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Functional equations in paranormed spaces

Abstract

In this paper, we prove the Hyers-Ulam stability of various functional equations in paranormed spaces.

MSC:35A17, 39B52, 39B72.

1 Introduction and preliminaries

The concept of statistical convergence for sequences of real numbers was introduced by Fast [1] and Steinhaus [2] independently, and since then several generalizations and applications of this notion have been investigated by various authors (see [37]). This notion was defined in normed spaces by Kolk [8].

We recall some basic facts concerning Fréchet spaces.

Definition 1.1 ([9])

Let X be a vector space. A paranorm P:X[0,) is a function on X such that

  1. (1)

    P(0)=0;

  2. (2)

    P(x)=P(x);

  3. (3)

    P(x+y)P(x)+P(y) (triangle inequality);

  4. (4)

    If { t n } is a sequence of scalars with t n t and { x n }X with P( x n x)0, then P( t n x n tx)0 (continuity of multiplication).

The pair (X,P) is called a paranormed space if P is a paranorm on X.

The paranorm is called total if, in addition, we have

  1. (5)

    P(x)=0 implies x=0.

A Fréchet space is a total and complete paranormed space.

The stability problem of functional equations originated from a question of Ulam [10] concerning the stability of group homomorphisms. Hyers [11] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [12] for additive mappings and by Th. M. Rassias [13] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Th. M. Rassias’ theorem was obtained by Găvruta [14] by replacing the unbounded Cauchy difference by a general control function in the spirit of Th. M. Rassias’ approach.

In 1990 during the 27th International Symposium on Functional Equations, Th. M. Rassias [15] asked the question whether such a theorem can also be proved for p1. In 1991 Gajda [16], following the same approach as in Th. M. Rassias [13], gave an affirmative solution to this question for p>1. It was shown by Gajda [16], as well as by Th. M. Rassias and Šemrl [17] that one cannot prove a Th. M. Rassias’ type theorem when p=1 (cf. the books of P. Czerwik [18], D. H. Hyers, G. Isac and Th. M. Rassias [19]).

In 1982 J. M. Rassias [20] followed the innovative approach of the Th. M. Rassias’ theorem [13] in which he replaced the factor x p + y p by x p y q for p,qR with p+q1. Găvruta [14] provided a further generalization of Th. M. Rassias’ theorem.

The functional equation

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

is called a quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. A Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [21] for mappings f:XY, where X is a normed space and Y is a Banach space. Cholewa [22] noticed that the theorem of Skof is still true if the relevant domain X is replaced by an Abelian group. Czerwik [23] proved the Hyers-Ulam stability of the quadratic functional equation. The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [2433]).

In [34], Jun and Kim considered the following cubic functional equation

f(2x+y)+f(2xy)=2f(x+y)+2f(xy)+12f(x).
(1.1)

It is easy to show that the function f(x)= x 3 satisfies the functional equation (1.1), which is called a cubic functional equation and every solution of the cubic functional equation is said to be a cubic mapping.

In [35], Lee et al. considered the following quartic functional equation

f(2x+y)+f(2xy)=4f(x+y)+4f(xy)+24f(x)6f(y).
(1.2)

It is easy to show that the function f(x)= x 4 satisfies the functional equation (1.2), which is called a quartic functional equation, and every solution of the quartic functional equation is said to be a quartic mapping.

Throughout this paper, assume that (X,P) is a Fréchet space and that (Y,) is a Banach space.

In this paper, we prove the Hyers-Ulam stability of the Cauchy additive functional equation, the quadratic functional equation, the cubic functional equation (1.1) and the quartic functional equation (1.2) in paranormed spaces.

2 Hyers-Ulam stability of the Cauchy additive functional equation

In this section, we prove the Hyers-Ulam stability of the Cauchy additive functional equation in paranormed spaces.

Note that P(2x)2P(x) for all xY.

Theorem 2.1 Let r, θ be positive real numbers with r>1, and let f:YX be an odd mapping such that

P ( f ( x + y ) f ( x ) f ( y ) ) θ ( x r + y r )
(2.1)

for all x,yY. Then there exists a unique Cauchy additive mapping A:YX such that

P ( f ( x ) A ( x ) ) 2 θ 2 r 2 x r
(2.2)

for all xY.

