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Stability of quadratic functional equations in tempered distributions

Journal of Inequalities and Applications20122012:177

https://doi.org/10.1186/1029-242X-2012-177

Received: 12 September 2011

Accepted: 6 August 2012

Published: 16 August 2012

Abstract

We reformulate the following quadratic functional equation:

f ( k x + y ) + f ( k x y ) = 2 k 2 f ( x ) + 2 f ( y )

as the equation for generalized functions. Using the fundamental solution of the heat equation, we solve the general solution of this equation and prove the Hyers-Ulam stability in the spaces of tempered distributions and Fourier hyperfunctions.

Keywords

quadratic functional equationstabilitytempered distributionheat kernelGauss transform

1 Introduction

In 1940, Ulam [31] raised a question concerning the stability of group homomorphisms as follows: The case of approximately additive mappings was solved by Hyers [16] under the assumption that G 2 is a Banach space. In 1978, Rassias [25] generalized Hyers’ result to the unbounded Cauchy difference.

Let G 1 be a group and let G 2 be a metric group with the metric d ( , ) . Given ϵ > 0 , does there exist a δ > 0 such that if a function h : G 1 G 2 satisfies the inequality d ( h ( x y ) , h ( x ) h ( y ) ) < δ for all x , y G 1 , then there exists a homomorphism H : G 1 G 2 with d ( h ( x ) , H ( x ) ) < ϵ for all x G 1 ?

During the last decades, stability problems of various functional equations have been extensively studied and generalized by a number of authors (see [13, 14, 17, 19, 24, 27, 30]). In particular, the stability problem of the following quadratic functional equation
f ( x + y ) + f ( x y ) = 2 f ( x ) + 2 f ( y )
(1.1)
was proved by Skof [29]. Thereafter, many authors studied the stability problems of (1.1) in various settings (see [3, 4, 12, 18]). Usually, quadratic functional equations are used to characterize the inner product spaces. Note that a square norm on an inner product space satisfies the parallelogram equality
x + y 2 + x y 2 = 2 x 2 + 2 y 2
for all vectors x, y. By virtue of this equality, the quadratic functional equation (1.1) is induced. It is well known that a function f between real vector spaces satisfies (1.1) if and only if there exists a unique symmetric biadditive function B such that f ( x ) = B ( x , x ) (see [1, 13, 17, 19, 27]). The biadditive function B is given by
B ( x , y ) = 1 4 ( f ( x + y ) f ( x y ) ) .
Recently, Lee et al. [21] introduced the following quadratic functional equation which is equivalent to (1.1):
f ( k x + y ) + f ( k x y ) = 2 k 2 f ( x ) + 2 f ( y ) ,
(1.2)

where k is a fixed positive integer. They proved the Hyers-Ulam-Rassias stability of this equation in Banach spaces. Wang [32] considered the intuitionistic fuzzy stability of (1.2) by using the fixed-point alternative. Saadati and Park [26] proved the Hyers-Ulam-Rassias stability of (1.2) in non-Archimedean L -fuzzy normed spaces.

In this paper, we solve the general solution and the stability problem of (1.2) in the spaces of generalized functions such as S of tempered distributions and F of Fourier hyperfunctions. Using pullbacks, Chung and Lee [8] reformulated (1.1) as the equation for generalized functions and proved that every solution of (1.1) in S (or F , resp.) is a quadratic form. Also, Chung [7, 11] proved the stability problem of (1.1) in the spaces S and F . Making use of the similar methods as in [711, 22], we reformulate (1.2) and the related inequality in the spaces of generalized functions as follows:
(1.3)
(1.4)
where A, B, P, and Q are the functions defined by
A ( x , y ) = k x + y , B ( x , y ) = k x y , P ( x , y ) = x , Q ( x , y ) = y .

Here, denotes the pullback of generalized functions and the inequality v ϵ in (1.4) means that | v , φ | ϵ φ L 1 for all test functions φ. We refer to [15] for pullbacks and to [2, 711] for more details of the spaces of generalized functions.

