Gen. Math. Notes, Vol. 15, No. 2, April, 2013, pp.1-13
c
ISSN 2219-7184; Copyright ICSRS
Publication, 2013
www.i-csrs.org
Available free online at http://www.geman.in
Stability of Quadratic Functional
Equations in 2-Banach Space
B.M. Patel1 and A.B. Patel2
1,2
Department of Mathematics, Sardar Patel University
Vallabh Vidyanagar-388120
Gujarat, India
1
E-mail: bhavinramani@yahoo.com
2
E-mail: abp1908@gmail.com
(Received: 17-1-13 / Accepted: 23-2-13)
Abstract
In this paper, we investigate the Hyers-Ulam stability of the functional equation f (2x + y) − f (x + 2y) = 3f (x) − 3f (y) in 2-Banach space.
Keywords: Hyers-Ulam stability, 2-Banach space, Quadratic functional
equation.
1
Introduction
Stability of for a function from a normed space to a Banach space has been
studied by Hyers [4]. Skof [12] has proved Hyers-Ulam stability of the
functional equation
f (x + y) + f (x − y) = 2f (x) + 2f (y)
(1)
He has proved that for a function f : X −→ Y , a function between normed
space X to Banach space Y satisfying
kf (x + y) + f (x − y) = 2f (x) + 2f (y)k ≤ δ
for each x, y ∈ X and δ > 0, there exists a unique quadratic function
Q : X −→ Y such that
δ
kf (x) − Q(x)k <
2
2
B.M. Patel et al.
The quadratic function f (x) = cx2 satisfies the functional equation (1) and
therefore Equation (1) is called the quadratic functional equation. Every solution of Equation (1) is said to be a quadratic mapping.
In fact several authors have studied the stability of different types of functional equations for functions from normed space to Banach space. (see [1, 2,
5, 6, 7, 8, 9, 10]).
Our aim is to study the Hyers-Ulam stability of the functional equation
f (2x + y) − f (x + 2y) = 3f (x) − 3f (y)
(2)
introduced by [15], for a function from 2-normed space (normed space) to
2-Banach space.
Theorem 1.1 [15] Let X and Y be real vector spaces, and let f : X −→ Y be
a function satisfies (2) if and only if f (x) = B(x, x) + C, for some symmetric
bi-additive function B : X × X −→ Y , for some C in Y . Therefore every
solution f of functional equation (2) with f (0) = 0 is also a quadratic function.
In the 1960s, S. Gähler [3] introduced the concept of 2-normed spaces. We
first introduce 2-normed space and topology on it.
Definition 1.2 Let X be a linear space over R with dim X > 1 and let
k·, ·k : X × X −→ R be a function satisfying the following properties:
1. kx, yk = 0 if and only if x and y are linearly dependent,
2. kx, yk = ky, xk,
3. kax, yk = |a|kx, yk,
4. kx, y + zk ≤ kx, yk + kx, zk
for each x, y, z ∈ X and a ∈ R. Then the function k·, ·k is called a 2-norm on
X and (X, k·, ·k) is called a 2-normed space.
We introduce a basic property of 2-normed spaces as follows. Let (X, k·, ·k) be
a linear 2-normed space, x ∈ X and kx, yk = 0 for each y ∈ X. Suppose x 6= 0,
since dim X > 1, choose y ∈ X such that {x, y} is linearly independent so we
have kx, yk =
6 0, which is a contradiction. Therefore, we have the following
lemma.
Lemma 1.3 Let (X, k·, ·k) be a 2-normed space. If x ∈ X and kx, yk = 0, for
each y ∈ X, then x = 0.
3
Stability of Quadratic Functional...
Let (X, k·, ·k) be a 2-normed space. For x, z ∈ X, let pz (x) = kx, zk,
x ∈ X. Then for each z ∈ X, pz is a real-valued function on X such that
pz (x) = kx, zk ≥ 0, pz (αx) = |α|kx, zk = |α|pz (x) and
pz (x + y) = kx + y, zk = kz, x + yk ≤ kz, xk + kz, yk = kx, zk + ky, zk =
pz (x) + pz (y), for each α ∈ R and all x, y ∈ X. Thus pz is a a semi-norm for
each z ∈ X.
For x ∈ X, let kx, zk = 0, for each z ∈ X. By Lemma 1.3, x = 0. Thus
for 0 6= x ∈ X, there is z ∈ X such that pz (x) = kx, zk =
6 0. Hence the family
{pz (x) : z ∈ X} is a separating family of semi-norms.
