# Fuzzy Stability of Generalized Mixed Type Cubic, Quadratic, and Additive Functional Equation

## Abstract

In this paper, we prove the generalized Hyers-Ulam stability of generalized mixed type cubic, quadratic, and additive functional equation, in fuzzy Banach spaces.

2010 Mathematics Subject Classification: 39B82; 39B52.

## 1. Introduction

The stability problem of functional equations originated from a question of Ulam  concerning the stability of group homomorphisms. Hyers  gave a first affirmative answer to the question of Ulam for Banach spaces. Hyers' theorem was generalized by Aoki  for additive mappings and by Rassias  for linear mappings by considering an unbounded Cauchy difference. The paper of Rassias  has provided a lot of influence in the development of what we now call generalized Hyers-Ulam stability of functional equations. In 1994, a generalization of the Rassias' theorem was obtained by Găvruta  by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias' approach.

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 [6, 7]. It is natural that this equation is called a quadratic functional equation. In particular, every solution of the quadratic 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 [6, 7]). The bi-additive mapping B is given by

$B ( x , y ) = 1 4 ( f ( x + y ) - f ( x - y ) )$
(1.2)

A generalized Hyers-Ulam stability problem for the quadratic functional equation (1.1) was proved by Skof for mappings f : AB, where A is normed space and B is a Banach space  (see ).

Jun and Kim  introduced the following cubic functional equation

$f ( 2 x + y ) + f ( 2 x - y ) = 2 f ( x + y ) + 2 f ( x - y ) + 1 2 f ( x )$
(1.3)

and they established the general solution and the generalized Hyers-Ulam stability for the functional equation (1.3). 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 mapping C : X × X × XY such that f (x) = C(x, x, x) for all x X, moreover, C is symmetric for each fixed one variable and is additive for fixed two variables. The mapping C is given by

$C ( x , y , z ) = 1 2 4 ( f ( x + y + z ) + f ( x - y - z ) - f ( x + y - z ) - f ( x - y + z ) )$
(1.4)

for all x, y, z X. During the last decades, several stability problems for various functional equations have been investigated by many mathematicians; .

Eshaghi and Khodaei  have established the general solution and investigated the generalized Hyers-Ulam stability for a mixed type of cubic, quadratic, and additive functional equation with f (0) = 0,

$f ( x + k y ) + f ( x - k y ) = k 2 f ( x + y ) + k 2 f ( x - y ) + 2 ( 1 - k 2 ) f ( x )$
(1.5)

in quasi-Banach spaces, where k is nonzero integer numbers with k ≠ ± 1. Obviously, the function f (x) = ax + bx2 + cx3 is a solution of the functional equation (1.5). Interesting new results concerning mixed functional equations has recently been obtained by Najati et. al. [23, 24], Jun and Kim [25, 26] as well as for the fuzzy stability of a mixed type of additive and quadratic functional equation by Park  (see also ).

This paper is organized as follows: In Section 3, we prove the generalized Hyers-Ulam stability of the functional equation (1.5) in fuzzy Banach spaces for an even case. In Section 4, we prove the generalized Hyers-Ulam stability of the functional equation (1.5) in fuzzy Banach spaces for an odd case. In Section 5, we prove the generalized Hyers-Ulam stability of generalized mixed cubic, quadratic, and additive functional equation (1.5) in fuzzy Banach spaces.

## 2. Preliminaries

We use the definition of fuzzy normed spaces given in  to investigate a fuzzy version of the generalized Hyers-Ulam stability for the functional equation (1.5) in the fuzzy normed space setting.

Definition 2.1. (Bag and Samanta , Mirmostafaee ). Let X be a real linear space. A function N : X × → [0, 1] is said to be a fuzzy norm on X if for all x, y X and all s, t ;

(N1) N(x, t) = 0 for all t ≤ 0;

(N2) x = 0 if and only if N(x, t) = 1 for all t > 0;

(N3) $N ( c x , t ) =N x , t | c |$ if c ≠ 0;

(N4) N(x + y, s + t) ≥ min{N(x, s), N(y, t)};

(N5) N(x, ·) is non-decreasing function on and limt→∞N(x, t) = 1;

(N6) N(x, ·) is left continuous on for every x ≠ 0.

The pair (X, N) is called a fuzzy normed linear space.

The properties of fuzzy normed vector spaces and examples of fuzzy norms are given in ([3, 4547]).

Definition 2.2. (Bag and Samanta , Mirmostafaee ). Let (X, N) be a fuzzy normed linear space. A sequence {x n } in X is said to be convergent if there exists x X such that limn→∞N(x n - x, t) = 1 for all t > 0. In that case, x is called the limit of the sequence (x n ) and we write N - limn→∞x n = x.

Definition 2.3. (Bag and Samanta , Mirmostafaee ). Let (X, N) be a fuzzy normed linear space. A sequence {x n } in X is called Cauchy if for each ϵ > 0 and each δ > 0, there exists n0 such that N(x m - x n , δ) > 1 - ϵ (m, nn0).

It is well known that every convergent sequence in a fuzzy normed vector space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed vector space is called a fuzzy Banach space.

We say that a function f : XY between fuzzy normed vector spaces X and Y is continuous at a point x0 X if for each sequence {x k } converging to x0 in X, then the sequence {f(x k )} converges to f(x0). If f : XY is continuous at each x X, then f : XY is said to be continuous on X (see ).

In the rest of this paper, unless otherwise explicitly stated, we will assume that X is a vector space, (Z, N') is a fuzzy normed space, and (Y, N) is a fuzzy Banach space. For convenience, we use the following abbreviation for a given function f : XY,

$D f ( x , y ) = f ( x + k y ) + f ( x - k y ) - k 2 f ( x + y ) - k 2 f ( x - y ) - 2 ( 1 - k 2 ) f ( x )$

for all x, y X, where k is nonzero integer numbers with k ≠ ± 1.