Proof Letting y=x in (2.1), we get

P ( f ( 2 x ) 2 f ( x ) ) 2θ x r

for all xY. So

P ( f ( x ) 2 f ( x 2 ) ) 2 2 r θ x r

for all xY. Hence

P ( 2 l f ( x 2 l ) 2 m f ( x 2 m ) ) j = l m 1 P ( 2 j f ( x 2 j ) 2 j + 1 f ( x 2 j + 1 ) ) 2 2 r j = l m 1 2 j 2 r j θ x r
(2.3)

for all nonnegative integers m and l with m>l and all xY. It follows from (2.3) that the sequence { 2 n f( x 2 n )} is a Cauchy sequence for all xY. Since X is complete, the sequence { 2 n f( x 2 n )} converges. So one can define the mapping A:YX by

A(x):= lim n 2 n f ( x 2 n )

for all xY. Moreover, letting l=0 and passing the limit m in (2.3), we get (2.2).

It follows from (2.1) that

P ( A ( x + y ) A ( x ) A ( y ) ) = lim n P ( 2 n ( f ( x + y 2 n ) f ( x 2 n ) f ( y 2 n ) ) ) lim n 2 n P ( f ( x + y 2 n ) f ( x 2 n ) f ( y 2 n ) ) lim n 2 n θ 2 n r ( x r + y r ) = 0

for all x,yY. Hence A(x+y)=A(x)+A(y) for all x,yY and so the mapping A:YX is Cauchy additive.

Now, let T:YX be another Cauchy additive mapping satisfying (2.2). Then we have

P ( A ( x ) T ( x ) ) = P ( 2 n ( A ( x 2 n ) T ( x 2 n ) ) ) 2 n P ( A ( x 2 n ) T ( x 2 n ) ) 2 n ( P ( A ( x 2 n ) f ( x 2 n ) ) + P ( T ( x 2 n ) f ( x 2 n ) ) ) 4 2 n ( 2 r 2 ) 2 n r θ x r ,

which tends to zero as n for all xY. So we can conclude that A(x)=T(x) for all xY. This proves the uniqueness of A. Thus the mapping A:YX is a unique Cauchy additive mapping satisfying (2.2). □

Theorem 2.2 Let r be a positive real number with r<1, and let f:XY be an odd mapping such that

f ( x + y ) f ( x ) f ( y ) P ( x ) r +P ( y ) r
(2.4)

for all x,yX. Then there exists a unique Cauchy additive mapping A:XY such that

f ( x ) A ( x ) 2 2 2 r P ( x ) r
(2.5)

for all xX.

Proof Letting y=x in (2.4), we get

2 f ( x ) f ( 2 x ) 2P ( x ) r

and so

f ( x ) 1 2 f ( 2 x ) P ( x ) r

for all xX. Hence

1 2 l f ( 2 l x ) 1 2 m f ( 2 m x ) j = l m 1 1 2 j f ( 2 j x ) 1 2 j + 1 f ( 2 j + 1 x ) j = l m 1 2 r j 2 j P ( x ) r
(2.6)

for all nonnegative integers m and l with m>l and all xX. It follows from (2.6) that the sequence { 1 2 n f( 2 n x)} is a Cauchy sequence for all xX. Since Y is complete, the sequence { 1 2 n f( 2 n x)} converges. So one can define the mapping A:XY by

A(x):= lim n 1 2 n f ( 2 n x )

for all xX. Moreover, letting l=0 and passing the limit m in (2.6), we get (2.5).

It follows from (2.4) that

A ( x + y ) A ( x ) A ( y ) = lim n 1 2 n f ( 2 n ( x + y ) ) f ( 2 n x ) f ( 2 n y ) lim n 2 n r 2 n ( P ( x ) r + P ( y ) r ) = 0

for all x,yX. Thus A(x+y)=A(x)+A(y) for all x,yX and so the mapping A:XY is Cauchy additive.

