Open Access

On the stability of an AQCQ-functional equation in random normed spaces

  • Choonkil Park1,
  • Sun Young Jang2,
  • Jung Rye Lee3 and
  • Dong Yun Shin4Email author
Journal of Inequalities and Applications20112011:34

https://doi.org/10.1186/1029-242X-2011-34

Received: 18 March 2011

Accepted: 18 August 2011

Published: 18 August 2011

Abstract

In this paper, we prove the Hyers-Ulam stability of the following additive-quadratic-cubic-quartic functional equation

f ( x + 2 y ) + f ( x - 2 y ) = 4 f ( x + y ) + 4 f ( x - y ) - 6 f ( x ) (1) + f ( 2 y ) + f ( - 2 y ) - 4 f ( y ) - 4 f ( - y ) (2) (3)

in random normed spaces.

2010 Mathematics Subject Classification: 46S40; 39B52; 54E70

Keywords

random normed space additive-quadratic-cubic-quartic functional equation Hyers-Ulam stability

1. Introduction

The stability problem of functional equations originated from a question of Ulam [1] in 1940, concerning the stability of group homomorphisms. Let (G1, ·) be a group and let (G2, *, d) be a metric group with the metric d(· , ·). Given ε > 0, does there exist a δ > 0 such that if a mapping h : G1G2 satisfies the inequality d(h(x·y), h(x) * h(y)) < δ for all x, y G1, then there exists a homomorphism H : G1G2 with d(h(x), H(x)) < ε for all x G1? In the other words, under what condition does there exists a homomorphism near an approximate homomorphism? The concept of stability for functional equation arises when we replace the functional equation by an inequality which acts as a perturbation of the equation. Hyers [2] gave a first affirmative answer to the question of Ulam for Banach spaces. Let f : EE' be a mapping between Banach spaces such that
f ( x + y ) - f ( x ) - f ( y ) δ
for all x, y E and some δ > 0. Then, there exists a unique additive mapping T : EE' such that
| | f ( x ) - T ( x ) | | δ

for all x E. Moreover, if f(tx) is continuous in t for each fixed x E, then T is -linear. In 1978, Th.M. Rassias [3] provided a generalization of the Hyers' theorem that allows the Cauchy difference to be unbounded. In 1991, Gajda [4] answered the question for the case p > 1, which was raised by Th.M. Rassias (see [511]).

On the other hand, in 1982-1998, J.M. Rassias generalized the Hyers' stability result by presenting a weaker condition controlled by a product of different powers of norms.

Theorem 1.1. ([1218]). Assume that there exist constants Θ ≥ 0 and p1, p2 such that p = p1 + p2 1, and f : EE' is a mapping from a normed space E into a Banach space E' such that the inequality
| | f ( x + y ) - f ( x ) - f ( y ) | | ε | | x | | p 1 | | y | | p 2
for all x, y E. Then, there exists a unique additive mapping T : EE' such that
| | f ( x ) - L ( x ) | | Θ 2 - 2 p | | x | | p

for all × E.

The control function ||x|| p · ||y|| q + ||x||p+q+ ||y||p+qwas introduced by Rassias [19] and was used in several papers (see [2025]).

The functional equation
f ( x + y ) + f ( x - y ) = 2 f ( x ) + 2 f ( y )
(1.1)
is related to a symmetric bi-additive mapping. It is natural that this equation is called a quadratic functional equation. In particular, every solution of the quadratic functional equation (1.1) is said to be a quadratic mapping. It is well known that a mapping f between real vector spaces is quadratic if and only if there exists a unique symmetric bi-additive mapping B such that f(x) = B(x, x) for all x (see [5, 26]). The bi-additive mapping B is given by
B ( x , y ) = 1 4 ( f ( x + y ) - f ( x - y ) ) .

The Hyers-Ulam stability problem for the quadratic functional equation (1.1) was proved by Skof for mappings f : AB, where A is a normed space and B is a Banach space (see [27]). Cholewa [28] noticed that the theorem of Skof is still true if relevant domain A is replaced by an abelian group. In [29], Czerwik proved the Hyers-Ulam stability of the functional equation (1.1). Grabiec [30] has generalized these results mentioned above.

In [31], Jun and Kim considered the following cubic functional equation:
f ( 2 x + y ) + f ( 2 x - y ) = 2 f ( x + y ) + 2 f ( x - y ) + 1 2 f ( x ) .
(1.2)

It is easy to show that the function f(x) = x3 satisfies the functional equation (1.2), which is called a cubic functional equation and every solution of the cubic functional equation is said to be a cubic mapping.