## 3. Fuzzy stability of the functional equation (1.5): an even case

In this section, we prove the generalized Hyers-Ulam stability of the functional equation (1.5) in fuzzy Banach spaces, for an even case. From now on, V1 and V2 will be real vector spaces.

Lemma 3.1. . If an even mapping f : V1V2 satisfies (1.5), then f(x) is quadratic.

Theorem 3.2. Let ℓ {-1, 1} be fixed and let φ q : X × XY be a mapping such that

$φ q ( k x , k y ) = α φ q ( x , y )$
(3.1)

for all x, y X and for some positive real number α with αℓ < k2ℓ. Suppose that an even mapping f : XY with f(0) = 0 satisfies the inequality

$N ( D f ( x , y ) , t ) ≥ N ′ ( φ q ( x , y ) , t )$
(3.2)

for all x, y X and all t > 0. Then, the limit

$Q ( x ) = N − lim n → ∞ f ( k ℓ n x ) k 2 ℓ n )$

exists for all x X and Q : XY is a unique quadratic mapping satisfying

$N ( f ( x ) - Q ( x ) , t ) ≥ N ′ φ q ( 0 , x ) , ℓ ( k 2 - α ) t 2$
(3.3)

for all x X and all t > 0.

Proof. Case (1): = 1. By putting x = 0 in (3.2) and then using evenness of f and f(0) = 0, we obtain

$N ( 2 f ( k y ) - 2 k 2 f ( y ) , t ) ≥ N ′ ( φ q ( 0 , y ) , t )$
(3.4)

for all x, y X and all t > 0. If we replace y in (3.4) by x, we get

$N f ( k x ) - k 2 f ( x ) , t 2 ≥ N ′ ( φ q ( 0 , x ) , t )$
(3.5)

for all x X. So

$N f ( k x ) k 2 - f ( x ) , t 2 k 2 ≥ N ′ ( φ q ( 0 , x ) , t )$
(3.6)

for all x X and all t > 0. Then by our assumption

$N ′ ( φ q ( 0 , k x ) , t ) = N ′ φ q ( 0 , x ) , t α$
(3.7)

for all x X and all t > 0. Replacing x by knx in (3.6) and using (3.7), we obtain

$N f ( k n + 1 x ) k 2 ( n + 1 ) - f ( k n x ) k 2 n , t k 2 ( k 2 n ) ≥ N ′ ( φ q ( 0 , k n x ) , t ) = N ′ φ q ( 0 , x ) , t α n$
(3.8)

for all x X, t > 0 and n ≥ 0. Replacing t by αnt in (3.8), we see that

$N f ( k n + 1 x ) k 2 ( n + 1 ) - f ( k n x ) k 2 n , α n t k 2 ( k 2 n ) ≥ N ′ ( φ q ( 0 , x ) , t )$
(3.9)

for all x X, t > 0 and n > 0. It follows from $f ( k n x ) k 2 n -f ( x ) = ∑ j = 0 n - 1 f ( k j + 1 x ) k 2 ( j + 1 ) - f ( k j x ) k 2 j$ and (3.9) that

$N f ( k n x ) k 2 n - f ( x ) , ∑ j = 0 n - 1 α j t k 2 ( k 2 ) j ≥ min ⋃ j = 0 n - 1 N f ( k j + 1 x ) k 2 ( j + 1 ) - f ( k j x ) k 2 j , α j t k 2 ( k 2 ) j ≥ N ′ ( φ q ( 0 , x ) , t )$
(3.10)

for all x X, t > 0 and n > 0. Replacing x by kmx in (3.10), we observe that

$N f ( k n + m x ) k 2 ( n + m ) - f ( k m x ) k 2 m , ∑ j = 0 n - 1 α j t k 2 ( k 2 ) j + m ≥ N ′ ( φ q ( 0 , k m x ) , t ) = N ′ φ ( 0 , x ) , t α m$

for all x X, all t > 0 and all m ≥ 0, n > 0. Hence

$N f ( k n + m x ) k 2 ( n + m ) - f ( k m x ) k 2 m , ∑ j = m n + m - 1 α j t k 2 ( k 2 ) j ≥ N ′ ( φ q ( 0 , x ) , t )$

for all x X, all t > 0 and all m ≥ 0, n > 0. By last inequality, we obtain

$N f ( k n + m x ) k 2 ( n + m ) - f ( k m x ) k 2 m , t ≥ N ′ φ q ( 0 , x ) , t ∑ j = m n + m - 1 α j k 2 ( k 2 ) j$
(3.11)

for all x X, all t > 0 and all m ≥ 0, n > 0. Since 0 < α < k2 and $∑ j = 0 ∞ α k 2 j <∞$, the Cauchy criterion for convergence and (N5) imply that $f ( k n x ) k 2 n$ is a Cauchy sequence in Y. Since Y is a fuzzy Banach space, this sequence converges to some point Q(x) Y. So one can define the function Q : XY by

$Q ( x ) = N − lim n → ∞ f ( k n x ) k 2 n )$
(3.12)

for all x X. Fix x X and put m = 0 in (3.11) to obtain

$N f ( k n x ) k 2 n - f ( x ) , t ≥ N ′ φ q ( 0 , x ) , t ∑ j = 0 n - 1 α j k 2 ( k 2 ) j$

for all x X, all t > 0 and all n > 0. From which we obtain

$N ( Q ( x ) - f ( x ) , t ) ≥ min N ( Q ( x ) - f ( k n x ) k 2 n , t 2 ) , N f ( k n x ) k 2 n - f ( x ) , t 2 ≥ N ′ φ q ( 0 , x ) , t ∑ j = 0 n - 1 2 α j k 2 ( k 2 ) j$
(3.13)

for n large enough. Taking the limit as n → ∞ in (3.13), we obtain

$N ( Q ( x ) - f ( x ) , t ) ≥ N ′ φ q ( 0 , x ) , ( k 2 - α ) t 2$
(3.14)

for all x X and all t > 0. It follows from (3.8) and (3.12) that

for all x X and all t > 0. Therefore,

$Q ( k x ) = k 2 Q ( x )$
(3.15)

for all x X. Replacing x, y by knx, kny in (3.2), respectively, we obtain

$N 1 k 2 n D f ( k n x , k n y ) , t ≥ N ′ ( φ q ( k n x , k n y ) , k 2 n t ) = N ′ φ q ( x , y ) , k 2 n t α n$

which tends to 1 as n → ∞ for all x, y X and all t > 0. So, we see that Q satisfies (1.5). Thus, by Lemma 3.1, the function x f(x) is quadratic. Therefore, (3.15) implies that the function Q is quadratic.

