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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 X
λ
Space
Abbas Najati
1
and Asghar Rahimi
2
1
Department of Mathematics, Faculty of Sciences, University of Mohaghegh Ardabili,
P.O. Box 56199-11367 Ardabili, Iran
2
Department of Mathematics, University of Maragheh, P.O. Box 55181-83111, Maragheh,
East Azarbayjan, Iran
Correspondence should be addressed to Abbas Najati,
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 X
λ
and Z
5
λ
. 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
X
λ
the linear space of all functions f : X→Y with pointwise operations for which there exists
a constant M
f
≥ 0with
fx≤M
f
e
λx
1.1
for all x ∈ X. It is easy to show that the space X
λ
with the norm
f : sup
x∈X
e
−λx
fx
1.2
is a normed space. Let us denote by X
n
λ
the linear space of all functions φ : X ×···×X
n times
→Y with
pointwise operations for which there exists a constant M
φ
≥ 0with
φx
1
, ,x
n
≤M
φ
e
λ
n
i1
x
i
1.3
2 Journal of Inequalities and Applications
for all x
1
, ,x
n
∈ X. It is not difficult to show that the space X
n
λ
with the norm
φ : sup
x
1
, ,x
n
∈X
φx
1
, ,x
n
e
−λ
n
i1
x
i
1.4
is a normed space.
We denote by Z
m
λ
the normed space
m
i1
X
λ
{f
1
, ,f
m
: f
1
, ,f
m
∈ X
λ
} with
pointwise operations together with the norm
f
1
, ,f
m
: max{f
1
, ,f
m
}. 1.5
The norms of the Pexiderized Cauchy, quadratic, and Jensen operators on the function
space X
λ
have been investigated by Czerwik and Dlutek 1, 2.In3, Moslehian et al. have
extended the results of 2 to the Pexiderized generalized Jensen and Pexiderized generalized
quadratic operators on the function space X
λ
and provided more general results regarding
their norms.
In 4, Jung investigated the norm of the cubic operator on the function space Z
5
λ
.
A function f : X→Y is called a cubic function if and only if f is a solution function of the
cubic functional equation
fx yfx − y2f
1
2
x y
2f
1
2
x − y
12f
1
2
x
. 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 fxBx, 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 yfx − yaf
1
a
x y
af
1
a
x − y
2aa
2
− 1f
1
a
x
1.7
for fixed integers a with a
/
0, ±1. Moreover, Jun and Kim 7 introduced the following Euler-
Lagrange-type cubic functional equation
f
1
a
x
1
b
y
f
1
b
x
1
a
y
aba−b
2
f
1
ab
x
f
1
ab
y
ababf
1
ab
x
1
ab
y
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,thenf 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 C
P
1
,C
P
2
: Z
5
λ
→X
2
λ
are defined by
C
P
1
f
1
, ,f
5
x, y : f
1
x yf
2
x − y − mf
3
1
m
x y
− mf
4
1
m
x − y
− 2m
m
2
− 1
f
5
1
m
x
,
C
P
2
f
1
, ,f
5
x, y : f
1
1
a
x
1
b
y
f
2
1
b
x
1
a
y
− a ba − b
2
f
3
1
ab
x
f
4
1
ab
y
− aba bf
5
1
ab
x
1
ab
y
,
1.9
where a, b,andm are fixed integers with a, b
/
0, a ± b
/
0, and m
/
0, ±1.
Definition 1.3. The operators C
1
,C
2
: X
λ
→X
2
λ
are defined by
C
1
fx, y : fx yfx − y − mf
1
m
x y
− mf
1
m
x − y
− 2mm
2
− 1f
1
m
x
,
C
2
fx, y : f
1
a
x
1
b
y
f
1
b
x
1
a
y
− a ba − b
2
f
1
ab
x
f
1
ab
y
− aba bf
1
ab
x
1
ab
y
,
1.10
where a, b,andm 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 C
P
1
,C
P
2
on the function space
Z
5
λ
, and norms of the operators C
1
,C
2
on the function space X
λ
. The results extend the results
of 4.
2. Main results
Throughout this section, a, b,andm are fixed integers with a, b
/
0, a ± b
/
0, and m
/
0, ±1.
The next theorems give us the exact norms of operators C
P
1
, C
P
2
, C
1
,andC
2
.
