Quadratic--quartic functional equations in RN--spaces

In this paper, we obtain the general solution and the stability result for the following functional equation in random normed spaces (in the sense of Sherstnev) under arbitrary $t$-norms $$f(2x+y)+f(2x-y)=4[f(x+y)+f(x-y)]+2[f(2x)-4f(x)]-6f(y).$$


Introduction
The stability problem of functional equations originated from a question of Ulam [33] 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 ǫ > 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)) < ǫ 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 [15] 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 : Moreover, if f (tx) is continuous in t ∈ R for each fixed x ∈ E, then T is R-linear. In 1978, Th. M. Rassias [27] provided a generalization of the Hyers' theorem which allows the Cauchy difference to be unbounded. In 1991, Z. Gajda [10] answered the question for the case p > 1, which was raised by Rassias. This new concept is known as Hyers-Ulam-Rassias stability of functional equations (see [1,2,3,11,16,17,28,29]). The functional equation f (x + y) + f (x − y) = 2f (x) + 2f (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 exits a unique symmetric bi-additive mapping B such that f (x) = B(x, x) for all x (see [1,18]). The bi-additive mapping B is given by The Hyers-Ulam-Rassias 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 [32]). Cholewa [5] noticed that the theorem of Skof is still true if relevant domain A is replaced an abelian group. In [7], Czerwik proved the Hyers-Ulam-Rassias stability of the functional equation (1.1). Grabiec [12] has generalized these results mentioned above.
In [26], W. Park and J. 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 : It is easy to show that the function f (x) = x 4 satisfies the functional equation (1.3), which is called a quartic functional equation (see also [6]). In addition, Kim [19] has obtained the Hyers-Ulam-Rassias stability for a mixed type of quartic and quadratic functional equation. The Hyers-Ulam-Rassias stability of different functional equations in random normed and fuzzy normed spaces has been recently studied in [20]- [25]. It should be noticed that in all these papers the triangle inequality is expressed by using the strongest triangular norm TM . The aim of this paper is to investigate the stability of the additive-quadratic 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 [4,21,22,30,31]. Throughout this paper, ∆ + 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 ∆ + consisting of all functions F ∈ ∆ + 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) = lim t→x − f (t). The space ∆ + 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 ∆ + in this order is the distribution function ε0 given by Typical examples of continuous t-norms are TP (a, b) = ab, TM (a, b) = min(a, b) and TL(a, b) = max(a + b − 1, 0) (the Lukasiewicz t-norm). Recall (see [13,14]) that if T is a t-norm and {xn} is a given sequence of numbers in [0, 1], then T n i=1 xi is defined recurrently by It is known ( [14]) that for the Lukasiewicz t-norm the following implication holds: where X is a vector space, T is a continuous t-norm and µ is a mapping from X into D + such that the following conditions hold: , µy(s)) for all x, y ∈ X and t, s ≥ 0.
Every normed space (X, . ) defines a random normed space (X, µ, TM ), where for all t > 0, and TM is the minimum t-norm. This space is called the induced random normed space.

T ) is said to be complete if and only if every Cauchy sequence in X is convergent to a point in X.
Theorem 1.4. ( [30]). If (X, µ, T ) is an RN-space and {xn} is a sequence such that xn → x, then limn→∞ µx n (t) = µx(t) almost everywhere.
Recently, M. Eshaghi Gordji et al. establish the stability of cubic, quadratic and additivequadratic functional equations in RN-spaces (see [8] and [9]). In this paper, we deal with the following functional equation on RN-spaces. It is easy to see that the function f (x) = ax 4 + bx 2 is a solution of (1.4).
In Section 2, we investigate the general solution of the functional equation (1.4) when f is a mapping between vector spaces and in Section 3, we establish the stability of the functional equation (1.4) in RN-spaces.

General solution
We need the following lemma for solution of (1.4). Throughout this section X and Y are vector spaces.
Proof. We show that the mappings g : for all x, y ∈ X. Interchanging x with y in (1.4), we obtain for all x, y ∈ X. Since f is even, by (2.2), one gets for all x, y ∈ X. It follows from (2.1) and (2.3) that for all x, y ∈ X. This means that for all x, y ∈ X. Therefore, the mapping g : for all x, y ∈ X. Since f is even, the mapping h is even. Now if we interchange x with y in the last equation, we get for all x, y ∈ X. Thus it is enough to prove that h satisfies in (2.4). Replacing x and y by 2x and 2y in (1.4), respectively, we obtain for all x, y ∈ X. Since g(2x) = 4g(x) for all x ∈ X, for all x ∈ X. By (2.5) and (2.6), we get for all x, y ∈ X. By multiplying both sides of (1.4) by 4, we get for all x, y ∈ X. If we subtract the last equation from (2.7), we obtain for all x, y ∈ X. Therefore, the mapping h : X → Y is quartic. This completes the proof of the lemma.
for all x ∈ X.
Proof. Let f satisfies (1.4) and assume that g, h : X → Y are mappings defined by for all x ∈ X. By Lemma 2.1, we obtain that the mappings g and h are quadratic and quartic, respectively, and for all x ∈ X. Therefore, there exist a unique symmetric multi-additive mapping M : X 4 → Y and a unique symmetric bi-additive mapping B : X × X → Y such that 1 12 h(x) = M (x, x, x, x) and −1 12 g(x) = B(x, x) for all x ∈ X(see [1,26]). So for all x ∈ X. The proof of the converse is obvious.

