61
Acta Math. Univ. Comenianae
Vol. LXXXI, 1 (2012), pp. 61–69
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 ,
∀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
Let us denote by
Xλn
the linear space of all functions ϕ : X × · · · × X → Y for
{z
}
|
n times
which there exists 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}.
S. Czerwik and K. Dlutek [1, 2] investigated some properties of Pexiderized
Cauchy, quadratic and Jensen operators on the function space Xλ . These results
Received May 10, 2011.
2010 Mathematics Subject Classification. Primary 47A30, 47B38, 39B52.
Key words and phrases. Pexiderized quartic operator; operator norm; Xλ spaces.
62
ZHIHUA WANG and LEI LIU
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
1
1
x + y + 4f
x−y
f (x + y) + f (x − y) = 4f
2
2
(1.1)
1
+ 24f
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),
1
1
x + y + a2 f
x−y
f (x + y) + f (x − y) = a2 f
a
a
(1.2)
1
+ 2a2 (a2 − 1)f
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
1
1
1
1
f
x+ y +f
x− y
b
a
b
a
(1.3)
1
1
1
1
= (ab) f
x+ y +f
x− y
ab
ab
ab
ab
2
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).
P
6
2
Definition 1.1. The operators QP
1 , Q2 : Zλ → Xλ are defined by
1
1
2
2
x+y
−m
f
x−y
QP
(f
,
·
·
·
,
f
)(x,
y)
:=
f
(x+y)+f
(x−y)−m
f
4
1
6
1
2
3
1
m
m
− 2m2 (m2 − 1)f5
1
x + 2(m2 − 1)f6 (y),
m
PEXIDER TYPE QUARTIC OPERATORS AND THEIR NORMS
QP
2 (f1 , · · ·
, f6 )(x, y) := f1
63
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
2
2
2
− 2a (a − b )f5
1
1
2 2
2
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.
Definition 1.2. The operators Q1 , Q2 : Xλ → Xλ2 are defined by
1
1
Q1 (f )(x, y) := f (x + y) + f (x − y) − m2 f ( x + y) − m2 f
x−y
m
m
2
2
− 2m (m − 1)f
Q2 (f )(x, y) := f
1
x + 2(m2 − 1)f (y),
m
1
1
1
1
x+ y +f
x− y
b
a
b
a
1
1
1
1
− (ab) f
x+ y +f
x− y
ab
ab
ab
ab
2
− 2a2 (a2 − b2 )f
1
1
x + 2b2 (a2 − b2 )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
In this paper, we will give the exact norms of the operators QP
1 , Q2 on the
6
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 the following theorems give us the exact norms of operators
P
QP
1 , Q2 , Q1 and Q2 .
6
2
Theorem 2.1. The operator QP
1 : Zλ → Xλ is a bounded linear operator with
(2.1)
2
2
kQP
1 k = 2m (m + 1).
64
ZHIHUA WANG and LEI LIU
2
2
Proof. First, we show that kQP
1 k ≤ 2m (m + 1). Since it holds that
1
1
, x − y , 1 x , kyk ≤ kxk + kyk
max kx + yk , kx − yk, x
+
y
m
m
m for all x, y ∈ X, we obtain
kQP
1 (f1 , · · · , f6 )k
−λ(kxk+kyk)
= sup e
x,y∈X
2
−m f4
f1 (x + y) + f2 (x − y) − m2 f3 1 x + y
m
1
1
2
2
2
x − y − 2m (m − 1)f5
x + 2(m − 1)f6 (y)
m
m
≤ sup e−λkx+yk kf1 (x + y)k + sup e−λkx−yk kf2 (x − y)k
x,y∈X
x,y∈X
1
1
+ m2 sup e−λk m x+yk f
x
+
y
3
m
x,y∈X
+m
2
sup e
x,y∈X
1
−λk m
x−y k
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}
= 2m2 (m2 + 1)k(f1 , · · · , f6 )k
for each (f1 , · · · , f6 ) ∈ Zλ6 . This implies that
(2.2)
2
2
kQP
1 k ≤ 2m (m + 1).
For a fixed ν ∈ Y with kνk = 1 and a sequence {ξn }n of positive real numbers
decreasing to 0, we define

2λξn
ν,
if kxk = 2ξn , kxk = 0 or kxk = ξn ,

 e



1
2λξn
(2.3) fn (x) =
1 ± 1 ξn
ξn ,
−
e
ν,
if
kxk
=
or
kxk
=
m


m



0,
otherwise
PEXIDER TYPE QUARTIC OPERATORS AND THEIR NORMS
for all x ∈ X. Then we have


















