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 K
Trang 1Volume 2008, Article ID 195137, 8 pages
doi:10.1155/2008/195137
Research Article
Euler-Lagrange Type Cubic Operators
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, 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 spacesX λ
andZ5
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
LetX and Y be complex normed spaces For a fixed nonnegative real number λ, we denote by
X λthe linear space of all functionsf : X→Y with pointwise operations for which there exists
a constantM f ≥ 0 with
fx ≤ M f e λx 1.1 for allx ∈ X It is easy to show that the space X λwith the norm
f : sup
x∈X
is a normed space Let us denote byX n
λthe linear space of all functionsφ : X × · · · × X
n times
→Y with
pointwise operations for which there exists a constant Mφ≥ 0 with
φx1, , x n ≤ M φ e λn
i1 x i 1.3
Trang 2for allx1, , x n ∈ X It is not difficult to show that the space X n
λ with the norm
φ : sup
x1, ,x n ∈X
φx1, , x n e −λn
i1 x i 1.4
is a normed space
We denote by Z m
λ the normed space m
i1 X λ {f1, , f m : f1, , f m ∈ X λ} with pointwise operations together with the norm
f1, , f m : max{f1, , 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 Dlutek1, 2 In 3, Moslehian et al have extended the results of2 to the Pexiderized generalized Jensen and Pexiderized generalized quadratic operators on the function space X λ and provided more general results regarding their norms
In4, Jung investigated the norm of the cubic operator on the function space Z5
λ
A functionf : X→Y is called a cubic function if and only if f is a solution function of the
cubic functional equation
fx y fx − y 2f
1
2x y 2f
1
2x − y 12f
1
2x 1.6 Jun and Kim5 proved that when both X and Y are real vector spaces, a function f : X→Y
satisfies1.6 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 to1.6,
fx y fx − y af
1
a x y af
1
a x − y 2aa2− 1f
1
a x 1.7
for fixed integersa 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−b2
f
1
ab x f
1
ab y ababf
1
ab x
1
ab y
1.8 for fixed integersa, 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 equation1.6, 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 equations1.7 and
1.8
Trang 3Definition 1.2 The operators C P
1, C P
2 :Z5
λ →X2
λare defined by
C P
1f1, , f5x, y : f1x y f2x − y − mf3
1
m x y
− mf4
1
m x − y − 2m
m2− 1f5
1
m x ,
C P
2f1, , f5x, y : f1
1
a x
1
b y f2
1
b x
1
a y − a ba − b2
f3
1
ab x f4
1
ab y
− aba bf5
1
ab x
1
ab y ,
1.9 wherea, b, and m are fixed integers with a, b / 0, a ± b / 0, and m / 0, ±1.
Definition 1.3 The operators C1, C2 :X λ →X2
λare defined by
C1fx, y : fx y fx − y − mf
1
m x y
− mf
1
m x − y − 2mm2− 1f
1
m x ,
C2fx, y : f
1
a x
1
b y f
1
b x
1
a y
− a ba − b2
f
1
ab x f
1
ab y − aba bf
1
ab x
1
ab y ,
1.10
wherea, 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 operatorsC P
1, C P
2 on the function space
Z5
λ , and norms of the operators C1, C2on the function spaceX λ The results extend the results
of4
2 Main results
Throughout this section,a, b, and m are fixed integers with a, b / 0, a ± b / 0, and m / 0, ±1.
The next theorems give us the exact norms of operatorsC P
1,C P
2,C1, andC2.
Theorem 2.1 The operator C P
1 :Z5
λ →X2
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,
m1x y
,m1x − y
,m1x
≤ x y 2.2
Trang 4for allx, y ∈ X, we get
C P
1f1, , f5 sup
x,y∈X e −λxy
f1x y f2x − y − mf3
1
m x y
− mf4
1
m x − y − 2mm2− 1f5
1
m x
≤ sup
x,y∈X e −λxy f1x y sup
x,y∈X e −λx−y f2x − y
|m| sup
x,y∈X e −λ1/mxy
f3
1
m x y
|m| sup
x,y∈X e −λ1/mx−y
f4
1
m x − y
2|m|m2− 1sup
x∈X e −λ1/mx
f5
1
m x
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
2.3
for eachf1, , f5 ∈ Z5
λ This implies that
C1P ≤ 2|m|3 2. 2.4
Now, letν ∈ Y be such that ν 1 and let {ξ n}nbe a sequence of positive real numbers decreasing to 0 We define
f n x
⎧
⎪
⎪
⎪
⎪
e2λξ n ν, ifx 2ξ n or x 0,
−|m| m e2λξ n ν, if mx |m 1|ξ n , mx |m − 1|ξ n or mx ξ n ,
0, otherwise
2.5
for allx ∈ X Hence we have
e −λx f n x
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
e2λξ n , ifx 0,
1, ifx 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 , ifmx ξ n ,
0, otherwise
2.6
Trang 5for allx ∈ X, so that f n ∈ X λ for all positive integersn, with
f n e2λξ n 2.7
Letu ∈ X be such that u 1 and take x0, y0∈ X as x0 y0 ξ n u Then it follows from
the definition off nthat
C P
1f n , , f n sup
x,y∈X e −λxy
f n x y f n x − y − mf n
1
m x y
− mf n
1
m x − y − 2mm2− 1f n
1
m x
≥ e −2λξ n e2λξ n ν e2λξ n ν |m|e2λξ n ν |m|e2λξ n ν 2|m|m2− 1e2λξ n ν
2|m|3 2.