Now, to prove the uniqueness property of Q, let Q': XY be another quadratic function satisfying (3.3). It follows from (3.3), (3.7) and (3.15) that

$N ( Q ( x ) - Q ′ ( x ) , t ) = N Q ( k n x ) k 2 n - Q ′ ( k n x ) k 2 n , t ≥ min N Q ( k n x ) k 2 n - f ( k n x ) k 2 n , t 2 , N f ( k n x ) k 2 n - Q ′ ( k n x ) k 2 n , t 2 ≥ N ′ φ q ( 0 , k n x ) , k 2 n ( k 2 - α ) t 4 = N ′ φ q ( 0 , x ) , k 2 n ( k 2 - α ) t 4 α n$

for all x X and all t > 0. Since α < k2, we obtain $lim n → ∞ N ′ φ q ( 0 , x ) , k 2 n ( k 2 - α ) t 4 α n =1$. Thus, Q(x) = Q'(x).

Case (2): = -1. We can state the proof in the same pattern as we did in the first case.

Replacing x by $x k$ in (3.5), we obtain

$N f ( x ) - k 2 f x k , t 2 ≥ N ′ φ q 0 , x k , t$
(3.16)

for all x X and all t > 0. Replacing x and t by $x k n$ and $t k 2 n$ in (3.16), respectively, we obtain

$N ( k 2 n f ( x k n ) − k 2 ( n + 1 ) f ( x k n + 1 ) , t 2 ) ≥ N ′ ( φ q ( 0, x k n + 1 ) , t k 2 n ) = N ′ ( φ q ( 0, x ) , ( α k 2 ) n α t )$

for all x X, all t > 0 and all n > 0. One can deduce

$N k 2 ( n + m ) f x k n + m - k 2 m f x k m , t ≥ N ′ φ q ( 0 , x ) , t ∑ j = m + 1 n + m 2 k 2 j k 2 α j$
(3.17)

for all x X, all t > 0 and all m ≥ 0, n ≥ 0. From which we conclude that $k 2 n f x k n$ is a Cauchy sequence in the fuzzy Banach space (Y, N). Therefore, there is a mapping Q : XY defined by $Q ( x ) :=N- lim n → ∞ k 2 n f x k n$. Employing (3.17) with m = 0, we obtain

$N ( Q ( x ) - f ( x ) , t ) ≥ N ′ φ q ( 0 , x ) , ( α - k 2 ) t 2$

for all x X and all t > 0. The proof for uniqueness of Q for this case proceeds similarly to that in the previous case, hence it is omitted.   □

Remark 3.3. Let 0 < α < k2. Suppose that the function t N(f(x) - Q(x), .) from (0, ∞) into [0, 1] is right continuous. Then, we obtain a better fuzzy (3.14) as follows.

We obtain

$N ( Q ( x ) − f ( x ) , t + s ) ≥ min { N ( Q ( x ) − f ( k n x ) k 2 n , s ) , N ( f ( k n x ) k 2 n − f ( x ) , t ) } ≥ N ′ ( φ q ( 0, x ) , t ∑ j = 0 n − 1 α j k 2 ( k ) 2 j ) ) ≥ N ′ ( φ q ( 0, x ) , ( k 2 − α ) t ) .$

Tending s to zero we infer that

$N ( Q ( x ) - f ( x ) , t ) ≥ N ′ ( φ q ( 0 , x ) , ( k 2 - α ) t )$

for all x X and all t > 0.

From Theorem 3.2, we obtain the following corollary concerning the generalized Hyers-Ulam stability  of quadratic mappings satisfying (1.5), in normed spaces.

Corollary 3.4. Let X be a normed space and Y be a Banach space. Let ε, λ be non-negative real numbers such that λ ≠ 2. Suppose that an even mapping f : XY with f(0) = 0 satisfies

$∥ D f ( x , y ) ∥ ≤ ε ( ∥ x ∥ λ + ∥ y ∥ λ )$
(3.18)

for all x, y X. Then, the limit

$Q ( x ) = N − lim n → ∞ f ( k ℓ n x ) k 2 ℓ n )$

exists for all x X and Q : XY is a unique quadratic mapping satisfying

$| | f ( x ) - Q ( x ) | | ≤ 2 ε ∥ x ∥ λ ℓ ( k 2 - k λ )$
(3.19)

for all x X, where λℓ < 2.

Proof. Define the function N by

$N ( x , t ) = t t + ∥ x ∥ , t > 0 0 , t ≤ 0$

It is easy to see that (X, N) is a fuzzy normed space and (Y, N) is a fuzzy Banach space. Denote φ q : X × X, the function sending each (x, y) to ε(||x||λ + ||y||λ). By assumption

$N ( D f ( x , y ) , t ) ≥ N ′ ( φ q ( x , y ) , t )$

note that N': × → [0, 1] given by

$N ′ ( x , t ) = t t + | x | , t > 0 0 , t ≤ 0$

is a fuzzy norm on . By Theorem 3.2, there exists a unique quadratic mapping Q : XY satisfying the equation (1.5) and

$t t + ∥ f ( x ) - Q ( x ) ∥ = N ( f ( x ) - Q ( x ) , t ) ≥ N ′ φ q ( 0 , x ) , ℓ ( k 2 - k λ ) t 2 = N ′ ε ∥ x ∥ λ , ℓ ( k 2 - k λ ) t 2 = ℓ ( k 2 - k λ ) t ℓ ( k 2 - k λ ) t + 2 ε ∥ x ∥ λ$

and thus

$t t + ∥ f ( x ) - Q ( x ) ∥ ≥ ℓ ( k 2 - k λ ) t ℓ ( k 2 - k λ ) t + 2 ε ∥ x ∥ λ$

which implies that, (k2 - kλ)||f(x) - Q(x)|| ≤ 2ε||x||λ for all x X.   □

In the following theorem, we will show that under some extra conditions on Theorem 3.2, the quadratic function r Q(rx) is fuzzy continuous. It follows that in such a case, Q(rx) = r2Q(x) for all x X and r .