In [32], Park and Bae considered the following quartic functional equation
f ( x + 2 y ) + f ( x - 2 y ) = 4 [ f ( x + y ) + f ( x - y ) + 6 f ( y ) ] - 6 f ( x ) .
(1.3)

In fact, they proved that a mapping f between two real vector spaces X and Y is a solution of (1:3) if and only if there exists a unique symmetric multi-additive mapping M : X4Y such that f(x) = M(x, x, x, x) for all x. It is easy to show that the function f(x) = x4 satisfies the functional equation (1.3), which is called a quartic functional equation (see also [33]). In addition, Kim [34] has obtained the Hyers-Ulam stability for a mixed type of quartic and quadratic functional equation.

It should be noticed that in all these papers, the triangle inequality is expressed by using the strongest triangular norm T M .

The aim of this paper is to investigate the Hyers-Ulam stability of the additive-quadratic-cubic-quartic functional equation
f ( x + 2 y ) + f ( x 2 y ) = 4 f ( x + y ) + 4 f ( x y ) 6 f ( x ) + f ( 2 y ) + f ( 2 y ) 4 f ( y ) 4 f ( y )
(1.4)

in random normed spaces in the sense of Sherstnev under arbitrary continuous t-norms.

In the sequel, we adopt the usual terminology, notations and conventions of the theory of random normed spaces, as in [3537]. Throughout this paper, Δ+ is the space of distribution functions, that is, the space of all mappings F : {-∞, ∞} → [0, 1] such that F is left-continuous and non-decreasing on , F(0) = 0 and F(+ ∞) = 1. D+ is a subset of Δ+ consisting of all functions F Δ+ for which l - F(+ ∞) = 1, where l - f (x) denotes the left limit of the function f at the point x, that is, l - f ( x ) = l i m t x - f ( t ) . The space Δ+ is partially ordered by the usual point-wise ordering of functions, i.e., FG if and only if F(t) ≤ G(t) for all t in . The maximal element for Δ+ in this order is the distribution function ε0 given by
ε 0 ( t ) = 0 , i f t 0 , 1 , i f t > 0 .

Definition 1.2. [36]A mapping T : [0, 1] × [0, 1] → [0, 1] is a continuous triangular norm (briefly, a continuous t-norm) if T satisfies the following conditions:

(a) T is commutative and associative;

(b) T is continuous;

(c) T(a, 1) = a for all a [0, 1];

(d) T(a, b) ≤ T(c, d) whenever ac and bd for all a, b, c, d [0, 1].

Typical examples of continuous t-norms are T P (a, b) = ab, T M (a, b) = min(a, b) and T L (a, b) = max(a+b - 1, 0) (the Lukasiewicz t-norm). Recall (see [38, 39]) that if T is a t-norm and {x n } is a given sequence of numbers in [0, 1], then T i = 1 n x i is defined recurrently by T i = 1 1 x i = x 1 and T i = 1 n x i = T ( T i = 1 n - 1 x i , x n ) for n ≥ 2. T i = n x i is defined as T i = 1 x n + i - 1 . It is known [39] that for the Lukasiewicz t-norm, the following implication holds:
lim n ( T L ) i = 1 x n + i 1 = 1 n = 1 ( 1 x n ) <

Definition 1.3. [37]A random normed space (briefly, RN-space) is a triple (X, μ, T), where × is a vector space, T is a continuous t-norm, and μ is a mapping from × into D+such that the following conditions hold:

(RN1) μ x (t) = ε0(t) for all t > 0 if and only if × = 0;

(RN2) μ α x ( t ) = μ x ( t | α | ) for all × X, α ≠ 0;

(RN3) μx+y(t + s) ≥ T (μ x (t), μ y (s)) for all x, y X and all t, s ≥ 0.

Every normed space (X, ||·||) defines a random normed space (X, μ, T M ),

where
μ x ( t ) = t t + | | x | |

for all t > 0, and T M is the minimum t-norm. This space is called the induced random normed space.

Definition 1.4. Let (X, μ, T) be an RN-space.
  1. (1)

    A sequence {x n } in × is said to be convergent to × in × if, for every ε > 0 and λ > 0, there exists a positive integer N such that μ x n - x ( ε ) > 1 - λ whenever nN.

     
  2. (2)

    A sequence {x n } in × is called a Cauchy sequence if, for every ε > 0 and λ > 0, there exists a positive integer N such that μ x n - x m ( ε ) > 1 - λ whenever nmN.