Let x0 ∈ X, for ε > 0, z ∈ X, let
Uz,ε (x0 ) := {x ∈ X : pz (x−x0 ) < ε} = {x ∈ X : kx−x0 , zk < ε}. Let S(x0 ) :=
{Uz,ε (x0 ) : ε > 0, z ∈ X} and β(x0 ) := {∩F : F is a finite subcollection of S(x0 )}.
Define a topology τ on X by saying that a set U is open if for every x ∈ U ,
there is some N ∈ β(x) such that N ⊂ U . That is, τ is the topology on X that
has subbase {Uz,ε (x0 ) : ε > 0, x0 ∈ X, z ∈ X}. The topology τ on X makes
X a topological vector space. Since for x ∈ X collection β(x) is a local base
whose members are convex, X is locally convex.
In the 1960s, S. Gähler and A. White [14] introduced the concept of
2-Banach spaces.
Definition 1.4 A sequence {xn } in a 2-normed space X is called a 2-Cauchy
sequence if
lim kxn − xm , xk = 0
m,n→∞
for each x ∈ X.
Definition 1.5 A sequence {xn } in a 2-normed space X is called a 2-convergent
sequence if there is an x ∈ X such that
lim kxn − x, yk = 0
n→∞
for each y ∈ X. If {xn } converges to x, we write limn→∞ xn = x.
Definition 1.6 We say that a 2-normed space (X, k·, ·k) is a 2-Banach space
if every 2-Cauchy sequence in X is 2-convergent in X.
By using (2) and (4) of Definition 1.2 one can see that k·, ·k is continuous in
each component. More precisely for a convergent sequence {xn } in a 2-normed
space X,
lim kxn , yk = lim xn , y n→∞
for each y ∈ X.
n→∞
4
2
B.M. Patel et al.
Stability of a Functional Equation for
Functions f : (X, k · k) −→ (X, k·, ·k)
Throughout this section, consider X a real normed linear space. We also
consider that there is a 2-norm on X which makes (X, k·, ·k) a 2-Banach space.
For a function f : (X, k · k) −→ (X, k·, ·k), define Df : X × X −→ X by
Df (x, y) = f (2x + y) − f (x + 2y) − 3f (x) + 3f (y)
for each x, y, ∈ X.
Theorem 2.1 Let ε ≥ 0, 0 < p, q < 2, r > 0. If f : X −→ X is a function
such that
kDf (x, y), zk ≤ ε(kxkp + kykq )kzkr
(3)
for each x, y, z ∈ X. Then there exists a unique quadratic function Q : X −→
X satisfying (2) and
kf (x) − Q(x) − f (0), zk ≤
εkxkp kzkr
4 − 2p
(4)
for each x, z ∈ X.
Proof 2.1 Let g : X −→ X be a function defined by g(x) = f (x) − f (0),
for each x ∈ X. Then g(0) = 0. Also
kDg (x, y), zk = kg(2x + y) − g(x + 2y) − 3g(x) + 3g(y), zk
≤ ε(kxkp + kykq )kzkr
(5)
for each x, z ∈ X. Putting y = 0 in (5), we get
kg(2x) − 4g(x), zk ≤ εkxkp kzkr
(6)
for each x, z ∈ X. Therefore
ε
1
g(x) − g(2x), z ≤ kxkp kzkr
4
4
(7)
for each x, z ∈ X. Replacing x by 2x in (7), we get
ε2p
1
kxkp kzkr
g(2x) − g(4x), z ≤
4
4
(8)
5
Stability of Quadratic Functional...
for each x, z ∈ X. By (7) and (8), we get
1
1
1
1
g(x) − g(4x), z ≤ g(x) − g(2x), z + g(2x) − g(4x), z 16
4
4
16
ε 2p
ε
kxkp kzkr
≤ kxkp kzkr +
4
44
εkxkp kzkr h
2p i
=
1+
4
4
for each x, z ∈ X. By using induction on n, we get
n−1 pj
εkxkp kzkr X
1
2
n
g(x) − n g(2 x), z ≤
4
4
4j
j=0
"
#
εkxkp kzkr 1 − 2(p−2)n
=
4
1 − 2p−2
(9)
for each x, z ∈ X. For m, n ∈ N, for x ∈ X
1
1
1
1
m
n
m+n−n
n
g(2
x)
−
g(2
x),
z
=
g(2
x)
−
g(2
x),
z
m
m+n−n
n
n
4
4
4
4
1 1
= n m−n g(2m−n · 2n x) − g(2n x), z 4 4
m−n−1
εk2n xkp kzkr X (p−2)j
≤
2
4 · 4n
j=0
=
m−n−1
εkxkp kzkr X (p−2)(n+j)
2
4
j=0
εkxkp kzkr 2(p−2)n 1 − 2(p−2)(m−n)
=
4
1 − 2p−2
−→ 0 as m, n → ∞
for each z ∈ X. Therefore, { 41n g(2n x)} is a 2-Cauchy sequence in X, for each
x ∈ X. Since X is a 2-Banach space, { 41n g(2n x)} 2-converges, for each x ∈ X.
