PEXIDER TYPE QUARTIC OPERATORS AND THEIR NORMS IN Xλ
SPACES
ZHIHUA WANG and LEI LIU
Abstract. In this paper, we introduce linear operators and obtain their exact norms defined on the
function spaces Xλ and Zλ6 . These operators are constructed from the quartic functional equations
and their Pexider versions.
1. Introduction
Let X and Y be complex normed spaces. Given λ ≥ 0, denote by Xλ the linear space of all
functions f : X → Y with the condition
kf (x)k ≤ Mf eλkxk ,
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∀x ∈ X,
where Mf ≥ 0 is a constant depending on f . It is easy to show that the space Xλ is a normed
space if it is equipped with the norm
kf k := sup {e−λkxk kf (x)k}.
x∈X
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Received May 10, 2011.
2010 Mathematics Subject Classification. Primary 47A30, 47B38, 39B52.
Key words and phrases. Pexiderized quartic operator; operator norm; Xλ spaces.
Let us denote by Xλn the linear space of all functions ϕ : X × · · · × X → Y for which there exists
|
{z
}
n times
a constant Mϕ ≥ 0 with
kϕ(x1 , · · · , xn )k ≤ Mϕ e
λ
n
P
kxi k
i=1
,
∀x1 , · · · , xn ∈ X.
It is easy to see that the space Xλn with the norm
kϕk :=
sup
{e
−λ
n
P
i=1
kxi k
kϕ(x1 , · · · , xn )k}
x1 ,··· ,xn ∈X
is a normed space.
We denote by Zλm the normed space
f1 , · · · , fm ∈ Xλ } together with the norm
Lm
i=1
Xλ = {(f1 , · · · , fm ) :
k(f1 , · · · , fm )k := max{kf1 k, · · · , kfm k}.
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S. Czerwik and K. Dlutek [1, 2] investigated some properties of Pexiderized Cauchy, quadratic
and Jensen operators on the function space Xλ . These results have extended in the paper [7]. In
fact, M. S. Moslehian, T. Riedel and A. Saadatpour [7] studied the Pexiderized generalized Jensen
and Pexiderized generalized quadratic operators on the function space Xλ and provided more
general results regarding their norms. S. M. Jung [3] investigated the norm of the cubic operators
on the function spaces Zλ5 . Recently, A. Najati and A. Rahimi [8] introduced Euler-Lagrange type
cubic operators and gave their exact norms defined on the function spaces Xλ and Zλ5 .
S. H. Lee, S. M. Im and I. S. Hwang [5] considered the following quartic functional equation
f (x + y) + f (x − y) = 4f
1
1
x + y + 4f
x−y
2
2
(1.1)
+ 24f
1
x − 6f (y).
2
They obtained the general solution of equation (1.1) and proved the Hyers-Ulam- -Rassias stability
of this equation. Y. S. Lee and S. Y. Chung [6] introduced the following quartic functional equation,
which is equivalent to (1.1),
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2
f (x + y) + f (x − y) = a f
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1
1
2
x+y +a f
x−y
a
a
(1.2)
2
2
+ 2a (a − 1)f
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1
x − 2(a2 − 1)f (y)
a
for fixed integers a with a 6= 0, ±1. Moreover, D. S. Kang [4] introduced the following generalized
quartic functional equation
f
1
1
1
1
x+ y +f
x− y
b
a
b
a
1
1
1
1
= (ab)2 f
x+ y +f
x− y
ab
ab
ab
ab
(1.3)
2
2
2
+ 2a (a − b )f
1
1
2 2
2
x − 2b (a − b )f
y
ab
ab