     
  3. (3)

    An RN-space (X, μ, T) is said to be complete if and only if every Cauchy sequence in × is convergent to a point in X.

     

Theorem 1.5. [36]If (X, μ, T) is an RN-space and {x n } is a sequence such that x n x, then lim n μ x n ( t ) = μ x ( t ) almost everywhere.

Recently, Eshaghi Gordji et al. establish the stability of cubic, quadratic and additive-quadratic functional equations in RN-spaces (see [4042]).

One can easily show that an odd mapping f : XY satisfies (1.4) if and only if the odd mapping f : XY is an additive-cubic mapping, i.e.,
f ( x + 2 y ) + f ( x - 2 y ) = 4 f ( x + y ) + 4 f ( x - y ) - 6 f ( x ) .

It was shown in [[43], Lemma 2.2] that g(x) := f (2x) - 8f (x) and h(x) := f (2x) - 2f (x) are additive and cubic, respectively, and that f ( x ) = 1 6 h ( x ) - 1 6 g ( x ) .

One can easily show that an even mapping f : XY satisfies (1.4) if and only if the even mapping f : XY is a quadratic-quartic mapping, i.e.,
f ( x + 2 y ) + f ( x - 2 y ) = 4 f ( x + y ) + 4 f ( x - y ) - 6 f ( x ) + 2 f ( 2 y ) - 8 f ( y ) .

It was shown in [[44], Lemma 2.1] that g (x) := f (2x) -16f (x) and h (x) := f (2x) -4f (x) are quadratic and quartic, respectively, and that f ( x ) = 1 1 2 h ( x ) - 1 1 2 g ( x )

Lemma 1.6. Each mapping f : XY satisfying (1.4) can be realized as the sum of an additive mapping, a quadratic mapping, a cubic mapping and a quartic mapping.

This paper is organized as follows: In Section 2, we prove the Hyers-Ulam stability of the additive-quadratic-cubic-quartic functional equation (1.4) in RN-spaces for an odd case. In Section 3, we prove the Hyers-Ulam stability of the additive-quadratic-cubic-quartic functional equation (1.4) in RN-spaces for an even case.

Throughout this paper, assume that X is a real vector space and that (X, μ, T) is a complete RN-space.

2.Hyers-Ulam stability of the functional equation (1.4): an odd mapping Case

For a given mapping f : XY , we define
D f ( x , y ) : = f ( x + 2 y ) + f ( x 2 y ) 4 f ( x + y ) 4 f ( x y ) + 6 f ( x ) f ( 2 y ) f ( 2 y ) + 4 f ( y ) + 4 f ( y )

for all x, y X.

In this section, we prove the Hyers-Ulam stability of the functional equation Df (x, y) = 0 in complete RN-spaces: an odd mapping case.

Theorem 2.1. Let f : XY be an odd mapping for which there is a ρ : X2D+ (ρ (x, y) is denoted by ρx, y) such that
μ D f ( x , y ) ( t ) ρ x , y ( t )
(2.1)
for all x, y X and all t > 0. If
lim n T k = 1 ( T ( ρ 2 k + n 1 x , 2 k + n 1 x ( 2 n 3 t ) , ρ 2 k + n x , 2 k + n 1 x ( 2 n 1 t ) ) ) = 1
(2.2)
and
lim n ρ 2 n x , 2 n y ( 2 n t ) = 1
(2.3)
for all x, y X and all t > 0, then there exist a unique additive mapping A : XY and a unique cubic mapping C : XY such that
μ f ( 2 x ) - 8 f ( x ) - A ( x ) ( t ) T k = 1 T ρ 2 k - 1 x , 2 k - 1 x t 8 , ρ 2 k x , 2 k - 1 x t 2 ,
(2.4)
μ f ( 2 x ) - 2 f ( x ) - C ( x ) ( t ) T k = 1 T ρ 2 k - 1 x , 2 k - 1 x t 8 , ρ 2 k x , 2 k - 1 x t 2
(2.5)

for all × X and all t > 0.

