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On the stability of an AQCQ-functional equation in random normed spaces
Journal of Inequalities and Applications volume 2011, Article number: 34 (2011)
Abstract
In this paper, we prove the Hyers-Ulam stability of the following additive-quadratic-cubic-quartic functional equation
in random normed spaces.
2010 Mathematics Subject Classification: 46S40; 39B52; 54E70
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 : G1 → G2 satisfies the inequality d(h(x·y), h(x) * h(y)) < δ for all x, y ∈ G1, then there exists a homomorphism H : G1 → G2 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 : E → E' be a mapping between Banach spaces such that
for all x, y ∈ E and some δ > 0. Then, there exists a unique additive mapping T : E → E' such that
for all x ∈ E. Moreover, if f(tx) is continuous in t ∈ ℝ for each fixed x ∈ E, then T is ℝ-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 [5–11]).
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. ([12–18]). Assume that there exist constants Θ ≥ 0 and p1, p2 ∈ ℝ such that p = p1 + p2≠ 1, and f : E → E' is a mapping from a normed space E into a Banach space E' such that the inequality
for all x, y ∈ E. Then, there exists a unique additive mapping T : E → E' such that
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 [20–25]).
The functional equation
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
The Hyers-Ulam stability problem for the quadratic functional equation (1.1) was proved by Skof for mappings f : A → B, 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:
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
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 : X4 → Y 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
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 [35–37]. Throughout this paper, Δ+ is the space of distribution functions, that is, the space of all mappings F : ℝ ∪ {-∞, ∞} → [0, 1] such that F is left-continuous and non-decreasing on ℝ, 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, . The space Δ+ is partially ordered by the usual point-wise ordering of functions, i.e., F ≤ G if and only if F(t) ≤ G(t) for all t in ℝ. The maximal element for Δ+ in this order is the distribution function ε0 given by
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 a ≤ c and b ≤ d 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 is defined recurrently by and for n ≥ 2. is defined as . It is known [39] that for the Lukasiewicz t-norm, the following implication holds:
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) 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
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)
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 whenever n ≥ N.
-
(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 whenever n ≥ m ≥ N.
-
(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, thenalmost everywhere.
Recently, Eshaghi Gordji et al. establish the stability of cubic, quadratic and additive-quadratic functional equations in RN-spaces (see [40–42]).
One can easily show that an odd mapping f : X → Y satisfies (1.4) if and only if the odd mapping f : X → Y is an additive-cubic mapping, i.e.,
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 .
One can easily show that an even mapping f : X → Y satisfies (1.4) if and only if the even mapping f : X → Y is a quadratic-quartic mapping, i.e.,
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
Lemma 1.6. Each mapping f : X → Y 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 : X → Y , we define
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 : X → Y be an odd mapping for which there is a ρ : X2 → D+ (ρ (x, y) is denoted by ρx, y) such that
for all x, y ∈ X and all t > 0. If
and
for all x, y ∈ X and all t > 0, then there exist a unique additive mapping A : X → Y and a unique cubic mapping C : X → Y such that
for all × ∈ X and all t > 0.
Proof. Putting x = y in (2.1), we get
for all y ∈ X and all t > 0. Replacing x by 2y in (2.1), we get
for all y ∈ X and all t > 0. It follows from (2.6) and (2.7) that
for all x ∈ X and all t > 0. Let g : X → Y be a mapping defined by g(x) := f (2x) - 8f (x). Then we conclude that
for all x ∈ X and all t > 0. Thus, we have
for all x ∈ X and all t > 0. Hence,
for all x ∈ X, all t > 0 and all k ∈ ℕ: From , it follows that
for all x ∈ X and all t > 0. In order to prove the convergence of the sequence , replacing x with 2 mx in (2.9), we obtain that
Since the right-hand side of the inequality (2.10) tends to 1 as m and n tend to infinity, the sequence is a Cauchy sequence. Thus, we may define for all x ∈ X.
Now, we show that A is an additive mapping. Replacing x and y with 2 nx and 2 ny in (2.1), respectively, we get
Taking the limit as n → ∞, we find that A : X → Y satisfies (1.4) for all x, y ∈ X. Since f : X → Y is odd, A : X → Y is odd. By [[43], Lemma 2.2], the mapping A : X → Y is additive. Letting the limit as n → ∞ in (2.9), we get (2.4).
Next, we prove the uniqueness of the additive mapping A : X → Y subject to (2.4). Let us assume that there exists another additive mapping L : X → Y which satisfies (2.4). Since A(2 nx) = 2 nA(x), L(2 nx) = 2 nL(x) for all x ∈ X and all n ∈ ℕ, from (2.4), it follows that
for all x ∈ X and all t > 0. Letting n → ∞ in (2.11), we conclude that A = L.
Let h : X → Y be a mapping defined by h(x) := f (2x) -2f (x). Then, we conclude that
for all x ∈ X and all t > 0. Thus, we have
for all x ∈ X and all t > 0. Hence,
for all x ∈ X, all t > 0 and all k ∈ ℕ: From , it follows that
for all x ∈ X and all t > 0. In order to prove the convergence of the sequence , replacing x with 2 mx in (2.12), we obtain that
Since the right-hand side of the inequality (2.13) tends to 1 as m and n tend to infinity, the sequence is a Cauchy sequence. Thus, we may define for all x ∈ X.