In the following result, we will assume that all conditions of the theorem 3.2 hold.

Theorem 3.5. Denote N1 the fuzzy norm obtained as Corollary 3.4 on . Let for all x X, the functions r f(rx) (from (, N1) into (Y, N)) and r φ q (0, rx) (from (, N1) into (Z, N')) be fuzzy continuous. Then, for all x X, the function r Q(rx) is fuzzy continuous and Q(rx) = r2Q(x) for all r .

Proof. Case (1): = 1. Let {r k } be a sequence in that converge to some r , and let t > 0. Let ε > 0 be given, since 0 < α < k2, so $lim n → ∞ ( k 2 - α ) k 2 n t 1 2 α n =∞$, there is m such that

$N ′ φ q ( 0 , r x ) , ( k 2 - α ) k 2 m t 1 2 α m > 1 - ε$
(3.20)

It follows from (3.14) and (3.20) that

$N f ( k m r x ) k 2 m - Q ( k m r x ) k 2 m , t 3 > 1 - ε$
(3.21)

By the fuzzy continuity of functions r f(rx) and r φ q (0, rx), we can find some $J∈ℕ$ such that for any nj,

$N f ( k m r k x ) k 2 m - f ( k m r x ) k 2 m , t 3 > 1 - ε$
(3.22)

and

$N ′ ( φ q ( 0 , r k x ) - φ q ( 0 , r x ) , ( k 2 - α ) k 2 m t 1 2 α m ) > 1 - ε$
(3.23)

It follows from (3.20) and (3.23) that

$N ′ ( φ q ( 0 , r k x ) , ( k 2 - α ) k 2 m t 6 α m ) > 1 - ε$
(3.24)

On the other hand,

$N Q ( r k x ) - f ( k m r k x ) k 2 m , t k 2 m = N Q ( k m r k x ) k 2 m - f ( k m r k x ) k 2 m , t k 2 m ≥ N ′ φ q ( 0 , r k x ) , ( k 2 - α ) t 2 α m$
(3.25)

It follows from (3.24) and (3.25) that

$N Q ( r k x ) - f ( k m r k x ) k 2 m , t 3 > 1 - ε$
(3.26)

So, it follows from (3.21), (3.22) and (3.26) that for any nj,

$N ( Q ( r k x ) - Q ( r x ) , t ) > 1 - ε$

Therefore, for every choice x X, t > 0, and ε > 0, we can find some $J∈ℕ$ such that N(Q(r k x) - Q(rx), t) > 1 - ε for every $n≥J$. This shows that Q(r k x) → Q(rx). The proof for = -1 proceeds similarly to that in the previous case.

It is not hard to see that Q(rx) = r2Q(x) for each rational number r. Since Q is a fuzzy continuous function, by the same reasoning as in the proof of , the quadratic mapping Q : XY satisfies Q(rx) = r2Q(x) for each r . □

## 4. Fuzzy stability of the functional equation (1.5): an odd case

In this section, we prove the generalized Hyers-Ulam stability of the functional equation (1.5) in fuzzy Banach spaces for an odd case.

Lemma 4.1. [22, 24]. If an odd mapping f : V1V2 satisfies (1.5), then the mapping g : V1V2, defined by g(x) = f(2x) - 8f(x), is additive.

Theorem 4.2. Let ℓ {-1, 1} be fixed and let φ q : X × XZ be a function such that

$φ a ( 2 x , 2 y ) = α φ a ( x , y )$
(4.1)

for all x, y X and for some positive real number α with αℓ < 2ℓ. Suppose that an odd mapping f : XY satisfies the inequality

$N ( D f ( x , y ) , t ) ≥ N ′ ( φ a ( x , y ) , t )$
(4.2)

for all x, y X and all t > 0. Then, the limit

$A ( x ) = N - lim n → ∞ 1 2 ℓ n ( f ( 2 ℓ n + 1 x ) - 8 f ( 2 ℓ n x ) )$

exists for all x X and A : XY is a unique additive mapping satisfying

$N ( f ( 2 x ) - 8 f ( x ) - A ( x ) , t ) ≥ M a x , ℓ ( 2 - α ) 2 t$
(4.3)

for all x X and all t > 0, where

Proof. Case (1): = 1. It follows from (4.2) and using oddness of f that

$N ( f ( k y + x ) - f ( k y - x ) - k 2 f ( x + y ) - k 2 f ( x - y ) + 2 ( k 2 - 1 ) f ( x ) , t ) ≥ N ′ ( φ a ( x , y ) , t )$
(4.4)

for all x, y X and all t > 0. Putting y = x in (4.4), we have

$N ( f ( ( k + 1 ) x ) - f ( ( k - 1 ) x ) - k 2 f ( 2 x ) + 2 ( k 2 - 1 ) f ( x ) , t ) ≥ N ′ ( φ a ( x , x ) , t )$
(4.5)

for all x X and all t > 0. It follows from (4.5) that

$N ( f ( 2 ( k + 1 ) x ) - f ( 2 ( k - 1 ) x ) - k 2 f ( 4 x ) + 2 ( k 2 - 1 ) f ( 2 x ) , t ) ≥ N ′ ( φ a ( 2 x , 2 x ) , t )$
(4.6)

for all x X and all t > 0. Replacing x and y by 2x and x in (4.4), respectively, we get