Stability
Throughout this section, assume that X is a real linear space and (Y, µ, T ) is a complete RN-space.
for all x, y ∈ X and all t > 0. If and lim n→∞ ρ2nx,2ny(2 2n t) = 1 (3.3) for all x, y ∈ X and all t > 0, then there exists a unique quadratic mapping Q1 : X → Y such that for all x ∈ X and all t > 0.
Proof. Putting y = x in (3.1), we obtain for all x ∈ X. Letting y = 2x in (3.1), we get for all x ∈ X. Putting x = 0 in (3.1), we obtain for all y ∈ X. Replacing y by x in (3.7), we see that for all x ∈ X. It follows from (3.6) and (3.8) that for all x ∈ X. If we add (3.5) to (3.9), then we have for all x ∈ X. Then we get for all x ∈ X and all t > 0. Let g : X → Y be a mapping defined by g( for all x ∈ X and all t > 0. Hence for all x ∈ X and all k ∈ N. This means that for all x ∈ X, t > 0 and all k ∈ N. By the triangle inequality, from 1 > 1 2 + 1 2 2 + · · · + 1 2 n , it follows for all x ∈ X and t > 0. In order to prove the convergence of the sequence { g(2 n x) 2 2n }, we replace x with 2 m x in (3.17) to obtain that µ g(2 n+m x) Since the right hand side of the inequality (3.18) tends to 1 as m and n tend to infinity, the sequence { g(2 n x) 2 2n } is a Cauchy sequence. Thus we may define Q1(x) = limn→∞ g(2 n x) for all x ∈ X. Now we show that Q1 is a quadratic mapping. Replacing x, y with 2 n x and 2 n y in (3.1), respectively, we get (3.19) Taking the limit as n → ∞, we find that Q1 satisfies (1.4) for all x, y ∈ X. By Lemma 2.1, the mapping Q1 : X → Y is quadratic. Letting the limit as n → ∞ in (3.17), we get (3.4) by (3.11).
Theorem 3.2. Let f : X → Y be a mapping with f (0) = 0 for which there is ρ : X×X → D + ( ρ(x, y) is denoted by ρx,y ) with the property: for all x, y ∈ X and all t > 0. If and lim n→∞ ρ2nx,2ny(2 4n t) = 1 (3.23) for all x, y ∈ X and all t > 0, then there exists a unique quartic mapping Q2 : X → Y such that (3.24) for all x ∈ X and all t > 0.
Proof. Putting y = x in (3.21), we obtain for all x ∈ X. Letting y = 2x in (3.21), we get for all x ∈ X. Putting x = 0 in (3.21), we obtain for all y ∈ X. Replacing y by x in (3.27), we get for all x ∈ X. It follows from (3.6) and (3.28) that for all x ∈ X. If we add (3.25) to (3.29), then we have for all x ∈ X. Then we get for all x ∈ X and all t > 0. Let h : X → Y be a mapping defined by h( Then we conclude that for all x ∈ X. Thus we have for all x ∈ X and all t > 0. Hence for all x ∈ X and all k ∈ N. This means that for all x ∈ X, t > 0 and all k ∈ N. By the triangle inequality, from 1 > 1 2 + 1 2 2 + · · · + 1 2 n , it follows for all x ∈ X and all t > 0. In order to prove the convergence of the sequence { h(2 n x) 2 4n }, we replace x with 2 m x in (3.37) to obtain that Since the right hand side of the inequality (3.38) tends to 1 as m and n tend to infinity, the sequence { h(2 n x) 2 4n } is a Cauchy sequence. Thus we may define Q2(x) = limn→∞ h(2 n x) for all x ∈ X. Now we show that Q2 is a quartic mapping. Replacing x, y with 2 n x and 2 n y in (3.21), respectively, we get (3.39) Taking the limit as n → ∞, we find that Q2 satisfies (1.4) for all x, y ∈ X. By Lemma 2.1 we get that the mapping Q2 : X → Y is quartic.
Theorem 3.3. Let f : X → Y be a mapping with f (0) = 0 for which there is ρ : X×X → D + ( ρ(x, y) is denoted by ρx,y ) with the property: for all x, y ∈ X and all t > 0. If for all x, y ∈ X and all t > 0, then there exist a unique quadratic mapping Q1 : X → Y and a unique quartic mapping Q2 : X → Y such that for all x ∈ X and all t > 0.
Proof. By Theorems 3.1 and 3.2, there exist a quadratic mapping Q ′ 1 : X → Y and a quartic mapping Q ′ 2 : X → Y such that and for all x ∈ X and all t > 0. So it follows from the last inequalities that for all x ∈ X and all t > 0. Hence we obtain (3.46) by letting Q1(x) = − 1 12 Q ′ 1 (x) and Q2(x) = 1 12 Q ′ 2 (x) for all x ∈ X. The uniqueness property of Q1 and Q2, are trivial.

Acknowledgement
The third author was supported by Korea Research Foundation Grant funded by the Korean Government (KRF-2008-313-C00041).