Proof. Putting x = y in (2.1), we get
μ f ( 3 y ) - 4 f ( 2 y ) + 5 f ( y ) ( t ) ρ y , y ( t )
(2.6)
for all y X and all t > 0. Replacing x by 2y in (2.1), we get
μ f ( 4 y ) - 4 f ( 3 y ) + 6 f ( 2 y ) - 4 f ( y ) ( t ) ρ 2 y , y ( t )
(2.7)
for all y X and all t > 0. It follows from (2.6) and (2.7) that
μ f ( 4 x ) - 1 0 f ( 2 x ) + 1 6 f ( x ) ( t ) = μ ( 4 f ( 3 x ) - 1 6 f ( 2 x ) + 2 0 f ( x ) ) + ( f ( 4 x ) - 4 f ( 3 x ) + 6 f ( 2 x ) - 4 f ( x ) ) ( t ) T μ 4 f ( 3 x ) - 1 6 f ( 2 x ) + 2 0 f ( x ) t 2 , μ f ( 4 x ) - 4 f ( 3 x ) + 6 f ( 2 x ) - 4 f ( x ) t 2 T ρ x , x t 8 , ρ 2 x , x t 2
(2.8)
for all x X and all t > 0. Let g : XY be a mapping defined by g(x) := f (2x) - 8f (x). Then we conclude that
μ g ( 2 x ) - 2 g ( x ) ( t ) T ρ x , x t 8 , ρ 2 x , x t 2
for all x X and all t > 0. Thus, we have
μ g ( 2 x ) 2 - g ( x ) ( t ) T ρ x , x t 4 , ρ 2 x , x t
for all x X and all t > 0. Hence,
μ g ( 2 k + 1 x ) 2 k + 1 g ( 2 k x ) 2 k ( t ) T ( ρ 2 k x , 2 k x ( 2 k 2 t ) , ρ 2 k + 1 x , 2 k x ( 2 k t ) )
for all x X, all t > 0 and all k : From 1 > 1 2 + 1 2 2 + + 1 2 n , it follows that
μ g ( 2 n x ) 2 n g ( x ) ( t ) T k = 1 n ( μ g ( 2 k x ) 2 k g ( 2 k 1 x ) 2 k 1 ( t 2 k ) ) T k = 1 n ( T ( ρ 2 k 1 x , 2 k 1 x ( t 8 ) , ρ 2 k x , 2 k 1 x ( t 2 ) ) )
(2.9)
for all x X and all t > 0. In order to prove the convergence of the sequence { g ( 2 n x ) 2 n } , replacing x with 2 m x in (2.9), we obtain that
μ g ( 2 n + m x ) 2 n + m g ( 2 m x ) 2 m ( t ) T k = 1 n ( T ( ρ 2 k + m 1 x , 2 k + m 1 x ( 2 m 3 t ) , ρ 2 k + m x , 2 k + m 1 x ( 2 m 1 t ) ) ) .
(2.10)

Since the right-hand side of the inequality (2.10) tends to 1 as m and n tend to infinity, the sequence { g ( 2 n x ) 2 n } is a Cauchy sequence. Thus, we may define A ( x ) = lim n g ( 2 n x ) 2 n for all x X.

Now, we show that A is an additive mapping. Replacing x and y with 2 n x and 2 n y in (2.1), respectively, we get
μ D f ( 2 n x , 2 n y ) 2 n ( t ) ρ 2 n x , 2 n y ( 2 n t ) .

Taking the limit as n → ∞, we find that A : XY satisfies (1.4) for all x, y X. Since f : XY is odd, A : XY is odd. By [[43], Lemma 2.2], the mapping A : XY is additive. Letting the limit as n → ∞ in (2.9), we get (2.4).

Next, we prove the uniqueness of the additive mapping A : XY subject to (2.4). Let us assume that there exists another additive mapping L : XY which satisfies (2.4). Since A(2 n x) = 2 n A(x), L(2 n x) = 2 n L(x) for all x X and all n , from (2.4), it follows that
μ A ( x ) L ( x ) ( 2 t ) = μ A ( 2 n x ) L ( 2 n x ) ( 2 n + 1 t ) T ( μ A ( 2 n x ) g ( 2 n x ) ( 2 n t ) , μ g ( 2 n x ) L ( 2 n x ) ( 2 n t ) ) T ( T k = 1 ( T ( ρ 2 n + k 1 x , 2 n + k 1 x ( 2 n 3 t ) , ρ 2 n + k x , 2 n + k 1 x ( 2 n 1 t ) ) ) , T k = 1 ( T ( ρ 2 n + k 1 x , 2 n + k 1 x ( 2 n 3 t ) , ρ 2 n + k x , 2 n + k 1 x ( 2 n 1 t ) ) )
(2.11)

for all x X and all t > 0. Letting n → ∞ in (2.11), we conclude that A = L.