for fixed integers a, b with a, b 6= 0, a ± b 6= 0.
Next, we will introduce linear operators which are constructed from the quartic and the Pexiderization of the quartic function equations (1.2) and (1.3).
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6
2
Definition 1.1. The operators QP
1 , Q2 : Zλ → Xλ are defined by
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QP
1 (f1 , · · ·
2
, f6 )(x, y) := f1 (x+y)+f2 (x−y)−m f3
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2
2
− 2m (m − 1)f5
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1
1
2
x+y −m f4
x−y
m
m
1
x + 2(m2 − 1)f6 (y),
m
QP
2 (f1 , · · ·
, f6 )(x, y) := f1
1
1
1
1
x + y + f2
x− y
b
a
b
a
1
1
1
1
− (ab) f3
x + y + f4
x− y
ab
ab
ab
ab
2
1
1
2 2
2
− 2a (a − b )f5
x + 2b (a − b )f6
y ,
ab
ab
where a, b and m are fixed integers with a, b 6= 0, a ± b 6= 0 and m 6= 0, ±1.
2
2
2
Definition 1.2. The operators Q1 , Q2 : Xλ → Xλ2 are defined by
1
Q1 (f )(x, y) := f (x + y) + f (x − y) − m f ( x + y) − m2 f
m
2
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1
x−y
m
1
− 2m (m − 1)f
x + 2(m2 − 1)f (y),
m
1
1
1
1
Q2 (f )(x, y) := f
x+ y +f
x− y
b
a
b
a
2
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1
1
1
1
− (ab) f
x+ y +f
x− y
ab
ab
ab
ab
2
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2
2
2
− 2a (a − b )f
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1
1
2 2
2
x + 2b (a − b )f
y
ab
ab
where a, b and m are fixed integers with a, b 6= 0, a ± b 6= 0 and m 6= 0, ±1.
P
6
In this paper, we will give the exact norms of the operators QP
1 , Q2 on the function space Zλ
and norms of the operators Q1 , Q2 on the function space Xλ .
2. Main results
Throughout this section, a, b and m are fixed integers with a, b 6= 0, a ± b 6= 0, and m 6= 0, ±1. In
P
the following theorems give us the exact norms of operators QP
1 , Q2 , Q1 and Q2 .
6
2
Theorem 2.1. The operator QP
1 : Zλ → Xλ is a bounded linear operator with
2
2
kQP
1 k = 2m (m + 1).
(2.1)
2
2
Proof. First, we show that kQP
1 k ≤ 2m (m + 1). Since it holds that
1
1
1 max kx + yk , kx − yk, x + y , x − y , x , kyk ≤ kxk + kyk
m
m
m
for all x, y ∈ X, we obtain
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kQP
1 (f1 , · · · , f6 )k
= sup e
−λ(kxk+kyk)
x,y∈X
f1 (x + y) + f2 (x − y) − m2 f3 1 x + y
m
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−m2 f4
1
1
x − y − 2m2 (m2 − 1)f5
x + 2(m2 − 1)f6 (y)
m
m
≤ sup e−λkx+yk kf1 (x + y)k + sup e−λkx−yk kf2 (x − y)k
x,y∈X
+m
2
x,y∈X
sup e
1
x+y k
−λk m
x,y∈X
+m
2
sup e
x,y∈X
1
−λk m
x−y k
f3 1 x + y m
f4 1 x − y m
1
1
+ 2(m2 − 1) sup e−λkyk kf6 (y)k
+ 2m2 (m2 − 1) sup e−λk m xk f
x
5
m
x∈X
y∈X
= kf1 k + kf2 k + m2 kf3 k + m2 kf4 k + 2m2 (m2 − 1)kf5 k + 2(m2 − 1)kf6 k
≤ 2m2 (m2 + 1) max{kf1 k, kf2 k, kf3 k, kf4 k, kf5 k, kf6 k}
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= 2m2 (m2 + 1)k(f1 , · · · , f6 )k
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for each (f1 , · · · , f6 ) ∈ Zλ6 . This implies that
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(2.2)
2
2
kQP
1 k ≤ 2m (m + 1).
For a fixed ν ∈ Y with kνk = 1 and
we define