2.8
If on the contraryC P
1 < 2|m|3 2, then there exists a δ > 0 such that
C P
1f n , , f n ≤ 2|m|3 2 − δf n , , f n 2.9 for all positive integersn So it follows from 2.7, 2.8, and 2.9 that
2|m|3 2 ≤ C P
1f n , , f n ≤ 2|m|3 2 − δe2λξ n 2.10
for all positive integersn Since lim n→∞ e2λξ 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 of4 is a result ofTheorem 2.1form 2.
Corollary 2.2 The operator C1:X λ →X2
λ is a bounded linear operator with
C1 2|m|3 2. 2.11
Proof The result follows from the proof ofTheorem 2.1
Theorem 2.3 The operator C P
2 :Z5
λ →X2
λ is a bounded linear operator with
C P
2 2|a b|a − b2 |aba b| 2. 2.12
Proof Since
max
a1x b1y
,b1x 1a y
,ab1 x
,ab1 y
,ab1 x ab1 y
≤ x y 2.13
Trang 6for allx, y ∈ X, we get
C P
2f1, , f5 sup
x,y∈X e −λxy
f1
1
a x
1
b y f2
1
b x
1
a y
− a ba − b2
f3
1
ab x f4
1
ab y
− aba bf5
1
ab x
1
ab y
≤ sup
x,y∈X e −λ1/ax1/by
f1
1
a x
1
b y
sup
x,y∈X e −λ1/bx1/ay
f2
1
b x
1
a y
|a b|a − b2sup
x∈X e −λ1/abx
f3
1
ab x
|a b|a − b2sup
y∈X e −λ1/aby
f4
1
ab y
|aba b| sup
x,y∈X e −λ1/abx1/aby
f5
1
ab x
1
ab y
≤ f1 f2 |a b|a − b2f3 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
2.14
for eachf1, , f5 ∈ Z5
λ This implies that
C P
2 ≤ 2|a b|a − b2 |aba b| 2. 2.15 Letη 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
Now, letu ∈ X, ν ∈ Y be such that u ν 1 and let {ξ n}n be a sequence of positive real numbers decreasing to 0 We define
f n x
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
e λ1|η|ξ n ν, ifx
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, ifx
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,
2.17
Trang 7for allx ∈ 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 , ifx ab1 ξ n u,
e 1|η|−|η/ab|λξ n , ifx ab η ξ n u,
e 1|η|−|1η/ab|λξ n , if x 1ab η ξ n u,
0, otherwise
2.18
for allx ∈ X, so that f n ∈ X λ for all positive integersn, 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
Letx0, y0∈ X be such that x0 ξ n u and y0 ηξ n u Then it follows from the definition of f nthat
C P
2f 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 − b2
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 − b2e λ1|η|ξ n |abab|e λ1|η|ξ n
2|a b|a − b2 |aba b| 2,
2.20
so that
C P
2f n , , f n ≥ 2|a b|a − b2 |aba b| 2. 2.21
If on the contraryC P
2 < 2|a b|a − b2 |aba b| 2, then there exists a δ > 0 such that
C P
2f n , , f n ≤ 2|a b|a − b2 |aba b| 2 − δf n , , f n 2.22
Trang 8for all positive integersn So it follows from 2.21 and 2.22 that
2|a b|a − b2 |aba b| 2 ≤ C P
2f n , , f n ≤ 2|a b|a − b2 |aba b| 2 − δf n
2.23
for all positive integersn Since lim n→∞ ξ n 0, it follows from 2.19 that limn→∞ f n 1, so
the right-hand side of2.23 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 haveC P
2 2|a b|a − b2 |aba b| 2.
Corollary 2.4 The operator C2:X λ →X2
λ is a bounded linear operator with
C2 2|a b|a − b2 |aba b| 2. 2.25
Proof The result follows from the proof ofTheorem 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.
... 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. .. Euler-Lagrange- type cubic functional equations1.7 and
1.8
Trang 3Definition 1.2 The operators C P
1,... ∈ 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