As results, we shall prove that every solution u in S (or F , resp.) of Eq. (1.3) is a quadratic form
u = 1 i j n a i j x i x j ,
where a i j C . Also, we shall prove that every solution u in S (or F , resp.) of the inequality (1.4) can be written uniquely in the form
u = 1 i j n a i j x i x j + μ ( x ) ,
where μ is a bounded measurable function such that
μ L { ϵ 2 , k = 1 , ( k 2 + 1 ) ϵ 2 k 2 ( k 2 1 ) , k 2 .

2 Preliminaries

In this section, we introduce the spaces of tempered distributions and Fourier hyperfunctions. Here, we use the n-dimensional notations. If α = ( α 1 , , α n ) N 0 n , where N 0 n is the set of nonnegative integers, then | α | = α 1 + + α n , α ! = α 1 ! α n ! . For x = ( x 1 , , x n ) R n , we denote x α = x 1 α 1 x n α n and α = ( / x 1 ) α 1 ( / x n ) α n .

2.1 Tempered distributions

We present a very useful space of test functions for the tempered distributions as follows.

Definition 2.1 ([15, 28])

An infinitely differentiable function φ in R n is called rapidly decreasing if
φ α , β = sup x R n | x α β φ ( x ) | <
(2.1)
for all α , β N 0 n . The vector space of such functions is denoted by S ( R n ) . A linear functional u on S ( R n ) is said to be a tempered distribution if there exists the constant C 0 and the nonnegative integer N such that
| u , φ | C | α | , | β | N sup x R n | x α β φ |

for all φ S ( R n ) . The set of all tempered distributions is denoted by S ( R n ) .

We note that, if φ S ( R n ) , then each derivative of φ decreases faster than | x | N for all N > 0 as | x | . It is easy to see that the function φ ( x ) = exp ( a | x | 2 ) , where a > 0 belongs to S ( R n ) , but ψ ( x ) = ( 1 + | x | 2 ) 1 is not a member of S ( R n ) . It is known from [5] that (2.1) is equivalent to
sup x R n | x α φ ( x ) | < , sup ξ R n | ξ β φ ˆ ( ξ ) | <

for all α , β N 0 n , where φ ˆ is the Fourier transform of φ.

For example, every polynomial p ( x ) = | α | m a α x α , where a α C , defines a tempered distribution by
p ( x ) , φ = R n p ( x ) φ ( x ) d x , φ S ( R n ) .

Note that tempered distributions are generalizations of L p -functions. These are very useful for the study of Fourier transforms in generality, since all tempered distributions have a Fourier transform.

2.2 Fourier hyperfunctions

Imposing the growth condition on α , β in (2.1) Sato and Kawai introduced the new space of test functions for the Fourier hyperfunctions as follows.

Definition 2.2 ([6])

We denote by F ( R n ) the set of all infinitely differentiable functions φ in R n such that
φ A , B = sup x , α , β | x α β φ ( x ) | A | α | B | β | α ! β ! <
(2.2)

for some positive constants A, B depending only on φ. The strong dual of F ( R n ) , denoted by F ( R n ) , is called the Fourier hyperfunction.

It can be verified that the seminorm (2.2) is equivalent to
φ h , k = sup x , α | α φ ( x ) | exp k | x | h | α | α ! <
for some constants h , k > 0 . Furthermore, it is shown in [6] that (2.2) is equivalent to
sup x R n | φ ( x ) | exp k | x | < , sup ξ R n | φ ˆ ( ξ ) | exp h | ξ | <

for some h , k > 0 .

Fourier hyperfunctions were introduced by Sato in 1958. The space F ( R n ) is a natural generalization of the space S ( R n ) and can be thought informally as distributions of a infinite order. Observing the above growth conditions, we can easily see the following topological inclusions:
F ( R n ) S ( R n ) , S ( R n ) F ( R n ) .