Now, let T:XY be another Cauchy additive mapping satisfying (2.5). Then we have

A ( x ) T ( x ) = 1 2 n A ( 2 n x ) T ( 2 n x ) 1 2 n ( A ( 2 n x ) f ( 2 n x ) + T ( 2 n x ) f ( 2 n x ) ) 4 2 n r ( 2 2 r ) 2 n P ( x ) r ,

which tends to zero as n for all xX. So we can conclude that A(x)=T(x) for all xX. This proves the uniqueness of A. Thus the mapping A:XY is a unique Cauchy additive mapping satisfying (2.5). □

3 Hyers-Ulam stability of the quadratic functional equation

In this section, we prove the Hyers-Ulam stability of the quadratic functional equation in paranormed spaces.

Note that P(2x)2P(x) for all xY.

Theorem 3.1 Let r, θ be positive real numbers with r>2, and let f:YX be a mapping satisfying f(0)=0 and

P ( f ( x + y ) + f ( x y ) 2 f ( x ) 2 f ( y ) ) θ ( x r + y r )
(3.1)

for all x,yY. Then there exists a unique quadratic mapping Q 2 :YX such that

P ( f ( x ) Q 2 ( x ) ) 2 θ 2 r 4 x r
(3.2)

for all xY.

Proof Letting y=x in (3.1), we get

P ( f ( 2 x ) 4 f ( x ) ) 2θ x r

for all xY. So

P ( f ( x ) 4 f ( x 2 ) ) 2 2 r θ x r

for all xY. Hence

P ( 4 l f ( x 2 l ) 4 m f ( x 2 m ) ) j = l m 1 P ( 4 j f ( x 2 j ) 4 j + 1 f ( x 2 j + 1 ) ) 2 2 r j = l m 1 4 j 2 r j θ x r
(3.3)

for all nonnegative integers m and l with m>l and all xY. It follows from (3.3) that the sequence { 4 n f( x 2 n )} is a Cauchy sequence for all xY. Since X is complete, the sequence { 4 n f( x 2 n )} converges. So one can define the mapping Q 2 :YX by

Q 2 (x):= lim n 4 n f ( x 2 n )

for all xY. Moreover, letting l=0 and passing the limit m in (3.3), we get (3.2).

It follows from (3.1) that

P ( Q 2 ( x + y ) + Q 2 ( x y ) 2 Q 2 ( x ) 2 Q 2 ( y ) ) = lim n P ( 4 n ( f ( x + y 2 n ) + f ( x y 2 n ) 2 f ( x 2 n ) 2 f ( y 2 n ) ) ) lim n 4 n P ( f ( x + y 2 n ) + f ( x y 2 n ) 2 f ( x 2 n ) 2 f ( y 2 n ) ) lim n 4 n θ 2 n r ( x r + y r ) = 0

for all x,yY. Hence Q 2 (x+y)+ Q 2 (xy)=2 Q 2 (x)+2 Q 2 (y) for all x,yY and so the mapping Q 2 :YX is quadratic.

Now, let T:YX be another quadratic mapping satisfying (3.2). Then we have

P ( Q 2 ( x ) T ( x ) ) = P ( 4 n ( Q 2 ( x 2 n ) T ( x 2 n ) ) ) 4 n P ( Q 2 ( x 2 n ) T ( x 2 n ) ) 4 n ( P ( Q 2 ( x 2 n ) f ( x 2 n ) ) + P ( T ( x 2 n ) f ( x 2 n ) ) ) 4 4 n ( 2 r 4 ) 2 n r θ x r ,

which tends to zero as n for all xY. So we can conclude that Q 2 (x)=T(x) for all xY. This proves the uniqueness of Q 2 . Thus the mapping Q 2 :YX is a unique quadratic mapping satisfying (3.2). □

Theorem 3.2 Let r be a positive real number with r<2, and let f:XY be a mapping satisfying f(0)=0 and

f ( x + y ) + f ( x y ) 2 f ( x ) 2 f ( y ) P ( x ) r +P ( y ) r
(3.4)

for all x,yX. Then there exists a unique quadratic mapping Q 2 :XY such that

f ( x ) Q 2 ( x ) 2 4 2 r P ( x ) r
(3.5)

for all xX.