e2λξn ν,





fn (x) =
(2.3)
− e2λξn ν,





0,
a sequence {ξn }n of positive real numbers decreasing to 0,
if kxk = 2ξn , kxk = 0 or kxk = ξn ,
1
1 if kxk = 1 ± ξn
or kxk = ξn ,
m
m
otherwise
for all x ∈ X. Then we have
(2.4)
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
















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e−λkxk kfn (x)k =


















e2λξn ,
eλξn ,
1,
if kxk = 0,
if kxk = ξn ,
if kxk = 2ξn ,
1
1 (2−|1+ m |)λξn
, if kxk = 1 + ξn ,
e
m
1
1 (2−|1− m |)λξn
, if kxk = 1 − ξn ,
e
m
1
1
e(2−| m |)λξn ,
if kxk = ξn ,
m
0,
otherwise
for all x ∈ X, so that fn ∈ Xλ for all positive integers n with
(2.5)
kfn k = e2λξn .
Let u ∈ X be such that kuk = 1 and take x, y ∈ X as x = y = ξn u. Then it follows from (2.3)
that
−λ(kxk+kyk)
kQP
kfn (x + y) + fn (x − y)
1 (fn , · · · , fn )k = sup e
x,y∈X
1
1
2
− m fn
x + y − m fn
x−y
m
m
1
x + 2(m2 − 1)fn (y)k
− 2m2 (m2 − 1)fn
m
2
(2.6)
≥ e−2λξn k2 e2λξn ν + 2m2 e2λξn ν + 2(m4 − 1) e2λξn νk
= 2m2 (m2 + 1).
2
2
If we assume that kQP
1 k < 2m (m + 1), then we can choose a positive constant ε with
(2.7)
2
2
kQP
1 (fn , · · · , fn )k ≤ (2m (m + 1) − ε)k(fn , · · · , fn )k
for all positive integers n. So it follows from (2.5), (2.6) and (2.7) that
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(2.8)
2
2
2λξn
2m2 (m2 + 1) ≤ kQP
1 (fn , · · · , fn )k ≤ (2m (m + 1) − ε) e
for all positive integers n. Since limn→∞ e2λξn = 1, the right-hand side of (2.8) tends to 2m2 (m2 +
1) − ε as n → ∞, whence 2m2 (m2 + 1) ≤ 2m2 (m2 + 1) − ε, which leads to a contradiction. Hence
2
2
we have kQP
1 k = 2m (m + 1). This completes the proof of the theorem.
Corollary 2.1. The operator Q1 : Xλ → Xλ2 is a bounded linear operator with
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(2.9)
kQ1 k = 2m2 (m2 + 1).
Proof. The result follows from the proof of Theorem 2.1.
The following corollary is a result of Theorem 2.1 for m = 2.
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2
