Volume 2008, Article ID 195137,8pages doi:10.1155/2008/195137
Research Article
Euler-Lagrange Type Cubic Operators
and Their Norms on
X
λ
Space
Abbas Najati1 and Asghar Rahimi2
1Department of Mathematics, Faculty of Sciences, University of Mohaghegh Ardabili,
P.O. Box 56199-11367 Ardabili, Iran
2Department of Mathematics, University of Maragheh, P.O. Box 55181-83111, Maragheh,
East Azarbayjan, Iran
Correspondence should be addressed to Abbas Najati,[email protected]
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 spacesXλ andZλ5. These operators are constructed from the Euler-Lagrange type cubic functional equations and their Pexider versions.
Copyrightq2008 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
LetXandY be complex normed spaces. For a fixed nonnegative real numberλ,we denote by
Xλthe linear space of all functionsf:X→Y with pointwise operationsfor which there exists a constantMf ≥0 with
fx ≤Mfeλx 1.1
for allx∈X.It is easy to show that the spaceXλwith the norm
f:sup x∈X
e−λxfx 1.2
is a normed space. Let us denote byXnλthe linear space of all functionsφ:X× · · · ×X
ntimes
→Y with
pointwise operationsfor which there exists a constantMφ≥0 with
φx1, . . . , xn ≤Mφeλ
n
for allx1, . . . , xn∈X.It is not difficult to show that the spaceXnλ with the norm
φ: sup x1,...,xn∈X
φx1, . . . , xne−λ
n i1xi
1.4
is a normed space.
We denote by Zλm the normed space mi1Xλ {f1, . . . , fm : f1, . . . , fm ∈ Xλ} with
pointwise operationstogether 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 Dlutek1, 2. In3, Moslehian et al. have extended the results of2to the Pexiderized generalized Jensen and Pexiderized generalized quadratic operators on the function space Xλ and provided more general results regarding their norms.
In4, Jung investigated the norm of the cubic operator on the function spaceZλ5. A functionf :X→Y is called a cubic function if and only iffis a solution function of the cubic functional equation
fxy fx−y 2f
1 2xy
2f
1 2x−y
12f
1 2x
. 1.6
Jun and Kim5proved that when bothX andY are real vector spaces, a functionf : X→Y satisfies1.6if and only if there exists a functionB:X×X×X→Y such thatfx 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,
fxy fx−y af
1
axy
af
1
ax−y
2aa2−1f
1
ax
1.7
for fixed integersawitha /0,±1.Moreover, Jun and Kim7introduced the following Euler-Lagrange-type cubic functional equation
f
1
ax
1
by
f
1
bx
1
ay
aba−b2f 1
abx
f
1
aby
ababf
1
abx
1
aby
1.8
for fixed integersa, bwitha, b /0, a±b /0,and they proved the following theorem.
Theorem 1.1 see7, Theorem 2.1. Let X and Y be real vector spaces. If a mapping f : X→Y
satisfies the functional equation1.6, thenf satisfies the functional equation1.8.
We will introduce linear operators which are constructed from the Euler-Lagrange-type cubic and the Pexiderization of the Euler-Lagrange-type cubic functional equations1.7and
Definition 1.2. The operatorsC1P, CP2 :Z5
λ→X2λare defined by
CP
1f1, . . . , f5x, y:f1xy f2x−y−mf3
1
mxy
−mf4
1
mx−y
−2mm2−1f5
1
mx
,
CP
2f1, . . . , f5x, y:f1
1
ax
1
by
f2
1
bx
1
ay
−aba−b2f 3
1
abx
f4
1
aby
−ababf5
1
abx
1
aby
,
1.9
wherea,b, andmare fixed integers witha, b /0,a±b /0, andm /0,±1.
Definition 1.3. The operatorsC1, C2 :Xλ→Xλ2are defined by
C1fx, y:fxy fx−y−mf
1
mxy
−mf
1
mx−y
−2mm2−1f
1
mx
,
C2fx, y:f
1
ax
1
by
f
1
bx
1
ay
−aba−b2
f
1
abx
f
1
aby
−ababf
1
abx
1
aby
,
1.10
wherea,b, andmare fixed integers witha, b /0,a±b /0, andm /0,±1.