Proof Letting y=x in (3.4), we get

4 f ( x ) f ( 2 x ) 2P ( x ) r

and so

f ( x ) 1 4 f ( 2 x ) 1 2 P ( x ) r

for all xX. Hence

1 4 l f ( 2 l x ) 1 4 m f ( 2 m x ) j = l m 1 1 4 j f ( 2 j x ) 1 4 j + 1 f ( 2 j + 1 x ) 1 2 j = l m 1 2 r j 4 j P ( x ) r
(3.6)

for all nonnegative integers m and l with m>l and all xX. It follows from (3.6) that the sequence { 1 4 n f( 2 n x)} is a Cauchy sequence for all xX. Since Y is complete, the sequence { 1 4 n f( 2 n x)} converges. So one can define the mapping Q 2 :XY by

Q 2 (x):= lim n 1 4 n f ( 2 n x )

for all xX. Moreover, letting l=0 and passing the limit m in (3.6), we get (3.5).

It follows from (3.4) that

Q 2 ( x + y ) + Q 2 ( x y ) 2 Q 2 ( x ) 2 Q 2 ( y ) = lim n 1 4 n f ( 2 n ( x + y ) ) + f ( 2 n ( x y ) ) 2 f ( 2 n x ) 2 f ( 2 n y ) lim n 2 n r 4 n ( P ( x ) r + P ( y ) r ) = 0

for all x,yX. Thus Q 2 (x+y)+ Q 2 (xy)=2 Q 2 (x)+2 Q 2 (y) for all x,yX and so the mapping Q 2 :XY is quadratic.

Now, let T:XY be another quadratic mapping satisfying (3.5). Then we have

Q 2 ( x ) T ( x ) = 1 4 n Q 2 ( 2 n x ) T ( 2 n x ) 1 4 n ( Q 2 ( 2 n x ) f ( 2 n x ) + T ( 2 n x ) f ( 2 n x ) ) 4 2 n r ( 4 2 r ) 4 n P ( x ) r ,

which tends to zero as n for all xX. So we can conclude that Q 2 (x)=T(x) for all xX. This proves the uniqueness of Q 2 . Thus the mapping Q 2 :XY is a unique quadratic mapping satisfying (3.5). □

4 Hyers-Ulam stability of the cubic functional equation

In this section, we prove the Hyers-Ulam stability of the cubic functional equation in paranormed spaces.

Note that P(2x)2P(x) for all xY.

Theorem 4.1 Let r, θ be positive real numbers with r>3, and let f:YX be a mapping such that

P ( 1 2 f ( 2 x + y ) + 1 2 f ( 2 x y ) f ( x + y ) f ( x y ) 6 f ( x ) ) θ ( x r + y r )
(4.1)

for all x,yY. Then there exists a unique cubic mapping C:YX such that

P ( f ( x ) C ( x ) ) θ 2 r 8 x r
(4.2)

for all xY.

Proof Letting y=0 in (4.1), we get

P ( f ( 2 x ) 8 f ( x ) ) θ x r

for all xY. So

P ( f ( x ) 8 f ( x 2 ) ) 1 2 r θ x r

for all xY. Hence

P ( 8 l f ( x 2 l ) 8 m f ( x 2 m ) ) j = l m 1 P ( 8 j f ( x 2 j ) 8 j + 1 f ( x 2 j + 1 ) ) 1 2 r j = l m 1 8 j 2 r j θ x r
(4.3)

for all nonnegative integers m and l with m>l and all xY. It follows from (4.3) that the sequence { 8 n f( x 2 n )} is a Cauchy sequence for all xY. Since X is complete, the sequence { 8 n f( x 2 n )} converges. So one can define the mapping C:YX by

C(x):= lim n 8 n f ( x 2 n )

for all xY. Moreover, letting l=0 and passing the limit m in (4.3), we get (4.2).

It follows from (4.1) that

P ( 1 2 C ( 2 x + y ) + 1 2 C ( 2 x y ) C ( x + y ) C ( x y ) 6 C ( x ) ) = lim n P ( 8 n ( 1 2 f ( 2 x + y 2 n ) + 1 2 f ( 2 x y 2 n ) f ( x + y 2 n ) f ( x y 2 n ) 6 f ( x 2 n ) ) ) lim n 8 n P ( 1 2 f ( 2 x + y 2 n ) + 1 2 f ( 2 x y 2 n ) f ( x + y 2 n ) f ( x y 2 n ) 6 f ( x 2 n ) ) lim n 8 n θ 2 n r ( x r + y r ) = 0

for all x,yY. Hence

1 2 C(2x+y)+ 1 2 C(2xy)=C(x+y)+C(xy)+6C(x)

for all x,yY and so the mapping C:YX is cubic.

