
f
x
y
n
n a
b
b
a
x,y∈X
1
1
2
− a ba − b fn
x fn
y
ab
ab
1
1
x
y − aba bfn
ab
ab
≥ e−λ1|η|ξn eλ1|η|ξn eλ1|η|ξn 2|ab|a − b2 eλ1|η|ξn |abab|eλ1|η|ξn 2|a b|a − b2 |aba b| 2,
2.20
so that
C2P fn , . . . , fn ≥ 2|a b|a − b2 |aba b| 2.
2.21
If on the contrary C2P < 2|a b|a − b2 |aba b| 2, then there exists a δ > 0 such that
C2P fn , . . . , fn ≤ 2|a b|a − b2 |aba b| 2 − δfn , . . . , fn 2.22
8
Journal of Inequalities and Applications
for all positive integers n. So it follows from 2.21 and 2.22 that
2|a b|a − b2 |aba b| 2 ≤ C2P fn , . . . , fn ≤ 2|a b|a − b2 |aba b| 2 − δfn 2.23
for all positive integers n. Since limn→∞ ξn 0, it follows from 2.19 that limn→∞ fn 1, so
the right-hand side of 2.23 tends to 2|a b|a − b2 |aba b| 2 − δ as n→∞, whence
2|a b|a − b2 |aba b| 2 ≤ 2|a b|a − b2 |aba b| 2 − δ,
2.24
which is a contradiction. Hence we have C2P 2|a b|a − b2 |aba b| 2.
Corollary 2.4. The operator C2 : Xλ →Xλ2 is a bounded linear operator with
C2 2|a b|a − b2 |aba b| 2.
2.25
Proof. The result follows from the proof of Theorem 2.3.
Acknowledgment
The authors would like to thank the referee for his/her useful comments.
References
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USA, 2002.
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Netherlands, 2003.
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