Define the function Q : X −→ X as
1
g(2n x)
n→∞ 4n
Q(x) = lim
for each x ∈ X. Now, from (9)
εkxkp kzkr
1
1
n
lim g(x) − n g(2 x), z ≤
n→∞
4
4
1 − 2p−2
6
B.M. Patel et al.
for each x, z ∈ X. Therefore
kf (x) − Q(x) − f (0), zk ≤
εkxkp kzkr
4 − 2p
for each x, z ∈ X. Next we show that Q satisfies (2). For x ∈ X
1
kDg (2n x, 2n y), zk
n→∞ 4n
ε
= lim n (k2n xkp + k2n ykq )kzkr
n→∞ 4
= lim ε 2(p−2)n kxkp + 2(q−2)n kykq kzkr
kDQ (x, y), zk = lim
n→∞
=0
for each z ∈ X. Therefore kDQ (x, y), zk = 0, for each z ∈ X. So we get
DQ (x, y) = 0. Next we prove the uniqueness of Q. Let Q0 be another quadratic
function satisfying (2) and (4). Since Q and Q0 are quadratic, Q(2n x) =
4n Q(x), Q0 (2n x) = 4n Q0 (x), for each x ∈ X. Now for x ∈ X
1
kQ(2n x) − Q0 (2n x), zk
n
4
1
≤ n [kQ(2n x) − g(2n x), zk + kg(2n x) − Q0 (2n x), zk]
4
1 2εk2n xkp kzkr
≤ n
4
4 − 2p
2ε
= 2(p−2)n
kxkp kzkr
4 − 2p
−→ 0 as n → ∞
kQ(x) − Q0 (x), zk =
for each z ∈ X. Therefore kQ(x) − Q0 (x), zk = 0, for each z ∈ X. Therefore
Q(x) = Q0 (x), for each x ∈ X.
Theorem 2.2 Let ε ≥ 0, p, q > 2, r > 0. If f : X −→ X is a function such
that
kDf (x, y), zk ≤ ε(kxkp + kykq )kzkr
(10)
for each x, y, z ∈ X. Then there exists a unique quadratic function Q : X −→
X satisfying (2) and
kf (x) − Q(x) − f (0), zk ≤
for each x, z ∈ X.
εkxkp kzkr
2p − 4
(11)
7
Stability of Quadratic Functional...
Proof 2.2 By (6) of Theorem 2.1, we have
kg(2x) − 4g(x), zk ≤ εkxkp kzkr
for each x, z ∈ X. Replacing x by x2 in (12), we get
x , z ≤ ε2−p kxkp kzkr
g(x) − 4g
2
for each x, z ∈ X. Replacing x by x2 in (13), we get
x
x − 4g
, z ≤ ε2−2p kxkp kzkr
g
2
4
(12)
(13)
(14)
for each x, z ∈ X. Combining (13) and (14), we get
x x
x x
, z + 4g
− 16g
, z
kg(x) − 16g( ), zk ≤ g(x) − 4g
4
2
2
4
≤ ε2−p kxkp kzkr + 4ε2−2p kxkp kzkr
= εkxkp kzkr [2−p + 2−p · 4]
for each x, z ∈ X. By using induction on n, we have
n−1
x X
n
p
r
4j 2p(−j−1)
g(x) − 4 g n , z ≤ εkxk kzk
2
j=0
p
= εkxk kzk
r
n−1
X
2(−p+2)j−p
j=0
= εkxkp kzkr
2−p (1 − 2(2−p)n ) 1 − 22−p
for each x, z ∈ X. For m, n ∈ N and for x ∈ X
x x x x m
m+n−n
n
n
g m+n−n − 4 g n , z 4 g m − 4 g n , z = 4
2
2
2 x
x2 n m−n
g m−n n − g n , z = 4 4
2
·2
2
m−n−1
x p
X
n
r
≤ 4 · ε n kzk
2(−p+2)j−p
2
j=0
p
r
= εkxk kzk
m−n−1
X
2(2−p)(n+j)−p
j=0
h 2(−p+2)n−p 1 − 2(−p+2)n i
= εkxkp kzkr
1 − 2−p+2
−→ 0 as n → ∞
(15)
8
B.M. Patel et al.