Corollary 2.2. The Pexiderized quartic operator QP
1 : Zλ → Xλ given by
1
1
QP
(f
,
·
·
·
,
f
)(x,
y)
:=
f
(x
+
y)
+
f
(x
−
y)
−
4f
x
+
y
−
4f
x
−
y
1
6
1
2
3
4
1
2
2
1
− 24f5
x + 6f6 (y)
2
is a bounded linear operator with kQP
1 k = 40.
The following corollary is a result of Corollary 2.1 for m = 2.
Corollary 2.3. The quartic operator Q1 : Xλ → Xλ2 given by
1
1
Q1 (f )(x, y) := f (x + y) + f (x − y) − 4f
x + y − 4f
x−y
2
2
1
− 24f
x + 6f (y)
2
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is a bounded linear operator with kQ1 k = 40.
6
2
Theorem 2.2. The operator QP
2 : Zλ → Xλ is a bounded linear operator with
(2.10)
4
4
2
kQP
2 k = 2|a − b | + 2(ab) + 2.
4
4
2
Proof. First, we prove that kQP
2 k ≤ 2|a − b | + 2(ab) + 2. By the assumption we obtain
1
1
1 1 1 1 max x ± y , x ± y , x , y ≤ kxk + kyk
b
a
ab
ab
ab
ab for all x, y ∈ X. Hence we obtain
kQP
2 (f1 , · · · , f6 )k
1
1
1
1
= sup e−λ(kxk+kyk) f
x
+
y
+
f
x
−
y
2
1 b
a
b
a
x,y∈X
1
1
1
1
− (ab)2 f3
x + y + f4
x− y
ab
ab
ab
ab
1
1
−2(a2 − b2 ) a2 f5
x − b2 f 6
y ab
ab
1 1
≤ sup e
f1 b x + a y x,y∈X
1
1
1 −λk 1b x− a
yk + sup e
f2 b x − a y x,y∈X
1
1
1
1
+ (ab)2 ( sup e−λk ab x+ ab yk f
x
+
y
3
ab
ab x,y∈X
1
1
1
1 f
x
−
y)
+ sup e−λk ab x− ab yk 4 ab
ab x,y∈X
1
1
1
1
2
2
2
−λk ab
xk 2
−λk ab
yk + 2|a − b | a sup e
x + b sup e
y f5
f6
ab
ab
x∈X
y∈X
1
−λk 1b x+ a
yk
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= kf1 k + kf2 k + (ab)2 (kf3 k + kf4 k) + 2a2 |a2 − b2 |kf5 k + 2b2 |a2 − b2 |kf6 k
≤ (2|a4 − b4 | + 2(ab)2 + 2) max{kf1 k, kf2 k, kf3 k, kf4 k, kf5 k, kf6 k}
= (2|a4 − b4 | + 2(ab)2 + 2)k(f1 , · · · , f6 )k
for each (f1 , · · · , f6 ) ∈ Zλ6 . This implies that
(2.11)
4
4
2
kQP
2 k ≤ 2|a − b | + 2(ab) + 2.
Let η be a real number such that
1
1−a 1−a 1−a a
a
(2.12)
η∈
/ 0, , 1, ±
,±
,±
,
,
.
2
1−b
1+b
b
1−b 1+b
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Let u ∈ X, ν ∈ Y be such that kuk = kνk =
numbers decreasing to 0. We define