3 General solution in generalized functions

In order to solve the general solution of (1.3), we employ the n-dimensional heat kernel, fundamental solution of the heat equation,
E t ( x ) = { ( 4 π t ) n / 2 exp ( | x | 2 / 4 t ) , x R n , t > 0 , 0 , x R n , t 0 .
Since for each t > 0 , E t ( ) belongs to the space F ( R n ) , the convolution
u ˜ ( x , t ) = ( u E t ) ( x ) = u y , E t ( x y )
is well defined for all u in F ( R n ) , which is called the Gauss transform of u. Subsequently, the semigroup property
( E t E s ) ( x ) = E t + s ( x )

of the heat kernel is very useful to convert Eq. (1.3) into the classical functional equation defined on upper-half plane. We also use the following famous result, the so-called heat kernel method, which is stated as follows.

Theorem 3.1 ([23])

Let u S ( R n ) . Then its Gauss transform u ˜ is a C -solution of the heat equation
( / t Δ ) u ˜ ( x , t ) = 0
satisfying
  1. (i)
    There exist positive constants C, M, and N such that
    | u ˜ ( x , t ) | C t M ( 1 + | x | ) N in R n × ( 0 , δ ) .
    (3.1)
     
  2. (ii)
    u ˜ ( x , t ) u as t 0 + in the sense that for every φ S ( R n ) ,
    u , φ = lim t 0 + u ˜ ( x , t ) φ ( x ) d x .
     

Conversely, every C -solution U ( x , t ) of the heat equation satisfying the growth condition (3.1) can be uniquely expressed as U ( x , t ) = u ˜ ( x , t ) for some u S ( R n ) .

Similarly, we can represent Fourier hyperfunctions as a special case of the results as in [20]. In this case, the estimate (3.1) is replaced by the following:

For every ϵ > 0 , there exists a positive constant C ϵ such that
| u ˜ ( x , t ) | C ϵ exp ( ϵ ( | x | + 1 / t ) ) in  R n × ( 0 , δ ) .

Here, we need the following lemma to solve the general solution of (1.3).

Lemma 3.2 Suppose that f : R n × ( 0 , ) C is a continuous function satisfying the equation
f ( k x + y , k 2 t + s ) + f ( k x y , k 2 t + s ) = 2 k 2 f ( x , t ) + 2 f ( y , s )
(3.2)
for all x , y R n , t , s > 0 . Then the solution f is the quadratic-additive function
f ( x , t ) = 1 i j n a i j x i x j + b t

for some a i j , b C .

Proof Define a function h : R n × ( 0 , ) C as h ( x , t ) : = f ( x , t ) f ( 0 , t ) . We immediately have h ( 0 , t ) = 0 and
h ( k x + y , k 2 t + s ) + h ( k x y , k 2 t + s ) = 2 k 2 h ( x , t ) + 2 h ( y , s )
(3.3)
for all x , y R n , t , s > 0 . Putting y = 0 in (3.3) yields
h ( k x , k 2 t + s ) = k 2 h ( x , t )
(3.4)
for all x R n , t , s > 0 . Letting s 0 + in (3.4) gives
h ( k x , k 2 t ) = k 2 h ( x , t )
(3.5)
for all x R n , t > 0 . Replacing s by k 2 s in (3.4) and then using (3.5), we obtain
h ( x , t + s ) = h ( x , t )
for all x R n , t , s > 0 . This shows that h ( x , t ) is independent with respect to the second variable. Thus, we see that H ( x ) : = h ( x , t ) satisfies (1.2). Using the induction argument on the dimension n, we verify that every continuous solution of (1.2) in R n is a quadratic form
H ( x ) = h ( x , t ) = 1 i j n a i j x i x j ,

where a i j C .

On the other hand, putting x = y = 0 in (3.2) yields
f ( 0 , k 2 t + s ) = k 2 f ( 0 , t ) + f ( 0 , s )
(3.6)
for all t , s > 0 . In view of (3.6), we verify that lim s 0 + f ( 0 , s ) = 0 and
f ( 0 , k 2 t ) = k 2 f ( 0 , t )
(3.7)
for all t > 0 . It follows from (3.6) and (3.7) that we see that f ( 0 , t ) satisfies the Cauchy functional equation
f ( 0 , t + s ) = f ( 0 , t ) + f ( 0 , s )
for all t , s > 0 . Given the continuity, we have
f ( 0 , t ) = b t
for some b C . Therefore, we finally obtain
f ( x , t ) = h ( x , t ) + f ( 0 , t ) = 1 i j n a i j x i x j + b t

for all x R n , t > 0 . □

As a direct consequence of the above lemma, we present the general solution of the quadratic functional equation (1.3) in the spaces of generalized functions.