In this paper, we will give the exact norms of the operatorsCP1, CP2 on the function space
Z5
λ,and norms of the operatorsC1, C2on the function spaceXλ.The results extend the results of4.
2. Main results
Throughout this section,a,b, andmare fixed integers witha, b /0,a±b /0, andm /0,±1. The next theorems give us the exact norms of operatorsCP1,CP2,C1, andC2.
Theorem 2.1. The operatorCP1 :Z5λ→Xλ2is a bounded linear operator with
CP
12|m|32. 2.1
Proof. First, we show thatCP1 ≤2|m|32.Since
max
xy,x−y,m1xy,m1x−y,m1x
for allx, y ∈X,we get
CP
1f1, . . . , f5 sup
x,y∈Xe
−λxyf
1xy f2x−y−mf3
1
mxy
−mf4
1
mx−y
−2mm2−1f5
1
mx
≤ sup x,y∈Xe
−λxyf
1xy sup
x,y∈Xe
−λx−yf
2x−y
|m|sup x,y∈Xe
−λ1/mxyf 3
1
mxy
|m|sup x,y∈Xe
−λ1/mx−yf
4
1
mx−y
2|m|m2−1sup
x∈Xe
−λ1/mxf
5
1
mx
f1f2|m|f3|m|f42|m|m2−1f5
≤2|m|32max{f
1,f2,f3,f4,f5}
2|m|32f
1, f2, f3, f4, f5
2.3
for eachf1, . . . , f5∈Zλ5.This implies that
C1P ≤2|m|32. 2.4
Now, letν∈Y be such thatν1 and let{ξn}nbe a sequence of positive real numbers decreasing to 0.We define
fnx
⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩
e2λξnν, ifx2ξn or x0,
−|mm|e2λξnν, ifmx|m1|ξn, mx|m−1|ξn or mxξn,
0, otherwise
2.5
for allx∈X.Hence we have
e−λxf
nx
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
e2λξn, ifx0,
1, ifx2ξn,
e2−|m1/m|λξn, ifmx|m1|ξn,
e2−|m−1/m|λξn, ifmx|m−1|ξn,
e2−1/|m|λξn, ifmxξn,
0, otherwise
for allx∈X,so thatfn∈Xλ for all positive integersn,with
fne2λξn. 2.7
Letu∈Xbe such thatu1 and takex0, y0∈Xasx0y0ξnu.Then it follows from
the definition offnthat
CP
1fn, . . . , fn sup
x,y∈Xe
−λxyf
nxy fnx−y−mfn
1
mxy
−mfn
1
mx−y
−2mm2−1fn
1
mx
≥e−2λξne2λξnνe2λξnν|m|e2λξnν|m|e2λξnν2|m|m2−1e2λξnν
2|m|32.
2.8
If on the contraryCP1<2|m|32,then there exists aδ >0 such that
CP
1fn, . . . , fn ≤2|m|32−δfn, . . . , fn 2.9
for all positive integersn.So it follows from2.7,2.8, and2.9that
2|m|32≤ CP
1fn, . . . , fn ≤2|m|32−δe2λξn 2.10
for all positive integersn.Since limn→∞e2λξn 1,the right-hand side of2.10tends to 2|m|3
2 −δ as n→∞, whence 2|m|3 2 ≤ 2|m|32 −δ, which is a contradiction. Hence we have CP
12|m|32.
Theorem 2.1 of4is a result ofTheorem 2.1form2.
Corollary 2.2. The operatorC1:Xλ→Xλ2is a bounded linear operator with
C12|m|32. 2.11
Proof. The result follows from the proof ofTheorem 2.1.