for each z ∈ X. Therefore {4n f ( 2xn )} is a 2-Cauchy sequence in X, for each
x ∈ X. Since X is a 2-Banach space, the sequence {4n f ( 2xn )} 2-converges, for
each x ∈ X. Define Q : X −→ X as
x
Q(x) := lim 4n f n
n→∞
2
for each x ∈ X. Now from (15),
x 2−p
n
lim g(x) − 4 g n , z ≤ εkxkp kzkr
n→∞
2
1 − 22−p
for each x, z ∈ X. Therefore
kf (x) − Q(x) − f (0), zk ≤
εkxkp kzkr
2p − 4
for each x, z ∈ X. The further part of the proof is similar to that of the proof
of Theorem 2.1.
3
Stability of a Functional Equation for
Function f : (X, k·, ·k) −→ (X, k·, ·k)
In this section we study similar problems which we have studied in section 2
for functions f : X −→ X, where (X, k·, ·k) is a 2-Banach space.
Theorem 3.1 Let ε ≥ 0, 0 < p, q < 2. If f : X −→ X is a function such that
kDf (x, y), zk ≤ ε(kx, zkp + ky, zkq )
(16)
for each x, y, z ∈ X. Then there exists a unique quadratic function Q : X −→
X satisfying (2) and
kf (x) − Q(x) − f (0), zk ≤
εkx, zkp
4 − 2p
(17)
for each x, z ∈ X.
Proof 3.1 Let g : X −→ X be a function defined by g(x) = f (x) − f (0),
for each x ∈ X. Then g(0) = 0. Also
kDg (x, y), zk = kg(2x + y) − g(x + 2y) − 3g(x) + 3g(y), zk
≤ ε(kx, zkp + ky, zkq )
(18)
for each x, z ∈ X. Putting y = 0 in (18), we get
kg(2x) − 4g(x), zk ≤ εkx, zkp
(19)
9
Stability of Quadratic Functional...
for each x, z ∈ X. Therefore
ε
1
g(x) − g(2x), z ≤ kx, zkp
4
4
(20)
for each x, z ∈ X. Replacing x by 2x in (20), we get
ε2p
1
kx, zkp
g(2x) − g(4x), z ≤
4
4
(21)
for each x, z ∈ X. By (20) and (21), we get
1
1
1
1
g(x) − g(4x), z ≤ g(x) − g(2x), z + g(2x) − g(4x), z 16
4
4
16
p
ε
ε
2
≤ kxkp kzkr +
kx, zkp
4
44
εkx, zkp h
2p i
=
1+
4
4
for each x, z ∈ X. By using induction on n, we get
n−1 pj
εkx, zkp X
1
2
n
g(x) − n g(2 x), z ≤
4
4
4j
j=0
n−1
εkx, zkp X (p−2)j
=
2
4
j=0
"
#
p
εkx, zk 1 − 2(p−2)n
=
4
1 − 2p−2
(22)
for each x, z ∈ X. For m, n ∈ N for x ∈ X
1
1
1
1
m
n
m+n−n
n
g(2
x)
−
g(2
x),
z
=
g(2
x)
−
g(2
x),
z
m
m+n−n
4
4n
4
4n
1 1
= n m−n g(2m−n · 2n x) − g(2n x), z 4 4
m−n−1
εk2n x, zkp X (p−2)j
≤
2
4 · 4n
j=0
m−n−1
εkx, zkp X (p−2)(n+j)
=
2
4
j=0
εkx, zkp 2(p−2)n 1 − 2(p−2)(m−n)
=
4
1 − 2p−2
−→ 0 as m, n → ∞
10
B.M. Patel et al.
for each z ∈ X. Therefore, { 41n g(2n x)} is a 2-Cauchy sequence in X, for each
x ∈ X. Since X is a 2-Banach space, { 41n g(2n x)} 2-converges, for each x ∈ X.