Now, let T:YX be another cubic mapping satisfying (4.2). Then we have

P ( C ( x ) T ( x ) ) = P ( 8 n ( C ( x 2 n ) T ( x 2 n ) ) ) 8 n P ( C ( x 2 n ) T ( x 2 n ) ) 8 n ( P ( C ( x 2 n ) f ( x 2 n ) ) + P ( T ( x 2 n ) f ( x 2 n ) ) ) 2 8 n ( 2 r 8 ) 2 n r θ x r ,

which tends to zero as n for all xY. So we can conclude that C(x)=T(x) for all xY. This proves the uniqueness of C. Thus the mapping C:YX is a unique cubic mapping satisfying (4.2). □

Theorem 4.2 Let r be a positive real number with r<3, and let f:XY be a mapping such that

1 2 f ( 2 x + y ) + 1 2 f ( 2 x y ) f ( x + y ) f ( x y ) 6 f ( x ) P ( x ) r +P ( y ) r
(4.4)

for all x,yX. Then there exists a unique cubic mapping C:XY such that

f ( x ) C ( x ) 1 8 2 r P ( x ) r
(4.5)

for all xX.

Proof Letting y=0 in (4.4), we get

8 f ( x ) f ( 2 x ) P ( x ) r

and so

f ( x ) 1 8 f ( 2 x ) 1 8 P ( x ) r

for all xX. Hence

1 8 l f ( 2 l x ) 1 8 m f ( 2 m x ) j = l m 1 1 8 j f ( 2 j x ) 1 8 j + 1 f ( 2 j + 1 x ) 1 8 j = l m 1 2 r j 8 j P ( x ) r
(4.6)

for all nonnegative integers m and l with m>l and all xX. It follows from (4.6) that the sequence { 1 8 n f( 2 n x)} is a Cauchy sequence for all xX. Since Y is complete, the sequence { 1 8 n f( 2 n x)} converges. So one can define the mapping C:XY by

C(x):= lim n 1 8 n f ( 2 n x )

for all xX. Moreover, letting l=0 and passing the limit m in (4.6), we get (4.5).

It follows from (4.4) that

1 2 C ( 2 x + y ) + 1 2 C ( 2 x y ) C ( x + y ) C ( x y ) 6 C ( x ) = lim n 1 8 n 1 2 f ( 2 n ( 2 x + y ) ) + 1 2 f ( 2 n ( 2 x y ) ) f ( 2 n ( x + y ) ) f ( 2 n ( x y ) ) 6 f ( 2 n x ) lim n 2 n r 8 n ( P ( x ) r + P ( y ) r ) = 0

for all x,yX. Thus

1 2 C(2x+y)+ 1 2 C(2xy)=C(x+y)+C(xy)+6C(x)

for all x,yX and so the mapping C:XY is cubic.

Now, let T:XY be another cubic mapping satisfying (4.5). Then we have

C ( x ) T ( x ) = 1 8 n C ( 2 n x ) T ( 2 n x ) 1 8 n ( C ( 2 n x ) f ( 2 n x ) + T ( 2 n x ) f ( 2 n x ) ) 2 2 n r ( 8 2 r ) 8 n P ( x ) r ,

which tends to zero as n for all xX. So we can conclude that C(x)=T(x) for all xX. This proves the uniqueness of C. Thus the mapping C:XY is a unique cubic mapping satisfying (4.5). □

5 Hyers-Ulam stability of the quartic functional equation

In this section, we prove the Hyers-Ulam stability of the quartic functional equation in paranormed spaces.

Note that P(2x)2P(x) for all xY.

Theorem 5.1 Let r, θ be positive real numbers with r>4, and let f:YX be a mapping satisfying f(0)=0 and

(5.1)

for all x,yY. Then there exists a unique quartic mapping Q 4 :YX such that

P ( f ( x ) Q 4 ( x ) ) θ 2 r 16 x r
(5.2)

for all xY.