eλ(1+|η|)ξn ν,







− eλ(1+|η|)ξn ν,
(2.13)
fn (x) =



|a2 −b2 | λ(1+|η|)ξn


−
ν,

a2 −b2 e



0,
1 and let {ξn }n be a sequence of positive real
1 η
if x = ( ± )ξn u,
b a
1
η
if x = ( ± )ξn u,
ab ab
η
1
if x = ξn u, or x = ξn u,
ab
ab
otherwise
for all x ∈ X. Then we obtain
(2.14)
e−λkxk kfn (x)k =

η
1


e(1+|η|−| b + a |)λξn ,







η
1


e(1+|η|−| b − a |)λξn ,







η
1


e(1+|η|−| ab + ab |)λξn ,


























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1 η
+
ξn u,
b a
1 η
if x =
−
ξn u,
b a
1
η
if x =
+
ξn u,
ab ab
η
1
η
1
e(1+|η|−| ab − ab |)λξn , if x =
−
ξn u,
ab ab
1
1
e(1+|η|−| ab |)λξn ,
if x = ξn u,
ab
η
η
e(1+|η|−| ab |)λξn ,
if x = ξn u,
ab
0,
otherwise
if x =
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for all x ∈ X, so that fn ∈ Xλ for all positive integers n with
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1
η
1
η
1
η
kfn k = max{e(1+|η|−| b + a |)λξn , e(1+|η|−| b − a |)λξn , e(1+|η|−| ab + ab |)λξn ,
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(2.15)
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η
1
η
e(1+|η|−| ab − ab |)λξn , e(1+|η|−| ab |)λξn , e(1+|η|−| ab |)λξn }.
Let x, y ∈ X be such that x = ξn u and y = ηξn u. Then it follows from the definition of fn that
1
1
1
1
P
−λ(kxk+kyk) kQ2 (fn , · · · , fn )k = sup e
f1 b x + a y + f2 b x − a y
x,y∈X
1
1
1
1
− (ab) f3
x + y + f4
x− y
ab
ab
ab
ab
2
−2a2 (a2 − b2 )f5
(2.16)
1
1
x + 2b2 (a2 − b2 )f6
y ab
ab ≥ e−λ(1+|η|)ξn k2e−λ(1+|η|)ξn + 2(ab)2 e−λ(1+|η|)ξn
+ 2(a2 + b2 )|a2 − b2 |e−λ(1+|η|)ξn k
= 2|a4 − b4 | + 2(ab)2 + 2.
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4
4
2
If on the contrary kQP
2 k < 2|a − b | + 2(ab) + 2, then there exists ε > 0 such that
(2.17)
4
4
2
kQP
2 (fn , · · · , fn )k ≤ (2|a − b | + 2(ab) + 2 − ε)k(fn , · · · , fn )k
for all positive integers n. So it follows from (2.16) and (2.17) that
2|a4 − b4 | + 2(ab)2 + 2 ≤ kQP
2 (fn , · · · , fn )k
(2.18)
≤ (2|a4 − b4 | + 2(ab)2 + 2 − ε)kfn k
for all positive integers n. Since lim ξn = 0, it follows from (2.15) that lim kfn k = 1, so the
n→∞
n→∞
right-hand side of (2.18) tends to 2|a4 − b4 | + 2(ab)2 + 2 − ε as n → ∞, whence
(2.19)
2|a4 − b4 | + 2(ab)2 + 2 ≤ 2|a4 − b4 | + 2(ab)2 + 2 − ε,
4
4
2
which is a contradiction. Hence we have kQP
2 k = 2|a − b | + 2(ab) + 2. This completes the proof
of the theorem.
Corollary 2.4. The operator Q2 : Xλ → Xλ2 is a bounded linear operator with
(2.20)
kQ2 k = 2|a4 − b4 | + 2(ab)2 + 2.
Proof. The result follows from the proof of Theorem 2.2.
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1. Czerwik S., Functional Equations and Inequalities in Several Variables, World Scientific, River Edge, NJ, USA,
2002.
2. Czerwik S. and Dlutek K., Cauchy and Pexider operators in some function spaces, In Functional Equations,
Inequalities and Applications, (Edited by Th. M. Rassias), Kluwer Acad. Publ., Dordrecht, 2003, 11–19.
3. Jung S. M., Cubic operator norm on Xλ space, Bull. Korean Math. Soc. 44 (2007), 309–313.
4. Kang D. S., On the stability of generalized quartic mappings in quasi-β normed spaces, J. Ineq. Appl. 2010,
Article ID 198098, 11 pages.
5. Lee S. H., Im S. M. and Hwang I. S., Quartic functional equations, J. Math. Anal. Appl. 307 (2005), 387–394.
6. Lee Y. S. and Chung S. Y., Stability of quartic functional equations in the spaces of generalized functions,
Advances in Difference Equations, 2009, Article ID 838347, 16 pages.
7. Moslehian M. S., Riedel T. and Saadatpour A., Norms of operators in Xλ spaces, Appl. Math. Lett. 20(2007),
1082–1087.
8. Najati A. and Rahimi A., Euler-Lagrange type cubic operators and their norms on Xλ space, J. Ineq. Appl.
2008, Article ID 195137, 8 pages.
Zhihua Wang, School of Science, Hubei University of Technology, Wuhan, Hubei 430068, PR China, e-mail:
matwzh2000@126.com
Lei Liu, Department of Mathematics, Shangqiu Normal University, Shangqiu, Henan 476000, PR China, e-mail:
liugh105@163.com
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