−λkxk
(2.4)
e
kfn (x)k =

















65
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
e(2−|1− m |)λξn , if kxk = 1 − ξn ,
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
kfn k = e2λξn .
(2.5)
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
(2.6)
1
1
x + y − m2 fn
x−y
m
m
1
− 2m2 (m2 − 1)fn
x + 2(m2 − 1)fn (y)k
m
− m2 fn
≥ 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
(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
2
2
leads to a contradiction. Hence 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
(2.9)
kQ1 k = 2m2 (m2 + 1).
66
ZHIHUA WANG and LEI LIU
Proof. The result follows from the proof of Theorem 2.1.
The following corollary is a result of Theorem 2.1 for m = 2.
6
2
Corollary 2.2. The Pexiderized quartic operator QP
1 : Zλ → Xλ given by
QP
1 (f1 , · · · , f6 )(x, y) := f1 (x + y) + f2 (x − y) − 4f3
1
− 24f5
x + 6f6 (y)
2
1
1
x + y − 4f4
x−y
2
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
Q1 (f )(x, y) := f (x + y) + f (x − y) − 4f
1
x + 6f (y)
− 24f
2
1
1
x + y − 4f
x−y
2
2
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 x ± 1 y , 1 x , 1 y ≤ kxk + kyk
max x
±
y
b
a
ab
ab
ab
ab PEXIDER TYPE QUARTIC OPERATORS AND THEIR NORMS
67
for all x, y ∈ X. Hence we obtain
kQP
2 (f1 , · · · , f6 )k
1
1
1
1
= sup e
f1 b x + a y + f2 b x − a y
x,y∈X
1
1
1
1
2
− (ab) f3
x + y + f4
x− y
ab
ab
ab
ab
1
1
−2(a2 − b2 ) a2 f5
x − b2 f 6
y ab
ab
−λ(kxk+kyk)
1
1 ≤ sup e
f1 b x + a y x,y∈X
1
1
1
1 + sup e−λk b x− a yk f
x
−
y
2 b
a x,y∈X
1
1
1
1
−λk ab
x+ ab
yk 2
+ (ab) ( sup e
f3
x+ y ab
ab x,y∈X
1
1
1 1
+ sup e−λk ab x− ab yk x − y)
f4
ab
ab x,y∈X
1
1
1
1
2
−λk ab
yk + 2|a2 − b2 | a2 sup e−λk ab xk +
b
sup
e
f
x
f
y
5
6
ab
ab x∈X
y∈X
1
−λk 1b x+ a
yk
= 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
Let u ∈ X, ν ∈ Y be such that kuk = kνk = 1 and let {ξn }n be a sequence of
positive real numbers decreasing to 0. We define

1 η

 eλ(1+|η|)ξn ν,
if x = ( ± )ξn u,


b a




1
η

λ(1+|η|)ξn
−e
ν,
if x = ( ± )ξn u,
(2.13) fn (x) =
ab ab


2
2

1
η
| λ(1+|η|)ξn


− |aa2 −b
ν, if x = ξn u, or x = ξn u,

−b2 e

ab
ab


0,
otherwise
68
ZHIHUA WANG and LEI LIU
for all x ∈ X. Then we obtain

η
1


e(1+|η|−| b + a |)λξn ,







η
1


e(1+|η|−| b − a |)λξn ,







η
1

+ ab
|)λξn
 e(1+|η|−| ab
,



(2.14) e−λkxk kfn (x)k =
η
1

e(1+|η|−| ab − ab |)λξn ,






 (1+|η|−| 1 |)λξn

ab

,
e





η
 (1+|η|−| ab
|)λξn

,
e






0,
1 η
if x =
+
ξn u,
b a
1 η
if x =
−
ξn u,
b a
1
η
if x =
+
ξn u,
ab ab
1
η
if x =
−
ξn u,
ab ab
1
if x = ξn u,
ab
η
if x = ξn u,
ab
otherwise
for all x ∈ X, so that fn ∈ Xλ for all positive integers n with
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 ,
(2.15)
1
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)2 f3
x + y + f4
x− y
ab
ab
ab
ab
−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.
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
PEXIDER TYPE QUARTIC OPERATORS AND THEIR NORMS
69
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 =
n→∞
n→∞
1, so the 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.
References
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