Proof Letting y=0 in (4.1), we get

P ( f ( 2 x ) 16 f ( x ) ) θ x r

for all xY. So

P ( f ( x ) 16 f ( x 2 ) ) 1 2 r θ x r

for all xY. Hence

(5.3)

for all nonnegative integers m and l with m>l and all xY. It follows from (5.3) that the sequence { 16 n f( x 2 n )} is a Cauchy sequence for all xY. Since X is complete, the sequence { 16 n f( x 2 n )} converges. So one can define the mapping Q 4 :YX by

Q 4 (x):= lim n 16 n f ( x 2 n )

for all xY. Moreover, letting l=0 and passing the limit m in (5.3), we get (5.2).

It follows from (5.1) that

P ( 1 2 Q 4 ( 2 x + y ) + 1 2 Q 4 ( 2 x y ) 2 Q 4 ( x + y ) 2 Q 4 ( x y ) 12 Q 4 ( x ) + 3 Q 4 ( y ) ) = lim n P ( 16 n ( 1 2 f ( 2 x + y 2 n ) + 1 2 f ( 2 x y 2 n ) 2 f ( x + y 2 n ) 2 f ( x y 2 n ) 12 f ( x 2 n ) + 3 f ( y 2 n ) ) ) lim n 16 n P ( 1 2 f ( 2 x + y 2 n ) + 1 2 f ( 2 x y 2 n ) 2 f ( x + y 2 n ) 2 f ( x y 2 n ) 12 f ( x 2 n ) + 3 f ( y 2 n ) ) lim n 16 n θ 2 n r ( x r + y r ) = 0

for all x,yY. Hence

1 2 Q 4 (2x+y)+ 1 2 Q 4 (2xy)=2 Q 4 (x+y)+2 Q 4 (xy)+12 Q 4 (x)3 Q 4 (y)

for all x,yY and so the mapping Q 4 :YX is quartic.

Now, let T:YX be another quartic mapping satisfying (5.2). Then we have

P ( Q 4 ( x ) T ( x ) ) = P ( 16 n ( Q 4 ( x 2 n ) T ( x 2 n ) ) ) 16 n P ( Q 4 ( x 2 n ) T ( x 2 n ) ) 16 n ( P ( Q 4 ( x 2 n ) f ( x 2 n ) ) + P ( T ( x 2 n ) f ( x 2 n ) ) ) 2 16 n ( 2 r 16 ) 2 n r θ x r ,

which tends to zero as n for all xY. So we can conclude that Q 4 (x)=T(x) for all xY. This proves the uniqueness of Q 4 . Thus the mapping Q 4 :YX is a unique quartic mapping satisfying (5.2). □

Theorem 5.2 Let r be a positive real number with r<4, and let f:XY be a mapping satisfying f(0)=0 and

(5.4)

for all x,yX. Then there exists a unique quartic mapping Q 4 :XY such that

f ( x ) Q 4 ( x ) 1 16 2 r P ( x ) r
(5.5)

for all xX.

Proof Letting y=0 in (5.4), we get

16 f ( x ) f ( 2 x ) P ( x ) r

and so

f ( x ) 1 16 f ( 2 x ) 1 16 P ( x ) r

for all xX. Hence

1 16 l f ( 2 l x ) 1 16 m f ( 2 m x ) j = l m 1 1 16 j f ( 2 j x ) 1 16 j + 1 f ( 2 j + 1 x ) 1 16 j = l m 1 2 r j 16 j P ( x ) r
(5.6)

for all nonnegative integers m and l with m>l and all xX. It follows from (5.6) that the sequence { 1 16 n f( 2 n x)} is a Cauchy sequence for all xX. Since Y is complete, the sequence { 1 16 n f( 2 n x)} converges. So one can define the mapping Q 4 :XY by

Q 4 (x):= lim n 1 16 n f ( 2 n x )

for all xX. Moreover, letting l=0 and passing the limit m in (5.6), we get (5.5).

It follows from (5.4) that

1 2 Q 4 ( 2 x + y ) + 1 2 Q 4 ( 2 x y ) 2 Q 4 ( x + y ) 2 Q 4 ( x y ) 12 Q 4 ( x ) + 3 Q 4 ( y ) = lim n 1 16 n 1 2 f ( 2 n ( 2 x + y ) ) + 1 2 f ( 2 n ( 2 x y ) ) 2 f ( 2 n ( x + y ) ) 2 f ( 2 n ( x y ) ) 12 f ( 2 n x ) + 3 f ( 2 n y ) lim n 2 n r 16 n ( P ( x ) r + P ( y ) r ) = 0

for all x,yX. Thus

1 2 Q 4 (2x+y)+ 1 2 Q 4 (2xy)=2 Q 4 (x+y)+2 Q 4 (xy)+12 Q 4 (x)3 Q 4 (y)

for all x,yX and so the mapping Q 4 :XY is quartic.