Let h : XY be a mapping defined by h(x) := f (2x) -2f (x). Then, we conclude that
μ h ( 2 x ) - 8 h ( x ) ( t ) T ρ x , x t 8 , ρ 2 x , x t 2
for all x X and all t > 0. Thus, we have
μ h ( 2 x ) 8 - h ( x ) ( t ) T ( ρ x , x ( t ) , ρ 2 x , x ( 4 t ) )
for all x X and all t > 0. Hence,
μ h ( 2 k + 1 x ) 8 k + 1 h ( 2 k x ) 8 k ( t ) T ( ρ 2 k x , 2 k x ( 8 k t ) , ρ 2 k + 1 x , 2 k x ( 4 · 8 k t ) )
for all x X, all t > 0 and all k : From 1 > 1 8 + 1 8 2 + + 1 8 n , it follows that
μ h ( 2 n x ) 8 n h ( x ) ( t ) T k = 1 n ( μ h ( 2 k x ) 8 k h ( 2 k 1 x ) 8 k 1 ( t 8 k ) ) T k = 1 n ( T ( ρ 2 k 1 x , 2 k 1 x ( t 8 ) , ρ 2 k x , 2 k 1 x ( t 2 ) ) )
(2.12)
for all x X and all t > 0. In order to prove the convergence of the sequence { h ( 2 n x ) 8 n } , replacing x with 2 m x in (2.12), we obtain that
μ h ( 2 n + m x ) 8 n + m h ( 2 m x ) 8 m ( t ) T k = 1 n ( T ( ρ 2 k + m 1 x , 2 k + m 1 x ( 8 m 1 t ) , ρ 2 k + m x , 2 k + m 1 x ( 4 · 8 m 1 t ) ) ) .
(2.13)

Since the right-hand side of the inequality (2.13) tends to 1 as m and n tend to infinity, the sequence { h ( 2 n x ) 8 n } is a Cauchy sequence. Thus, we may define C ( x ) = lim n h ( 2 n x ) 8 n for all x X.

Now, we show that C is a cubic mapping. Replacing x and y with 2 n x and 2 n y in (2.1), respectively, we get
μ D f ( 2 n x , 2 n y ) 8 n ( t ) ρ 2 n x , 2 n y ( 8 n t ) ρ 2 n x , 2 n y ( 2 n t ) .

Taking the limit as n → ∞, we find that C : XY satisfies (1.4) for all x, y X. Since f : XY is odd, C : XY is odd. By [[43], Lemma 2.2], the mapping C : XY is cubic. Letting the limit as n → ∞ in (2.12), we get (2.5).

Finally, we prove the uniqueness of the cubic mapping C : XY subject to (2.5). Let us assume that there exists another cubic mapping L : XY which satisfies (2.5). Since C(2 n x) = 8 n C(x), L(2 n x) = 8 n L(x) for all x X and all n , from (2.5), it follows that
μ C ( x ) L ( x ) ( 2 t ) = μ C ( 2 n x ) L ( 2 n x ) ( 2 · 8 n t ) T ( μ C ( 2 n x ) h ( 2 n x ) ( 8 n t ) , μ h ( 2 n x ) L ( 2 n x ) ( 8 n t ) ) T ( T k = 1 ( T ( ρ 2 n + k 1 x , 2 n + k 1 x ( 8 n 1 t ) , ρ 2 n + k x , 2 n + k 1 x ( 4 · 8 n 1 t ) ) ) , T k = 1 ( T ( ρ 2 n + k 1 x , 2 n + k 1 x ( 8 n 1 t ) , ρ 2 n + k x , 2 n + k 1 x ( 4 · 8 n 1 t ) ) ) T ( T k = 1 ( T ( ρ 2 n + k 1 x , 2 n + k 1 x ( 2 n 3 t ) , ρ 2 n + k x , 2 n + k 1 x ) ) ) , T k = 1 ( T ( ρ 2 n + k 1 x , 2 n + k 1 x ( 2 n 3 t ) , ρ 2 n + k x , 2 n + k 1 x ( 2 n 1 t ) ) )
(2.14)

for all x X and all t > 0. Letting n → ∞ in (2.14), we conclude that C = L, as desired. □

Similarly, one can obtain the following result.