$N ( f ( ( k + 2 ) x ) - f ( ( k - 2 ) x ) - k 2 f ( 3 x ) - k 2 f ( x ) + 2 ( k 2 - 1 ) f ( 2 x ) , t ) ≥ N ′ ( φ a ( 2 x , x ) , t )$
(4.7)

for all x X. Setting y = 2x in (4.4), we have

$N ( f ( ( 2 k + 1 ) x ) - f ( ( 2 k - 1 ) x ) - k 2 f ( 3 x ) - k 2 f ( - x ) + 2 ( k 2 - 1 ) f ( x ) , t ) ≥ N ′ ( φ a ( x , 2 x ) , t )$
(4.8)

for all x X and all t > 0. Putting y = 3x in (4.4), we obtain

$N ( f ( ( 3 k + 1 ) x ) - f ( ( 3 k - 1 ) x ) - k 2 f ( 4 x ) - k 2 f ( - 2 x ) + 2 ( k 2 - 1 ) f ( x ) , t ) ≥ N ′ ( φ a ( x , 3 x ) , t )$
(4.9)

for all x X and all t > 0. Replacing x and y by (k + 1)x and x in (4.4), respectively, we get

$N ( f ( ( 2 k + 1 ) x ) - f ( - x ) - k 2 f ( ( k + 2 ) x ) - k 2 f ( k x ) + 2 ( k 2 - 1 ) f ( ( k + 1 ) x ) , t ) ≥ N ′ ( φ a ( ( k + 1 ) x , x ) , t )$
(4.10)

for all x X and all t > 0. Replacing x and y by (k - 1)x and x in (4.4), respectively, one gets

$N ( f ( ( 2 k - 1 ) x ) - f ( x ) - k 2 f ( ( k - 2 ) x ) - k 2 f ( k x ) + 2 ( k 2 - 1 ) f ( ( k - 1 ) x ) , t ) ≥ N ′ ( φ a ( ( k - 1 ) x , x ) , t )$
(4.11)

for all x X and all t > 0. Replacing x and y by (2k + 1)x and x in (4.4), respectively, we obtain

$N ( f ( ( 3 k + 1 ) x ) - f ( - ( k + 1 ) x ) - k 2 f ( 2 ( k + 1 ) x ) - k 2 f ( 2 k x ) + 2 ( k 2 - 1 ) f ( ( 2 k + 1 ) x ) , t ) ≥ N ′ ( φ a ( ( 2 k + 1 ) x , x ) , t )$
(4.12)

for all x X and all t > 0. Replacing x and y by (2k - 1)x and x in (4.4), respectively, we have

$N ( f ( ( 3 k - 1 ) x ) - f ( - ( k - 1 ) x ) - k 2 f ( 2 ( k - 1 ) x ) - k 2 f ( 2 k x ) + 2 ( k 2 - 1 ) f ( ( 2 k - 1 ) x ) , t ) ≥ N ′ ( φ a ( ( 2 k - 1 ) x , x ) , t )$
(4.13)

for all x X and all t > 0. It follows from (4.5), (4.7), (4.8), (4.10) and (4.11) that

(4.14)

for all x X and all t > 0. And, from (4.5), (4.6), (4.8), (4.9), (4.12) and (4.14), we conclude that

$N f ( 4 x ) - 2 f ( 3 x ) - 2 f ( 2 x ) + 6 f ( x ) , 1 k 2 ( k 2 - 1 ) ( 2 ( k 2 - 1 ) + k 2 + 4 ) t ≥ min { N ′ ( φ a ( x , x ) , t ) , N ′ ( φ a ( 2 x , 2 x ) , t ) , N ′ ( φ a ( x , 2 x ) , t ) , N ′ ( φ a ( x , 3 x ) , t ) , N ′ ( φ a ( ( 2 k + 1 ) x , x ) , t ) , N ′ ( φ a ( ( 2 k - 1 ) x , x ) , t ) }$
(4.15)

for all x X and all t > 0. Finally, by using (4.14) and (4.15), we obtain that Similar to the proof Theorem 3.2, we have

$N f ( 4 x ) - 1 0 f ( 2 x ) + 1 6 f ( x ) , 9 k 2 + 4 k 2 ( k 2 - 1 ) t ≥ min { N ′ ( φ a ( x , x ) , t ) , N ′ ( φ a ( 2 x , x ) , t ) , N ′ ( φ a ( x , 2 x ) , t ) , N ′ ( φ a ( ( k + 1 ) x , x ) , t ) , N ′ ( φ a ( ( k - 1 ) x , x ) , t ) ) , N ′ ( φ a ( 2 x , 2 x ) , t ) , N ′ ( φ a ( x , 3 x ) , t ) , N ′ ( φ a ( ( 2 k + 1 ) x , x ) , t ) , N ′ ( φ a ( ( 2 k - 1 ) x , x ) , t ) }$
(4.16)

for all x X and all t > 0, where

for all x X and all t > 0. Thus, (4.16) means that

$N ( f ( 4 x ) - 1 0 f ( 2 x ) + 1 6 f ( x ) , t ) ≥ M a ( x , t )$
(4.17)

for all x X and all t > 0. Let g : XY be a mapping defined by g(x):= f(2x) - 8f(x) for all x X. From (4.17), we conclude that

$N ( g ( 2 x ) - 2 g ( x ) , t ) ≥ M a ( x , t )$
(4.18)

for all x X and all t > 0. So

$N g ( 2 x ) 2 - g ( x ) , t 2 ≥ M a ( x , t )$
(4.19)

for all x X and all t > 0. Then, by our assumption

$M a ( 2 x , t ) = M a x , t α$
(4.20)

for all x X and all t > 0. Replacing x by 2nx in (4.19) and using (4.20), we obtain