Now, let T:XY be another quartic mapping satisfying (5.5). Then we have

Q 4 ( x ) T ( x ) = 1 16 n Q 4 ( 2 n x ) T ( 2 n x ) 1 16 n ( Q 4 ( 2 n x ) f ( 2 n x ) + T ( 2 n x ) f ( 2 n x ) ) 2 2 n r ( 16 2 r ) 16 n P ( x ) r ,

which tends to zero as n for all xX. So we can conclude that Q 4 (x)=T(x) for all xX. This proves the uniqueness of Q 4 . Thus the mapping Q 4 :XY is a unique quartic mapping satisfying (5.5). □

References

  1. 1.

    Fast H: Sur la convergence statistique. Colloq. Math. 1951, 2: 241–244.

    MathSciNet  Google Scholar 

  2. 2.

    Steinhaus H: Sur la convergence ordinaire et la convergence asymptotique. Colloq. Math. 1951, 2: 34–73.

    MathSciNet  Google Scholar 

  3. 3.

    Fridy JA: On statistical convergence. Analysis 1985, 5: 301–313.

    MathSciNet  Article  Google Scholar 

  4. 4.

    Karakus S: Statistical convergence on probabilistic normed spaces. Math. Commun. 2007, 12: 11–23.

    MathSciNet  Google Scholar 

  5. 5.

    Mursaleen M: λ -statistical convergence. Math. Slovaca 2000, 50: 111–115.

    MathSciNet  Google Scholar 

  6. 6.

    Mursaleen M, Mohiuddine SA: On lacunary statistical convergence with respect to the intuitionistic fuzzy normed space. J. Comput. Appl. Math. 2009, 233: 142–149. 10.1016/j.cam.2009.07.005

    MathSciNet  Article  Google Scholar 

  7. 7.

    Šalát T: On the statistically convergent sequences of real numbers. Math. Slovaca 1980, 30: 139–150.

    MathSciNet  Google Scholar 

  8. 8.

    Kolk E: The statistical convergence in Banach spaces. Tartu ülik. Toim. 1991, 928: 41–52.

    MathSciNet  Google Scholar 

  9. 9.

    Wilansky A: Modern Methods in Topological Vector Space. McGraw-Hill, New York; 1978.

    Google Scholar 

  10. 10.

    Ulam SM: A Collection of the Mathematical Problems. Interscience, New York; 1960.

    Google Scholar 

  11. 11.

    Hyers DH: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 1941, 27: 222–224. 10.1073/pnas.27.4.222

    MathSciNet  Article  Google Scholar 

  12. 12.

    Aoki T: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Jpn. 1950, 2: 64–66. 10.2969/jmsj/00210064

    Article  Google Scholar 

  13. 13.

    Rassias TM: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 1978, 72: 297–300. 10.1090/S0002-9939-1978-0507327-1

    Article  Google Scholar 

  14. 14.

    Gǎvruta P: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 1994, 184: 431–436. 10.1006/jmaa.1994.1211

    MathSciNet  Article  Google Scholar 

  15. 15.

    Rassias TM: Problem 16; 2. Report of the 27th international symposium on functional equations. Aequ. Math. 1990, 39: 292–293.

    Google Scholar 

  16. 16.

    Gajda Z: On stability of additive mappings. Int. J. Math. Math. Sci. 1991, 14: 431–434. 10.1155/S016117129100056X

    MathSciNet  Article  Google Scholar 

  17. 17.

    Rassias TM, Šemrl P: On the behaviour of mappings which do not satisfy Hyers-Ulam stability. Proc. Am. Math. Soc. 1992, 114: 989–993. 10.1090/S0002-9939-1992-1059634-1

    Article  Google Scholar 

  18. 18.

    Czerwik P: Functional Equations and Inequalities in Several Variables. World Scientific, Singapore; 2002.

    Book  Google Scholar 

  19. 19.