Theorem 2.2. Let f : XY be an odd mapping for which there is a ρ : X2D+(x, y) is denoted by ρ x, y ) satisfying (2.1). If
lim n T k = 1 ( T ( ρ x 2 k + n , x 2 k + n ( t 8 n + 2 k ) , ρ x 2 k + n 1 , x 2 k + n ( 4 t 8 n + 2 k ) ) ) = 1
and
lim n ρ x 2 n , y 2 n ( t 8 n ) = 1
for all x, y X and all t > 0, then there exist a unique additive mapping A : XY and a unique cubic mapping C : XY such that
μ f ( 2 x ) - 8 f ( x ) - A ( x ) ( t ) T k = 1 T ρ x 2 k , x 2 k t 2 2 k + 1 , ρ x 2 k - 1 , x 2 k t 2 2 k - 1 , μ f ( 2 x ) - 2 f ( x ) - C ( x ) ( t ) T k = 1 T ρ x 2 k , x 2 k t 8 2 k , ρ x 2 k - 1 , x 2 k 4 t 8 2 k

for all × X and all t > 0.

3. Hyers-ulam stability of the functional equation (1.4): an even mapping case

In this section, we prove the Hyers-Ulam stability of the functional equation D f (x, y) = 0 in complete RN-spaces: an even mapping case.

Theorem 3.1. Let f : XY be an even mapping for which there is a ρ : X2D+ (ρ (x, y) is denoted by ρx, y) satisfying f (0) = 0 and (2.1). If
lim n T k = 1 ( T ( ρ 2 k + n 1 x , 2 k + n 1 x ( 2 · 4 n 2 t ) , ρ 2 k + n x , 2 k + n 1 x ( 2 · 4 n 1 t ) ) ) = 1
(3.1)
and
lim n ρ 2 n x , 2 n y ( 4 n t ) = 1
(3.2)
for all x, y X and all t > 0, then there exist a unique quadratic mapping P : XY and a unique quartic mapping Q : XY such that
μ f ( 2 x ) - 1 6 f ( x ) - P ( x ) ( t ) T k = 1 T ρ 2 k - 1 x , 2 k - 1 x t 8 , ρ 2 k x , 2 k - 1 x t 2 ,
(3.3)
μ f ( 2 x ) - 4 f ( x ) - Q ( x ) ( t ) T k = 1 T ρ 2 k - 1 x , 2 k - 1 x t 8 , ρ 2 k x , 2 k - 1 x t 2
(3.4)

for all × X and all t > 0.

Proof. Putting x = y in (2.1), we get
μ f ( 3 y ) - 6 f ( 2 y ) + 1 5 f ( y ) ( t ) ρ y , y ( t )
(3.5)
for all y X and all t > 0. Replacing x by 2y in (2.1), we get
μ f ( 4 y ) - 4 f ( 3 y ) + 4 f ( 2 y ) + 4 f ( y ) ( t ) ρ 2 y , y ( t )
(3.6)
for all y X and all t > 0. It follows from (3.5) and (3.6) that
μ f ( 4 x ) - 2 0 f ( 2 x ) + 6 4 f ( x ) ( t ) = μ ( 4 f ( 3 x ) - 2 4 f ( 2 x ) + 6 0 f ( x ) ) + ( f ( 4 x ) - 4 f ( 3 x ) + 4 f ( 2 x ) + 4 f ( x ) ) ( t ) T μ 4 f ( 3 x ) - 2 4 f ( 2 x ) + 6 0 f ( x ) t 2 , μ f ( 4 x ) - 4 f ( 3 x ) + 4 f ( 2 x ) + 4 f ( x ) t 2 T ρ x , x t 8 , ρ 2 x , x t 2
(3.7)
for all x X and all t > 0. Let g : XY be a mapping defined by g(x) := f (2x) - 16 f (x). Then we conclude that
μ g ( 2 x ) - 4 g ( x ) ( t ) T ρ x , x t 8 , ρ 2 x , x t 2
for all x X and all t > 0. Thus, we have
μ g ( 2 x ) 4 - g ( x ) ( t ) T ρ x , x t 2 , ρ 2 x , x 2 t
for all x X and all t > 0. Hence,
μ g ( 2 k + 1 x ) 4 k + 1 g ( 2 k x ) 4 k ( t ) T ( ρ 2 k x , 2 k x ( 2 · 4 k 1 t ) , ρ 2 k + 1 x , 2 k x ( 2 · 4 k t ) )
for all x X, all t > 0 and all k . From 1 > 1 4 + 1 4 2 + + 1 4 n , it follows that
μ g ( 2 n x ) 4 n g ( x ) ( t ) T k = 1 n ( μ g ( 2 k x ) 4 k g ( 2 k 1 x ) 4 k 1 ( t 4 k ) ) T k = 1 n ( T ( ρ 2 k 1 x , 2 k 1 x ( t 8 ) , ρ 2 k x , 2 k 1 x ( t 2 ) ) )
(3.8)
for all x X and all t > 0. In order to prove the convergence of the sequence { g ( 2 n x ) 4 n } , replacing x with 2 m x in (3.8), we obtain that
μ g ( 2 n + m x ) 4 n + m g ( 2 m x ) 4 m ( t ) T k = 1 n ( T ( ρ 2 k + m 1 x , 2 k + m 1 x ( 2 · 4 m 2 t ) , ρ 2 k + m x , 2 k + m 1 x ( 2 · 4 m 1 t ) ) ) .
(3.9)