Theorem 2.1. The operator C
P
1
: Z
5
λ
→X
2
λ
is a bounded linear operator with
C
P
1
2|m|
3
2. 2.1
Proof. First, we show that C
P
1
≤2|m|
3
2. Since
max
x y, x − y,
1
m
x y
,
1
m
x − y
,
1
m
x
≤x y 2.2
4 Journal of Inequalities and Applications
for all x, y ∈ X, we get
C
P
1
f
1
, ,f
5
sup
x,y∈X
e
−λxy
f
1
x yf
2
x − y − mf
3
1
m
x y
− mf
4
1
m
x − y
− 2mm
2
− 1f
5
1
m
x
≤ sup
x,y∈X
e
−λxy
f
1
x y sup
x,y∈X
e
−λx−y
f
2
x − y
|m| sup
x,y∈X
e
−λ1/mxy
f
3
1
m
x y
|m| sup
x,y∈X
e
−λ1/mx−y
f
4
1
m
x − y
2|m|m
2
− 1sup
x∈X
e
−λ1/mx
f
5
1
m
x
f
1
f
2
|m|f
3
|m|f
4
2|m|m
2
− 1f
5
≤ 2|m|
3
2 max{f
1
, f
2
, f
3
, f
4
, f
5
}
2|m|
3
2f
1
,f
2
,f
3
,f
4
,f
5
2.3
for each f
1
, ,f
5
∈ Z
5
λ
. This implies that
C
P
1
≤2|m|
3
2. 2.4
Now, let ν ∈ Y be such that ν 1andlet{ξ
n
}
n
be a sequence of positive real numbers
decreasing to 0. We define
f
n
x
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
e
2λξ
n
ν, if x 2ξ
n
or x 0,
−
|m|
m
e
2λξ
n
ν, if mx |m 1|ξ
n
, mx |m − 1|ξ
n
or mx ξ
n
,
0, otherwise
2.5
for all x ∈ X. Hence we have
e
−λx
f
n
x
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
e
2λξ
n
, if x 0,
1, if x 2ξ
n
,
e
2−|m1/m|λξ
n
, if mx |m 1|ξ
n
,
e
2−|m−1/m|λξ
n
, if mx |m − 1|ξ
n
,
e
2−1/|m|λξ
n
, if mx ξ
n
,
0, otherwise
2.6
A. Najati and A. Rahimi 5
for all x ∈ X, so that f
n
∈ X
λ
for all positive integers n, with
f
n
e
2λξ
n
. 2.7
Let u ∈ X be such that u 1 and take x
0
,y
0
∈ X as x
0
y
0
ξ
n
u. Then it follows from
the definition of f
n
that
C
P
1
f
n
, ,f
n
sup
x,y∈X
e
−λxy
f
n
x yf
n
x − y − mf
n
1
m
x y
− mf
n
1
m
x − y
− 2mm
2
− 1f
n
1
m
x
≥ e
−2λξ
n
e
2λξ
n
ν e
2λξ
n
ν |m|e
2λξ
n
ν |m|e
2λξ
n
ν 2|m|m
2
− 1e
2λξ
n
ν
2|m|
3
2.
2.8
If on the contrary C
P
1
< 2|m|
3
2, then there exists a δ>0 such that
C
P
1
f
n
, ,f
n
≤2|m|
3
2 − δf
n
, ,f
n
2.9
for all positive integers n. So it follows from 2.7, 2.8,and2.9 that
2|m|
3
2 ≤C
P
1
f
n
, ,f
n
≤2|m|
3
2 − δe
2λξ
n
2.10
for all positive integers n. Since lim
n→∞
e
2λξ
n
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
C
P
1
2|m|
3
2.
Theorem 2.1 of 4 is a result of Theorem 2.1 for m 2.
Corollary 2.2. The operator C
1
: X
λ
→X
2
λ
is a bounded linear operator with
C
1
2|m|
3
2. 2.11
Proof. The result follows from the proof of Theorem 2.1.
Theorem 2.3. The operator C
P
2
: Z
5
λ
→X
2
λ
is a bounded linear operator with
C
P
2
2|a b|a − b
2
|aba b| 2. 2.12
Proof. Since
max
1
a
x
1
b
y
,
1
b
x
1
a
y
,
1
ab
x
,
1
ab
y
,
1
ab
x
1
ab
y
≤x y 2.13
6 Journal of Inequalities and Applications
for all x, y ∈ X, we get
C
P
2
f
1
, ,f
5
sup
x,y∈X
e
−λxy
f
1
1
a
x
1
b
y
f
2
1
b
x
1
a
y
− a ba − b
2
f
3
1
ab
x
f
4
1
ab
y
− aba bf
5
1
ab
x
1
ab
y
≤ sup
x,y∈X
e
−λ1/ax1/by
f
1
1
a
x
1
b
y
sup
x,y∈X
e
−λ1/bx1/ay
f
2
1
b
x
1
a
y
|a b|a − b
2
sup
x∈X
e
−λ1/abx
f
3
1
ab
x
|a b|a − b
2
sup
y∈X
e
−λ1/aby
f
4
1
ab
y
|aba b| sup
x,y∈X
e
−λ1/abx1/aby
f
5
1
ab
x
1
ab
y
≤f
1
f
2
|a b|a − b
2
f
3
f
4
|aba b|f
5
≤ 2|a b|a − b
2
|aba b| 2 max{f
1
, f
2
, f
3
, f
4
, f
5
}
2|a b|a − b
2
|aba b| 2f
1
,f
2
,f
3
,f
4
,f
5
2.14
for each f
1
, ,f
5
∈ Z
5
λ
. This implies that
C
P
2
≤2|a b|a − b
2
|aba b| 2. 2.15
Let η be a real number such that
η
/
∈
0, 1,
1 − a
b
,
1 − b
a
,
a − 1
1 − b
,
b − 1
1 − a
,
a