Theorem 3.3 Every solution u in S ( R n ) (or F ( R n ) , resp.) of Eq. (1.3) is the quadratic form
u = 1 i j n a i j x i x j

for some a i j C .

Proof Convolving the tensor product E t ( ξ ) E s ( η ) of n-dimensional heat kernels in both sides of (1.3), we have
[ ( u A ) ( E t ( ξ ) E s ( η ) ) ] ( x , y ) = u A , E t ( x ξ ) E s ( y η ) = u ξ , k n E t ( x ξ η k ) E s ( y η ) d η = u ξ , k n E t ( k x + y ξ η k ) E s ( η ) d η = u ξ , E k 2 t ( k x + y ξ η ) E s ( η ) d η = u ξ , ( E k 2 t E s ) ( k x + y ξ ) = u ξ , E k 2 t + s ( k x + y ξ ) = u ˜ ( k x + y , k 2 t + s )
and similarly we get
Thus, (1.3) is converted into the classical functional equation
u ˜ ( k x + y , k 2 t + s ) + u ˜ ( k x y , k 2 t + s ) = 2 k 2 u ˜ ( x , t ) + 2 u ˜ ( y , s )
for all x , y R n , t , s > 0 . We note that the Gauss transform u ˜ is a C function and so, by Lemma 3.2, the solution u ˜ is of the form
u ˜ ( x , t ) = 1 i j n a i j x i x j + b t
(3.8)
for some a i j , b C . By the heat kernel method, we obtain
u = 1 i j n a i j x i x j

as t 0 + in (3.8). □

4 Stability in generalized functions

In this section, we are going to solve the stability problem of (1.4). For the case of k = 1 in (1.4), the result is known as follows.

Theorem 4.1 ([7, 10])

Suppose that u in S ( R n ) (or F ( R n ) , resp.) satisfies the inequality
u A + u B 2 u P 2 u Q ϵ .
Then there exists a unique quadratic form
T ( x ) = 1 i j n a i j x i x j
such that
u T ( x ) ϵ 2 .

We here need the following lemma to solve the stability problem of (1.4).