Since the right-hand side of the inequality (3.9) tends to 1 as m and n tend to infinity, the sequence { g ( 2 n x ) 4 n } is a Cauchy sequence. Thus, we may define P ( x ) = lim n g ( 2 n x ) 4 n for all x X.

Now, we show that P is a quadratic mapping. Replacing x and y with 2 n x and 2 n y in (2.1), respectively, we get
μ D f ( 2 n x , 2 n y ) 4 n ( t ) ρ 2 n x , 2 n y ( 4 n t ) .

Taking the limit as n → ∞, we find that P : XY satisfies (1.4) for all x, y X. Since f : XY is even, P : XY is even. By [[44], Lemma 2.1], the mapping P : XY is quadratic. Letting the limit as n → ∞ in (3.8), we get (3.3).

Next, we prove the uniqueness of the quadratic mapping P : XY subject to (3.3). Let us assume that there exists another quadratic mapping L : XY, which satisfies (3.3). Since P(2 n x) = 4 n P(x), L(2 n x) = 4 n L(x) for all x X and all n , from (3.3), it follows that
μ P ( x ) L ( x ) ( 2 t ) = μ P ( 2 n x ) L ( 2 n x ) ( 2 · 4 n t ) T ( μ P ( 2 n x ) g ( 2 n x ) ( 4 n t ) , μ g ( 2 n x ) L ( 2 n x ) ( 4 n t ) ) T ( T k = 1 ( T ( ρ 2 n + k 1 x , 2 n + k 1 x ( 2 · 4 n 2 t ) , ρ 2 n + k x , 2 n + k 1 x ( 2 · 4 n 1 t ) ) ) , T k = 1 ( T ( ρ 2 n + k 1 x , 2 n + k 1 x ( 2 · 4 n 2 t ) , ρ 2 n + k x , 2 n + k 1 x ( 2 · 4 n 1 t ) ) ) )
(3.10)

for all x X and all t > 0. Letting n → ∞ in (3.10), we conclude that P = L.

Let h : XY be a mapping defined by h(x) := f (2x) -4f (x). Then, we conclude that
μ h ( 2 x ) - 1 6 h ( x ) ( t ) T ρ x , x t 8 , ρ 2 x , x t 2
for all x X and all t > 0. Thus, we have
μ h ( 2 x ) 1 6 - h ( x ) ( t ) T ( ρ x , x ( 2 t ) , ρ 2 x , x ( 8 t ) )
for all x X and all t > 0. Hence,
μ h ( 2 k + 1 x ) 16 k + 1 h ( 2 k x ) 16 k ( t ) T ( ρ 2 k x , 2 k x ( 2 · 16 k t ) , ρ 2 k + 1 x , 2 k x ( 8 · 16 k t ) )
for all x X, all t > 0 and all k . From 1 > 1 1 6 + 1 1 6 2 + + 1 1 6 n , it follows that
μ h ( 2 n x ) 16 n h ( x ) ( t ) T k = 1 n ( μ h ( 2 k x ) 16 k h ( 2 k 1 x ) 16 k 1 ( t 16 k ) ) T k = 1 n ( T ( ρ 2 k 1 x , 2 k 1 x ( t 8 ) , ρ 2 k x , 2 k 1 x ( t 2 ) ) )
(3.11)
for all x X and all t > 0. In order to prove the convergence of the sequence { h ( 2 n x ) 16 n } , replacing x with 2 m x in (3.11), we obtain that
μ h ( 2 n + m x ) 16 n + m h ( 2 m x ) 16 m ( t ) T k = 1 n ( T ( ρ 2 k + m 1 x , 2 k + m 1 x ( 2 · 16 m 1 t ) , ρ 2 k + m x , 2 k + m 1 x ( 8 · 16 m 1 t ) ) ) .
(3.12)

Since the right-hand side of the inequality (3.12) tends to 1 as m and n tend to infinity, the sequence { h ( 2 n x ) 16 n } is a Cauchy sequence. Thus, we may define Q ( x ) = lim n h ( 2 n x ) 16 n x X.