1 − b
,
b
1 − a
. 2.16
Now, let u ∈ X, ν ∈ Y be such that u ν 1andlet{ξ
n
}
n
be a sequence of positive
real numbers decreasing to 0. We define
f
n
x
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
e
λ1|η|ξ
n
ν, if x
1
a
η
b
ξ
n
u, or x
1
b
η
a
ξ
n
u,
−
|a b|
a b
e
λ1|η|ξ
n
ν, if x
1
ab
ξ
n
u, or x
η
ab
ξ
n
u,
−
|aba b|
aba b
e
λ1|η|ξ
n
ν, if x
1 η
ab
ξ
n
u,
0, otherwise
2.17
A. Najati and A. Rahimi 7
for all x ∈ X. Hence we have
e
−λx
f
n
x
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
e
1|η|−|1/aη/b|λξ
n
, if x
1
a
η
b
ξ
n
u,
e
1|η|−|1/bη/a|λξ
n
, if x
1
b
η
a
ξ
n
u,
e
1|η|−|1/ab|λξ
n
, if x
1
ab
ξ
n
u,
e
1|η|−|η/ab|λξ
n
, if x
η
ab
ξ
n
u,
e
1|η|−|1η/ab|λξ
n
, if x
1 η
ab
ξ
n
u,
0, otherwise
2.18
for all x ∈ X, so that f
n
∈ X
λ
for all positive integers n, with
f
n
max{e
1|η|−|1/aη/b|λξ
n
,e
1|η|−|1/bη/a|λξ
n
,
e
1|η|−|1/ab|λξ
n
,e
1|η|−|η/ab|λξ
n
,e
1|η|−|1η/ab|λξ
n
}.
2.19
Let x
0
,y
0
∈ X be such that x
0
ξ
n
u and y
0
ηξ
n
u. Then it follows from the definition of f
n
that
C
P
2
f
n
, ,f
n
sup
x,y∈X
e
−λxy
f
n
1
a
x
1
b
y
f
n
1
b
x
1
a
y
− a ba − b
2
f
n
1
ab
x
f
n
1
ab
y
− aba bf
n
1
ab
x
1
ab
y
≥ e
−λ1|η|ξ
n
e
λ1|η|ξ
n
e
λ1|η|ξ
n
2|ab|a − b
2
e
λ1|η|ξ
n
|abab|e
λ1|η|ξ
n
2|a b|a − b
2
|aba b| 2,
2.20
so that
C
P
2
f
n
, ,f
n
≥2|a b|a − b
2
|aba b| 2. 2.21
If on the contrary C
P
2
< 2|a b|a − b
2
|aba b| 2, then there exists a δ>0 such that
C
P
2
f
n
, ,f
n
≤2|a b|a − b
2
|aba b| 2 − δf
n
, ,f
n
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 − b
2
|aba b| 2 ≤C
P
2
f
n
, ,f
n
≤2|a b|a − b
2
|aba b| 2 − δf
n
2.23
for all positive integers n. Since lim
n→∞
ξ
n
0, it follows from 2.19 that lim
n→∞
f
n
1, so
the right-hand side of 2.23 tends to 2|a b|a − b
2
|aba b| 2 − δ as n→∞, whence
2|a b|a − b
2
|aba b| 2 ≤ 2|a b|a − b
2
|aba b| 2 − δ, 2.24
which is a contradiction. Hence we have C
P
2
2|a b|a − b
2
|aba b| 2.
Corollary 2.4. The operator C
2
: X
λ
→X
2
λ
is a bounded linear operator with
C
2
2|a b|a − b
2
|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
1 S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific, River Edge, NJ,
USA, 2002.
2 S. Czerwik and K. Dlutek, “Cauchy and Pexider operators in some function spaces,” in Functional
Equations, Inequalities and Applications, pp. 11–19, Kluwer Academic Publishers, Dordrecht, The
Netherlands, 2003.
3 M. S. Moslehian, T. Riedel, and A. Saadatpour, “Norms of operators in X
λ
spaces,” Applied Mathematics
Letters, vol. 20, no. 10, pp. 1082–1087, 2007.
4 S M. Jung, “Cubic operator norm on X
λ
space,” Bulletin of the Korean Mathematical Society, vol. 44, no.
2, pp. 309–313, 2007.
5 K W. Jun and H M. Kim, “The generalized Hyers-Ulam-Rassias stability of a cubic functional
equation,” Journal of Mathematical Analysis and Applications, vol. 274, no. 2, pp. 867–878, 2002.
6 K W. Jun, H M. Kim, and I S. Chang, “On the Hyers-Ulam stability of an Euler-Lagrange type cubic
functional equation,” Journal of Computational Analysis and Applications, vol. 7, no. 1, pp. 21–33, 2005.
7 K W. Jun and H M. Kim, “On the stability of Euler-Lagrange type cubic mappings in quasi-Banach
spaces,” Journal of Mathematical Analysis and Applications, vol. 332, no. 2, pp. 1335–1350, 2007.