    Hyers DH, Isac G, Rassias TM: Stability of Functional Equations in Several Variables. Birkhäuser, Basel; 1998.

    Book  Google Scholar 

  20. 20.

    Rassias JM: On approximation of approximately linear mappings by linear mappings. J. Funct. Anal. 1982, 46: 126–130. 10.1016/0022-1236(82)90048-9

    MathSciNet  Article  Google Scholar 

  21. 21.

    Skof F: Proprietà locali e approssimazione di operatori. Rend. Semin. Mat. Fis. Milano 1983, 53: 113–129. 10.1007/BF02924890

    MathSciNet  Article  Google Scholar 

  22. 22.

    Cholewa PW: Remarks on the stability of functional equations. Aequ. Math. 1984, 27: 76–86. 10.1007/BF02192660

    MathSciNet  Article  Google Scholar 

  23. 23.

    Czerwik S: On the stability of the quadratic mapping in normed spaces. Abh. Math. Semin. Univ. Hamb. 1992, 62: 59–64. 10.1007/BF02941618

    MathSciNet  Article  Google Scholar 

  24. 24.

    Aczel J, Dhombres J: Functional Equations in Several Variables. Cambridge University Press, Cambridge; 1989.

    Book  Google Scholar 

  25. 25.

    Eshaghi Gordji M, Savadkouhi MB: Stability of a mixed type cubic-quartic functional equation in non-Archimedean spaces. Appl. Math. Lett. 2010, 23: 1198–1202. 10.1016/j.aml.2010.05.011

    MathSciNet  Article  Google Scholar 

  26. 26.

    Isac G, Rassias TM: On the Hyers-Ulam stability of ψ -additive mappings. J. Approx. Theory 1993, 72: 131–137. 10.1006/jath.1993.1010

    MathSciNet  Article  Google Scholar 

  27. 27.

    Jun K, Lee Y: A generalization of the Hyers-Ulam-Rassias stability of the pexiderized quadratic equations. J. Math. Anal. Appl. 2004, 297: 70–86. 10.1016/j.jmaa.2004.04.009

    MathSciNet  Article  Google Scholar 

  28. 28.

    Jung S: Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press, Palm Harbor; 2001.

    Google Scholar 

  29. 29.

    Park C:Homomorphisms between Poisson J C -algebras. Bull. Braz. Math. Soc. 2005, 36: 79–97. 10.1007/s00574-005-0029-z

    MathSciNet  Article  Google Scholar 

  30. 30.

    Rassias JM: Solution of a problem of Ulam. J. Approx. Theory 1989, 57: 268–273. 10.1016/0021-9045(89)90041-5

    MathSciNet  Article  Google Scholar 

  31. 31.

    Rassias TM: Functional Equations and Inequalities. Kluwer Academic, Dordrecht; 2000.

    Book  Google Scholar 

  32. 32.

    Rassias TM: On the stability of functional equations in Banach spaces. J. Math. Anal. Appl. 2000, 251: 264–284. 10.1006/jmaa.2000.7046

    MathSciNet  Article  Google Scholar 

  33. 33.

    Rassias TM: On the stability of functional equations and a problem of Ulam. Acta Appl. Math. 2000, 62: 23–130. 10.1023/A:1006499223572

    MathSciNet  Article  Google Scholar 

  34. 34.

    Jun K, Kim H: The generalized Hyers-Ulam-Rassias stability of a cubic functional equation. J. Math. Anal. Appl. 2002, 274: 867–878. 10.1016/S0022-247X(02)00415-8

    MathSciNet  Article  Google Scholar 

  35. 35.

    Lee S, Im S, Hwang I: Quartic functional equations. J. Math. Anal. Appl. 2005, 307: 387–394. 10.1016/j.jmaa.2004.12.062

    MathSciNet  Article  Google Scholar 

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Acknowledgements

C. Park was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2012R1A1A2004299). D. Y. Shin was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2010-0021792).

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Correspondence to Dong Yun Shin.

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All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.

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Park, C., Shin, D.Y. Functional equations in paranormed spaces. Adv Differ Equ 2012, 123 (2012). https://doi.org/10.1186/1687-1847-2012-123

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Keywords

  • Hyers-Ulam stability
  • paranormed space
  • functional equation