$N ( g ( 2 n + 1 x ) 2 n + 1 − g ( 2 n x ) 2 n , t 2 ( 2 n ) ) ≥ M a ( 2 n x , t ) = M a ( x , t α n )$
(4.21)

for all x X, t > 0 and n ≥ 0. Replacing t by αnt in (4.21), we see that

$N ( g ( 2 n + 1 x ) 2 n + 1 − g ( 2 n x ) 2 n , t α n 2 ( 2 n ) ) ≥ M a ( x , t )$
(4.22)

for all x X, t > 0 and n > 0. It follows from $g ( 2 n x ) 2 n − g ( x ) = ∑ j = 0 n − 1 ( g ( 2 j + 1 x ) 2 j + 1 − g ( 2 j x ) 2 j )$ and (4.22) that

$N ( g ( 2 n x ) 2 n − g ( x ) , ∑ j = 0 n − 1 α j t 2 ( 2 ) j ) ≥ min ∪ j = 0 n − 1 { N ( g ( 2 j + 1 x ) 2 j + 1 − g ( 2 j x ) 2 j , α j t 2 ( 2 ) j ) } ≥ M a ( x , t )$
(4.23)

for all x X, t > 0 and n > 0. Replacing x by 2mx in (4.23), we observe that

$N ( g ( 2 n + m x ) 2 n + m − g ( 2 m x ) 2 m , ∑ j = 0 n − 1 α j t 2 ( 2 ) j + m ) ≥ M a ( 2 m x , t ) = M a ( x , t α m )$

for all x X, all t > 0 and all m ≥ 0, n > 0. So

$N ( g ( 2 n + m x ) 2 n + m − g ( 2 m x ) 2 m , ∑ j = m n + m − 1 α j t 2 ( 2 ) j ) ≥ M a ( x , t )$

for all x X, all t > 0 and all m ≥ 0, n > 0. Hence

$N ( g ( 2 n + m x ) 2 n + m − g ( 2 m x ) 2 m , t ) ≥ M a ( x , t ∑ j = m n + m − 1 α j 2 ( 2 ) j )$
(4.24)

for all x X, all t > 0 and all m ≥ 0, n > 0. Since 0 < α < 2 and $∑ n = 0 ∞ α 2 n <∞$, the Cauchy criterion for convergence and (N5) imply that ${ g ( 2 n x ) 2 n }$ is a Cauchy sequence in (Y, N) to some point A(x) Y. So one can define the mapping A : XY by

$A ( x ) = N − lim n → ∞ g ( 2 n x ) 2 n )$
(4.25)

for all x X. Fix x X and put m = 0 in (4.24) to obtain

$N ( g ( 2 n x ) 2 n − g ( x ) , t ) ≥ M a ( x , t ∑ j = 0 n − 1 α j 2 ( 2 ) j )$

for all x X, t > 0 and n > 0. From which we obtain

$N ( A ( x ) − g ( x ) , t ) ≥ min { N ( A ( x ) − g ( 2 n x ) 2 n , t 2 ) , N ( g ( 2 n x ) 2 n − g ( x ) , t 2 ) } ≥ M a ( x , t ∑ j = 0 n − 1 α j 2 j )$
(4.26)

for n large enough. Taking the limit as n → ∞ in (4.26), we obtain

$N ( A ( x ) - g ( x ) , t ) ≥ M a x , t ( 2 - α ) 2$
(4.27)

for all x X and all t > 0. It follows from (4.21) and (4.25) that

$N ( A ( 2 x ) 2 − A ( x ) , t ) ≥ min { N ( A ( 2 x ) 2 − g ( 2 n + 1 x ) 2 n + 1 ) , t 3 ) , N ( g ( 2 n x ) 2 n − A ( x ) , t 3 ) , N ( g ( 2 n + 1 x ) 2 n + 1 − g ( 2 n x ) 2 n ) , t 3 ) = M a ( x , 2 ( 2 ) n t 3 α n )$

for all x X and all t > 0. Therefore,

$A ( 2 x ) = 2 A ( x )$
(4.28)

for all x X. Replacing x, y by 2nx, 2ny in (4.2), respectively, we obtain

$N ( 1 2 n D g ( 2 n x ,2 n y ) , t ) = N ( D f ( 2 n + 1 x ,2 n + 1 y ) − 8 D f ( 2 n x ,2 n y ) , 2 n t ) = min { N ( D f ( 2 n + 1 x ,2 n + 1 y ) , 2 n t 2 ) , N ( D f ( 2 n x ,2 n y ) , 2 n t 16 ) } ≥ m i n { N ′ ( φ a ( 2 n + 1 x ,2 n + 1 y ) , 2 n t 2 ) , N ′ ( φ a ( 2 n x ,2 n y ) , 2 n t 16 ) } = min { N ′ ( φ a ( x , y ) , 2 n t 2 α n + 1 ) , N ′ ( φ a ( x , y ) , 2 n t 16 α n ) }$

which tends to 1 as n → ∞ for all x, y X and all t > 0. So we see that A satisfies (1.5). Thus, by Lemma 4.1, the mapping x A(2x) - 8A(x) is additive. So (4.28) implies that the mapping A is additive.

The rest of the proof is similar to the proof of Theorem 3.2 and we omit the details.   □

Remark 4.3. Let 0 < α < 2. Suppose that the function t N(f(2x) - 8f(x) - A(x), .) from (0, ∞) into [0, 1] is right continuous. Then, we obtain a better fuzzy approximation than (4.27).

Corollary 4.4. Let X be a normed space and Y be a Banach space. Let ε, λ be non-negative real numbers such that λ ≠ 1. Suppose that an odd mapping f : XY satisfies the inequality (3.18) for all x, y X. Then, the limit

$A ( x ) = lim n → ∞ 1 2 ℓ n ( f ( 2 ℓ n + 1 x ) - 8 f ( 2 ℓ n x ) )$

exists for all x X and A : XY is a unique additive mapping satisfying

$f ( 2 x ) - 8 f ( x ) - A ( x ) ≤ ( | 2 k + ℓ | λ + 1 ) ( 9 k 2 + 4 ) ε ∥ x ∥ λ ℓ k 2 ( k 2 - 1 ) ( 1 - 2 λ - 1 )$
(4.29)

for all x X, where λℓ < .