Now, we show that Q is a quartic mapping. Replacing x and y with 2 n x and 2 n y in (2.1), respectively, we get
μ D f ( 2 n x , 2 n y ) 16 n ( t ) ρ 2 n x , 2 n y ( 16 n t ) ρ 2 n x , 2 n y ( 4 n t ) .

Taking the limit as n → ∞, we find that Q : XY satisfies (1.4) for all x, y X. Since f : XY is even, Q : XY is even. By [[44], Lemma 2.1], the mapping Q : XY is quartic. Letting the limit as n → ∞ in (3.11), we get (3.4).

Finally, we prove the uniqueness of the quartic mapping Q : XY subject to (3.4). Let us assume that there exists another quartic mapping L : XY , which satisfies (3.4). Since Q(2 n x) = 16 n Q(x), L(2 n x) = 16 n L(x) for all x X and all n , from (3.4), it follows that
μ Q ( x ) L ( x ) ( 2 t ) = μ Q ( 2 n x ) L ( 2 n x ) ( 2 · 16 n t ) T ( μ Q ( 2 n x ) h ( 2 n x ) ( 16 n t ) , μ h ( 2 n x ) L ( 2 n x ) ( 16 n t ) ) T ( T k = 1 ( T ( ρ 2 n + k 1 x , 2 n + k 1 x ( 2 · 16 n 1 t ) , ρ 2 n + k x , 2 n + k 1 x ( 8 · 16 n 1 t ) ) ) , T k = 1 ( T ( ρ 2 n + k 1 x , 2 n + k 1 x ( 2 · 16 n 1 t ) , ρ 2 n + k x , 2 n + k 1 x ( 8 · 16 n 1 t ) ) ) T ( T k = 1 ( T ( ρ 2 n + k 1 x , 2 n + k 1 x ( 2 · 4 n 2 t ) , ρ 2 n + k x , 2 n + k 1 x ( 2 · 4 n 1 t ) ) ) , T k = 1 ( T ( ρ 2 n + k 1 x , 2 n + k 1 x ( 2 · 4 n 2 t ) , ρ 2 n + k x , 2 n + k 1 x ( 2 · 4 n 1 t ) ) ) )
(3.13)

for all x X and all t > 0. Letting n → ∞ in (3.13), we conclude that Q = L, as desired. □

Similarly, one can obtain the following result.

Theorem 3.2. Let f : XY be an even mapping for which there is a ρ : X2D+ (x, y) is denoted by ρ x, y) satisfying f (0) = 0 and (2.1). If
lim n T k = 1 ( T ( ρ x 2 k + n , x 2 k + n ( 2 t 16 n + 2 k ) , ρ x 2 k + n 1 , x 2 k + n ( 8 t 16 n + 2 k ) ) ) = 1
and
lim n ρ x 2 n , y 2 n ( t 16 n ) = 1
for all x, y X and all t > 0, then there exist a unique quadratic mapping P : XY and a unique quartic mapping Q : XY such that
μ f ( 2 x ) - 1 6 f ( x ) - P ( x ) ( t ) T k = 1 T ρ x 2 k , x 2 k 2 t 4 2 k + 1 , ρ x 2 k - 1 , x 2 k 2 t 4 2 k , μ f ( 2 x ) - 4 f ( x ) - Q ( x ) ( t ) T k = 1 T ρ x 2 k , x 2 k 2 t 1 6 2 k , ρ x 2 k - 1 , x 2 k 8 t 1 6 2 k

for all × X and all t > 0.

Declarations

Acknowledgements

Choonkil Park, Jung Rye Lee and Dong Yun Shin were supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2009-0070788), (NRF-2010-0009232) and (NRF-2010-0021792), respectively. Sun Young Jang was supported by NRF Research Fund 2010-0013211 and has written during visiting the research Institute of Mathematics, Seoul National University.

Authors’ Affiliations

(1)
Department of Mathematics, Hanyang University
(2)
Department of Mathematics, University of Ulsan
(3)
Department of Mathematics, Daejin University
(4)
Department of Mathematics, University of Seoul

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© Park et al; licensee Springer. 2011

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