Lemma 4.2 Let k be a fixed positive integer with k 2 . Suppose that f : R n × ( 0 , ) C is a continuous function satisfying the inequality
f ( k x + y , k 2 t + s ) + f ( k x y , k 2 t + s ) 2 k 2 f ( x , t ) 2 f ( y , s ) L ϵ .
(4.1)
Then there exist a unique function g ( x , t ) satisfying the quadratic-additive functional equation
g ( k x + y , k 2 t + s ) + g ( k x y , k 2 t + s ) = 2 k 2 g ( x , t ) + 2 g ( y , s )
such that
f ( x , t ) g ( x , t ) L k 2 + 1 2 k 2 ( k 2 1 ) ϵ .
Proof Putting x = y = 0 in (4.1) yields
| f ( 0 , k 2 t + s ) k 2 f ( 0 , t ) f ( 0 , s ) | ϵ 2
(4.2)
for all t , s > 0 . In view of (4.2), we see that
c : = lim sup t 0 + f ( 0 , t )
exists. Letting t = t n 0 + so that f ( 0 , t n ) c in (4.2) gives
| c | ϵ 2 k 2 .
(4.3)
Putting y = 0 and letting s = s n 0 + so that f ( 0 , s n ) c in (4.1) we have
| f ( k x , k 2 t ) k 2 f ( x , t ) c | ϵ 2
(4.4)
for all x R n , t > 0 . Using (4.3), we can rewrite (4.4) as
| f ( k x , k 2 t ) k 2 f ( x , t ) | k 2 + 1 2 k 4 ϵ
for all x R n , t > 0 . By the induction argument yields
| f ( k n x , k 2 n t ) k 2 n f ( x , t ) | k 2 + 1 2 k 2 ( k 2 1 ) ϵ
(4.5)
for all n N , x R n , t > 0 . We claim that the sequence { k 2 n f ( k n x , k 2 n t ) } converges. Replacing x by k m x and t by k 2 m t in (4.5), respectively, where m n , we get
| f ( k m + n x , k 2 ( m + n ) t ) k 2 ( m + n ) f ( k m x , k 2 m t ) k 2 m | k 2 + 1 2 k 2 ( m + 1 ) ( k 2 1 ) ϵ .
Letting n , by Cauchy convergence criterion, we see that the sequence { k 2 n f ( k n x , k 2 n t ) } is a Cauchy sequence. We can now define a function h : R n × ( 0 , ) C by
g ( x , t ) : = lim n f ( k n x , k 2 n t ) k 2 n .
Letting n in (4.5) we obtain
f ( x , t ) g ( x , t ) L k 2 + 1 2 k 2 ( k 2 1 ) ϵ .
(4.6)
Replacing x, y, t, s by k n x , k n y , k 2 n t , k 2 n s in (4.1), dividing both sides by k 2 n and letting n we have
g ( k x + y , k 2 t + s ) + g ( k x y , k 2 t + s ) = 2 k 2 g ( x , t ) + 2 g ( y , s )
(4.7)
for all x , y R n , t , s > 0 . Next, we shall prove that g is unique. Suppose that there exists another function h : R n × ( 0 , ) C such that h satisfies (4.6) and (4.7). Since g and h satisfy (4.7), we see from Lemma 3.2 that
g ( k n x , k 2 n t ) = k 2 n g ( x , t ) , h ( k n x , k 2 n t ) = k 2 n h ( x , t )
for all n N , x R n , t > 0 . One gets from (4.6) that

for all n N , x R n , t > 0 . Taking the limit as n , we conclude that g ( x , t ) = h ( x , t ) for all x R n , t > 0 . □

We now state and prove the main theorem of this paper.

Theorem 4.3 Suppose that u in S ( R n ) (or F ( R n ) , resp.) satisfies the inequality (1.4). Then there exists a unique quadratic form
T ( x ) = 1 i j n a i j x i x j
such that
u T ( x ) { ϵ 2 , k = 1 , ( k 2 + 1 ) ϵ 2 k 2 ( k 2 1 ) , k 2 .
Proof As discussed above, it is done for the case of k = 1 . We assume that k is a fixed-positive integer with k 2 . Convolving the tensor product E t ( ξ ) E s ( η ) of n-dimensional heat kernels in both sides of (1.4), we have
u ˜ ( k x + y , k 2 t + s ) + u ˜ ( k x y , k 2 t + s ) 2 k 2 u ˜ ( x , t ) 2 u ˜ ( y , s ) L ϵ .
By Lemma 4.2, there exists a unique function g ( x , t ) satisfying the quadratic-additive functional equation
g ( k x + y , k 2 t + s ) + g ( k x y , k 2 t + s ) = 2 k 2 g ( x , t ) + 2 g ( y , s )
such that
u ˜ ( x , t ) g ( x , t ) L k 2 + 1 2 k 2 ( k 2 1 ) ϵ .
(4.8)
It follows from Lemma 3.2 that g ( x , t ) is of the form
g ( x , t ) = 1 i j n a i j x i x j + b t
for some a i j , b C . Letting t 0 + in (4.8), we have
u 1 i j n a i j x i x j k 2 + 1 2 k 2 ( k 2 1 ) ϵ .

This completes the proof. □

Remark 4.4 The resulting inequality in Theorem 4.3 implies that u T ( x ) is a measurable function. Thus, all of the solution u in S ( R n ) (or F ( R n ) , resp.) can be written uniquely in the form
u = T ( x ) + μ ( x ) ,
where
μ L { ϵ 2 , k = 1 , ( k 2 + 1 ) ϵ 2 k 2 ( k 2 1 ) , k 2 .

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, Sogang University, Seoul, Republic of Korea

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