Proof. The proof is similar to the proof of Corollary 3.4 and the result follows from Theorem 4.2.   □

Theorem 4.5. Denote N1 the fuzzy norm obtained as Corollary 3.4 on R. Let for all x X, the functions r f(2rx) - 8f(rx) (from (R, N1) into (Y, N)) and r φ a (ι1rx, ι2ry) (from (R, N1) into (Z, N')) be fuzzy continuous, where ι1 {1, 2, (k + 1), (k - 1), (2k + 1), (2k - 1)} and ι2 {1, 2, 3}. Then, for all x X, the function r A(rx) is fuzzy continuous and A(rx) = rA(x) for all r R.

Proof. The proof is similar to the proof of Theorem 3.5 and the result follows from Theorem 4.2.   □

Lemma 4.6. [22, 24]. If an odd mapping f : V1V2 satisfies (1.5), then the mapping h : V1V2 defined by h(x) = f(2x) - 2f(x) is cubic.

Theorem 4.7. Let ℓ {-1, 1} be fixed and let φ c : X × XZ be a mapping such that

$φ c ( 2 x , 2 y ) = α φ c ( x , y )$
(4.30)

for all x, y X and for some positive real number α with αℓ < 8ℓ. Suppose that an odd mapping f : XY satisfies the inequality

$N ( D f ( x , y ) , t ) ≥ N ′ ( φ c ( x , y ) , t )$
(4.31)

for all x, y X and all t > 0. Then, the limit

$C ( x ) = N - lim n → ∞ 1 8 ℓ n ( f ( 2 ℓ n + 1 x ) - 2 f ( 2 ℓ n x ) )$

exists for all x X and C : XY is a unique cubic mapping satisfying

$N ( f ( 2 x ) - 2 f ( x ) - C ( x ) , t ) ≥ M c x , ℓ ( 8 - α ) 2 t$
(4.32)

for all x X and all t > 0, where

$M c ( x , t ) = min { N ′ ( φ c ( x , x ) , k 2 ( k 2 − 1 ) 9 k 2 + 4 t ) , N ′ ( φ c ( 2 x , x ) , k 2 ( k 2 − 1 ) 9 k 2 + 4 t ) , N ′ ( φ c ( x ,2 x ) , k 2 ( k 2 − 1 ) 9 k 2 + 4 t ) , N ′ ( φ c ( ( k + 1 ) x , x ) , k 2 ( k 2 − 1 ) 9 k 2 + 4 t ) , N ′ ( φ c ( ( k − 1 ) x , x ) , k 2 ( k 2 − 1 ) 9 k 2 + 4 t ) ) , N ′ ( φ c ( 2 x ,2 x ) , k 2 ( k 2 − 1 ) 9 k 2 + 4 t ) , N ′ ( φ c ( x ,3 x ) , k 2 ( k 2 − 1 ) 9 k 2 + 4 t ) , N ′ ( φ c ( ( 2 k + 1 ) x , x ) , k 2 ( k 2 − 1 ) 9 k 2 + 4 t ) , N ′ ( φ c ( ( 2 k − 1 ) x , x ) , k 2 ( k 2 − 1 ) 9 k 2 + 4 t ) } .$

Proof. Case (1): = 1. Similar to the proof of Theorem 4.2, we have

$N ( f ( 4 x ) - 1 0 f ( 2 x ) + 1 6 f ( x ) , t ) ≥ M c ( x , t )$

for for all x X and all t > 0, where M c (x, t) is defined as in above. Letting h : XY be a mapping defined by h(x):= f(2x) - 2f(x). Then, we conclude that

$N ( h ( 2 x ) - 8 h ( x ) , t ) ≥ M c ( x , t )$
(4.33)

for all x X and all t > 0. So

$N h ( 2 x ) 8 - h ( x ) , t 8 ≥ M c ( x , t )$
(4.34)

for all x X and all t > 0. Then, by our assumption

$M c ( 2 x , t ) = M c x , t α$
(4.35)

for all x X and all t > 0. Replacing x by 2nx in (4.34) and using (4.35), we obtain

$N ( h ( 2 n + 1 x ) 8 n + 1 − h ( 2 n x ) 8 n , t 8 ( 8 n ) ) ≥ M c ( 2 n x , t ) = M c ( x , t α n )$
(4.36)

for all x X, t > 0 and n ≥ 0. Replacing t by αnt in (4.36), we see that

$N ( h ( 2 n + 1 x ) 8 n + 1 − h ( 2 n x ) 8 n , t α n 8 ( 8 n ) ) ≥ M c ( x , t )$
(4.37)

for all x X, t > 0 and n > 0. It follows from $h ( 2 n x ) 8 n − h ( x ) = ∑ j = 0 n − 1 ( h ( 2 j + 1 x ) 8 j + 1 − h ( 2 j x ) 8 j )$ and (4.37) that

$N ( h ( 2 n x ) 8 n − h ( x ) , ∑ j = 0 n − 1 α j t 8 ( 8 ) j ) ≥ min ∪ j = 0 n − 1 { N ( h ( 2 j + 1 x ) 8 j + 1 − h ( 2 j x ) 8 j , α j t 8 ( 8 ) j ) } ≥ M c ( x , t )$
(4.38)

for all x X, t > 0 and n > 0. Replacing x by 2mx in (4.38), we observe that

$N ( h ( 2 n + m x ) 8 n + m − h ( 2 m x ) 8 m , ∑ j = 0 n − 1 α j t 8 ( 8 ) j + m ) ≥ M c ( 2 m x , t ) = M c ( x , t α m )$

for all x X, all t > 0 and all m ≥ 0, n > 0. So

$N ( h ( 2 n + m x ) 8 n + m − h ( 2 m x ) 8 m , ∑ j = m n + m − 1 α j t 8 ( 8 ) j ) ≥ M c ( x , t )$

for all x X, all t > 0 and all m ≥ 0, n > 0. Hence

$N ( h ( 2 n + m x ) 8 n + m − h ( 2 m x ) 8 m , t ) ≥ M c ( x , t ∑ j = m n + m − 1 α j 8 ( 8 ) j ) )$
(4.39)

for all x X, all t > 0 and all m ≥ 0, n > 0. Since 0 < α < 8 and $∑ n = 0 ∞ α 8 n <∞$, the Cauchy criterion for convergence and (N5) imply that ${ h ( 2 n x ) 8 n }$ is a Cauchy sequence in (Y, N) to some point C(x) Y. So one can define the mapping C : XY by