Now, we show that C is a cubic mapping. Replacing x and y with 2 nx and 2 ny in (2.1), respectively, we get
Taking the limit as n → ∞, we find that C : X → Y satisfies (1.4) for all x, y ∈ X. Since f : X → Y is odd, C : X → Y is odd. By [[43], Lemma 2.2], the mapping C : X → Y is cubic. Letting the limit as n → ∞ in (2.12), we get (2.5).
Finally, we prove the uniqueness of the cubic mapping C : X → Y subject to (2.5). Let us assume that there exists another cubic mapping L : X → Y which satisfies (2.5). Since C(2 nx) = 8 nC(x), L(2 nx) = 8 nL(x) for all x ∈ X and all n ∈ ℕ, from (2.5), it follows that
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 : X → Y be an odd mapping for which there is a ρ : X2 → D+(ρ(x, y) is denoted by ρ x, y ) satisfying (2.1). If
and
for all x, y ∈ X and all t > 0, then there exist a unique additive mapping A : X → Y and a unique cubic mapping C : X → Y such that
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 : X → Y be an even mapping for which there is a ρ : X2 → D+ (ρ (x, y) is denoted by ρx, y) satisfying f (0) = 0 and (2.1). If
and
for all x, y ∈ X and all t > 0, then there exist a unique quadratic mapping P : X → Y and a unique quartic mapping Q : X → Y such that
for all × ∈ X and all t > 0.
Proof. Putting x = y in (2.1), we get
for all y ∈ X and all t > 0. Replacing x by 2y in (2.1), we get
for all y ∈ X and all t > 0. It follows from (3.5) and (3.6) that
for all x ∈ X and all t > 0. Let g : X → Y be a mapping defined by g(x) := f (2x) - 16 f (x). Then we conclude that
for all x ∈ X and all t > 0. Thus, we have
for all x ∈ X and all t > 0. Hence,
for all x ∈ X, all t > 0 and all k ∈ ℕ. From , it follows that
for all x ∈ X and all t > 0. In order to prove the convergence of the sequence , replacing x with 2 mx in (3.8), we obtain that
Since the right-hand side of the inequality (3.9) tends to 1 as m and n tend to infinity, the sequence is a Cauchy sequence. Thus, we may define for all x ∈ X.
Now, we show that P is a quadratic mapping. Replacing x and y with 2 nx and 2 ny in (2.1), respectively, we get
Taking the limit as n → ∞, we find that P : X → Y satisfies (1.4) for all x, y ∈ X. Since f : X → Y is even, P : X → Y is even. By [[44], Lemma 2.1], the mapping P : X → Y is quadratic. Letting the limit as n → ∞ in (3.8), we get (3.3).
Next, we prove the uniqueness of the quadratic mapping P : X → Y subject to (3.3). Let us assume that there exists another quadratic mapping L : X → Y, which satisfies (3.3). Since P(2 nx) = 4 nP(x), L(2 nx) = 4 nL(x) for all x ∈ X and all n ∈ ℕ, from (3.3), it follows that
for all x ∈ X and all t > 0. Letting n → ∞ in (3.10), we conclude that P = L.
Let h : X → Y be a mapping defined by h(x) := f (2x) -4f (x). Then, we conclude that
for all x ∈ X and all t > 0. Thus, we have
for all x ∈ X and all t > 0. Hence,
for all x ∈ X, all t > 0 and all k ∈ ℕ. From , it follows that
for all x ∈ X and all t > 0. In order to prove the convergence of the sequence , replacing x with 2 mx in (3.11), we obtain that
Since the right-hand side of the inequality (3.12) tends to 1 as m and n tend to infinity, the sequence is a Cauchy sequence. Thus, we may define x ∈ X.
Now, we show that Q is a quartic mapping. Replacing x and y with 2 nx and 2 ny in (2.1), respectively, we get
Taking the limit as n → ∞, we find that Q : X → Y satisfies (1.4) for all x, y ∈ X. Since f : X → Y is even, Q : X → Y is even. By [[44], Lemma 2.1], the mapping Q : X → Y is quartic. Letting the limit as n → ∞ in (3.11), we get (3.4).
Finally, we prove the uniqueness of the quartic mapping Q : X → Y subject to (3.4). Let us assume that there exists another quartic mapping L : X → Y , which satisfies (3.4). Since Q(2 nx) = 16 nQ(x), L(2 nx) = 16 nL(x) for all x ∈ X and all n ∈ ℕ, from (3.4), it follows that
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 : X → Y be an even mapping for which there is a ρ : X2 → D+(ρ (x, y) is denoted by ρ x, y) satisfying f (0) = 0 and (2.1). If
and
for all x, y ∈ X and all t > 0, then there exist a unique quadratic mapping P : X → Y and a unique quartic mapping Q : X → Y such that
for all × ∈ X and all t > 0.
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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.
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Park, C., Jang, S.Y., Lee, J.R. et al. On the stability of an AQCQ-functional equation in random normed spaces. J Inequal Appl 2011, 34 (2011). https://doi.org/10.1186/1029-242X-2011-34
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DOI: https://doi.org/10.1186/1029-242X-2011-34