Theorem 2.3. The operatorCP2 :Z5λ→X2
λis a bounded linear operator with
CP
22|ab|a−b2|abab|2. 2.12
Proof. Since
max1
ax
1
by
,b1x a1y,ab1 x,ab1 y,ab1 xab1 y
for allx, y ∈X,we get
CP
2f1, . . . , f5 sup
x,y∈Xe
−λxyf 1 1 ax 1 by f2 1 bx 1 ay
−aba−b2 f3 1 abx f4 1 aby
−ababf5 1 abx 1 aby ≤ sup x,y∈Xe
−λ1/ax1/byf
1 1 ax 1 by sup
x,y∈Xe
−λ1/bx1/ayf 2 1 bx 1 ay
|ab|a−b2sup
x∈Xe
−λ1/abxf 3 1 abx
|ab|a−b2sup
y∈Xe
−λ1/abyf 4 1 aby
|abab|sup x,y∈Xe
−λ1/abx1/abyf
5 1 abx 1 aby
≤ f1f2|ab|a−b2f3f4 |abab|f5
≤2|ab|a−b2|abab|2max{f
1,f2,f3,f4,f5}
2|ab|a−b2|abab|2f
1, f2, f3, f4, f5
2.14
for eachf1, . . . , f5∈Zλ5.This implies that
CP
2 ≤2|ab|a−b2|abab|2. 2.15
Letηbe 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, letu∈X, ν∈Y be such thatuν 1 and let{ξn}n be a sequence of positive real numbers decreasing to 0.We define
fnx
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
eλ1|η|ξnν, ifx
1 a η b
ξnu, or x
1 b η a
ξnu,
−|ab|
ab eλ1|η|ξnν, ifx
1
abξnu, or x η abξnu,
−|abab|
abab eλ1|η|ξnν, ifx
1η
ab ξnu,
0, otherwise
for allx∈X.Hence we have
e−λxf
nx
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
e1|η|−|1/aη/b|λξn, ifx
1
a η b
ξnu,
e1|η|−|1/bη/a|λξn, ifx
1
b η a
ξnu,
e1|η|−|1/ab|λξn, ifx 1
abξnu, e1|η|−|η/ab|λξn, ifx η
abξnu,
e1|η|−|1η/ab|λξn, ifx 1η
ab ξnu,
0, otherwise
2.18
for allx∈X,so thatfn∈Xλ for all positive integersn,with
fnmax{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
Letx0, y0∈Xbe such thatx0ξnuandy0ηξnu.Then it follows from the definition offnthat
CP
2fn, . . . , fn sup
x,y∈Xe
−λxyf
n
1
ax
1
by
fn
1
bx
1
ay
−aba−b2
fn
1
abx
fn
1
aby
−ababfn
1
abx
1
aby
≥e−λ1|η|ξneλ1|η|ξneλ1|η|ξn2|ab|a−b2eλ1|η|ξn|abab|eλ1|η|ξn
2|ab|a−b2|abab|2,
2.20
so that
CP
2fn, . . . , fn ≥2|ab|a−b
2|abab|2. 2.21
If on the contraryCP2<2|ab|a−b2|abab|2,then there exists aδ >0 such that
CP
for all positive integersn.So it follows from2.21and2.22that
2|ab|a−b2|abab|2≤ CP
2fn, . . . , fn ≤2|ab|a−b
2|abab|2−δf
n
2.23
for all positive integersn.Since limn→∞ξn 0,it follows from 2.19that limn→∞fn 1, so the right-hand side of2.23tends to 2|ab|a−b2|abab|2−δasn→∞,whence
2|ab|a−b2|abab|2≤2|ab|a−b2|abab|2−δ, 2.24
which is a contradiction. Hence we haveCP22|ab|a−b2|abab|2.
Corollary 2.4. The operatorC2:Xλ→Xλ2is a bounded linear operator with
C22|ab|a−b2|abab|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
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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 inXλspaces,”Applied Mathematics Letters, vol. 20, no. 10, pp. 1082–1087, 2007.
4 S.-M. Jung, “Cubic operator norm onXλspace,”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.
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