Define the function Q : X −→ X as
Q(x) := lim
n→∞
1
g(2n x)
n
4
for each x ∈ X. Now, by (22)
εkx, zkp
1
1
n
lim g(x) − n g(2 x), z ≤
n→∞
4
4
1 − 2p−2
for each x, z ∈ X. Therefore
kf (x) − Q(x) − f (0), zk ≤
εkx, zkp
4 − 2p
for each x, z ∈ X. Next we show that Q satisfies (2). For x ∈ X
1
kDg (2n x, 2n y), zk
n→∞ 4n
ε
= lim n (k2n x, zkp + k2n y, zkq )
n→∞ 4
= lim ε 2(p−2)n kx, zkp + 2(q−2)n ky, zkq
kDQ (x, y), zk = lim
n→∞
=0
for each z ∈ X. Therefore kDQ (x, y), zk = 0, for each z ∈ X. So we get
DQ (x, y) = 0. Next we prove the uniqueness of Q. Let Q0 be another quadratic
function satisfying (2) and (17). Since Q and Q0 are quadratic,
Q(2n x) = 4n Q(x), Q0 (2n x) = 4n Q0 (x), for each x ∈ X. Now for x ∈ X
1
kQ(2n x) − Q0 (2n x), zk
4n
1
≤ n [kQ(2n x) − g(2n x), zk + kg(2n x) − Q0 (2n x), zk]
4
1 2εk2n x, zkp
≤ n
4
4 − 2p
2ε kx, zkp
= 2(p−2)n
4 − 2p
−→ 0 as n → ∞
kQ(x) − Q0 (x), zk =
for each z ∈ X. Therefore kQ(x) − Q0 (x), zk = 0, for each z ∈ X. Therefore
Q(x) = Q0 (x), for each x ∈ X.
11
Stability of Quadratic Functional...
Theorem 3.2 Let ε ≥ 0, p, q > 2, r > 0. If f : X −→ X is a function such
that
kDf (x, y), zk ≤ ε(kx, zkp + ky, zkq )
(23)
for each x, y, z ∈ X. Then there exists a unique quadratic function Q : X −→
X satisfying (2) and
kf (x) − Q(x) − f (0), zk ≤
εkx, zkp
2p − 4
(24)
for each x, z ∈ X.
Proof 3.2 By (19) of Theorem 3.1, we have
kg(2x) − 4g(x), zk ≤ εkx, zkp
for each x, z ∈ X. Replacing x by
x
2
in (25), we get
x , z ≤ ε2−p kx, zkp
g(x) − 4g
2
for each x, z ∈ X. Replacing x by
x
2
(25)
(26)
in (26), we get
x
x − 4g
, z ≤ ε2−2p kx, zkp
g
2
4
(27)
for each x, z ∈ X. Combining (26) and (27), we get
x x
x , zk ≤ g(x) − 4g
, z + 4g
− 16g
, z
kg(x) − 16g
4
2
2
4
≤ ε2−p kx, zkp + 4ε2−2p kx, zkp
= εkx, zkp [2−p + 2−p · 4]
x
for each x, z ∈ X. By using induction on n, we have
n−1
x X
n
p
4j 2p(−j−1)
g(x) − 4 g n , z ≤ εkx, zk
2
j=0
= εkx, zk
p
n−1
X
2(−p+2)j−p
j=0
= εkx, zkp
2−p (1 − 2(2−p)n ) 1 − 22−p
(28)
12
B.M. Patel et al.
for each x, z ∈ X. For m, n ∈ N, For x ∈ X
x x x x m
4 g m − 4n g n , z = 4m+n−n g m+n−n − 4n g n , z 2
2
2 x
x2 n m−n
= 4 4
g m−n n − g n , z 2
·2
2
m−n−1
x p
X
≤ 4n · ε n kzkr
2(−p+2)j−p
2
j=0
= εkx, zkp
m−n−1
X
2(2−p)(n+j)−p
j=0
h (−p+2)n−p 1 − 2(−p+2)n i
p 2
= εkx, zk
1 − 2−p+2
−→ 0 as n → ∞
for each z ∈ X. Therefore {4n f ( 2xn )} is a 2-Cauchy sequence in X, for each
x ∈ X. Since X is a 2-Banach space, the sequence {4n f ( 2xn )} 2-converges, for
each x ∈ X. Define Q : X −→ X as
x
n
Q(x) := lim 4 f n
n→∞
2
for each x ∈ X. Now, by (28)
x 2−p
lim g(x) − 4n g n , z ≤ εkx, zkp
n→∞
2
1 − 22−p
for each x, z ∈ X. Therefore
εkx, zkp
kf (x) − Q(x) − f (0), zk ≤ p
2 −4
for each x, z ∈ X. The further part of the proof is similar to that of the proof
of Theorem 3.1.
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Stability of Quadratic Functional...
13
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