$C ( x ) = N − lim n → ∞ h ( 2 n x ) 8 n )$
(4.40)

for all x X. Fix x X and put m = 0 in (4.39) to obtain

$N ( h ( 2 n x ) 8 n − h ( x ) , t ) ≥ M c ( x , t ∑ j = 0 n − 1 α j 8 ( 8 ) j ) )$

for all x X, t > 0 and n > 0. From which we obtain

$N ( C ( x ) − h ( x ) , t ) ≥ min { N ( C ( x ) − h ( 2 n x ) 8 n , t 2 ) , N ( h ( 2 n x ) 8 n − h ( x ) , t 2 ) } ≥ M c ( x , t ∑ j = 0 n − 1 α j 4 ( 8 j ) ) )$
(4.41)

for n large enough. Taking the limit as n → ∞ in (4.41), we obtain

$N ( C ( x ) - h ( x ) , t ) ≥ M a x , t ( 8 - α ) 2$
(4.42)

for all x X and all t > 0. It follows from (4.36) and (4.40) that

$N ( C ( 2 x ) 8 − C ( x ) , t ) ≥ min { N ( C ( 2 x ) 8 − h ( 2 n + 1 x ) 8 n + 1 ) , t 3 ) , N ( h ( 2 n x ) 8 n − C ( x ) , t 3 ) , N ( h ( 2 n + 1 x ) 8 n + 1 − h ( 2 n x ) 8 n ) , t 3 ) = M c ( x , 8 ( 8 ) n t 3 α n )$

for all x X and all t > 0. Therefore,

$C ( 2 x ) = 8 C ( x )$
(4.43)

for all x X. Replacing x, y by 2nx, 2ny in (4.31), respectively, we obtain

$N ( 1 8 n D h ( 2 n x ,2 n y ) , t ) = N ( D f ( 2 n + 1 x ,2 n + 1 y ) − 2 D f ( 2 n x ,2 n y ) , 8 n t ) = min { N ( D f ( 2 n + 1 x ,2 n + 1 y ) , 8 n t 2 ) , N ( D f ( 2 n x ,2 n y ) , 8 n t 4 ) } ≥ min { N ′ ( φ c ( 2 n + 1 x ,2 n + 1 y ) , 8 n t 2 ) , N ′ ( φ c ( 2 n x ,2 n y ) , 8 n t 4 ) } = min { N ′ ( φ c ( x , y ) , 8 n t 2 α n + 1 ) , N ′ ( φ c ( x , y ) , 8 n t 4 α n ) }$

which tends to 1 as n → ∞ for all x, y X and all t > 0. So we see that C, satisfies (1.5). Thus, by Lemma 4.6, the mapping x C(2x) - 2C(x) is cubic. So (4.43) implies that the function C is cubic. The rest of the proof is similar to the proof of Theorem 3.2 and we omit the details.   □

Remark 4.8. Let 0 < α < 8. Suppose that the function t N(f(2x) - 2f(x) - C(x), .) from (0, ∞) into [0, 1] is right continuous. Then, we obtain a better fuzzy approximation than(4.42).

Corollary 4.9. Let X be a normed space and Y be a Banach space. Let ε, λ be non-negative real numbers such that λ ≠ 3. Suppose that an odd mapping f : XY satisfies the inequality (3.18) for all x, y X. Then, the limit

$C ( x ) = lim n → ∞ 1 8 ℓ n ( f ( 2 ℓ n + 1 x ) - 2 f ( 2 ℓ n x ) )$

exists for all x X and C : XY is a unique cubic mapping satisfying

$∥ f ( 2 x ) - 2 f ( x ) - C ( x ) | | ≤ ( | 2 k + ℓ | λ + 1 ) ( 9 k 2 + 4 ) ε ∥ x ∥ λ ℓ k 2 ( k 2 - 1 ) ( 4 - 2 λ - 1 )$
(4.44)

for all x X, where λℓ < 3.

Theorem 4.10. Denote N1 the fuzzy norm obtained as Corollary 3.4 on R. Let for all x X, the functions r f(2rx) - 2f(rx) (from (R, N1) into (Y, N)) and r φ q (ι1rx, ι2ry) (from (R, N1) into (Z, N')) be fuzzy continuous, where ι1 {1, 2, (k + 1), (k - 1), (2k + 1), (2k - 1)} and ι2 {1, 2, 3}. Then, for all x X, the function r C(rx) is fuzzy continuous and C(rx) = r3C(x) for all r R.

Proof. The proof is similar to the proof of Theorem 3.5 and the result follows from Theorem 4.7.   □

Theorem 4.11. Let φ : X × XZ be a mapping such that

$φ ( 2 x , 2 y ) = α φ ( x , y )$
(4.45)

for all x, y X and for some positive real number α. Suppose that an odd mapping f : XY satisfies the inequality

$N ( D f ( x , y ) , t ) ≥ N ′ ( φ ( x , y ) , t )$
(4.46)

for all x, y X and all t > 0. Then, there exist a unique cubic mapping C : XY and a unique additive mappingA : XY such that

$N ( f ( x ) - A ( x ) - C ( x ) , t ) ≥ min M ( x , 3 t ( 2 - α ) 2 ) , M ( x , 3 t ( 8 - α ) 2 ) , 0 < α < 2 min M ( x , 3 t ( α - 2 ) 2 ) , M ( x , 3 t ( 8 - α ) 2 ) , 2 < α < 8 min M ( x , 3 t ( α - 2 ) 2 ) , M ( x , 3 t ( α - 8 ) 2 ) , α > 8$
(4.47)

for all x X and all t > 0, where

$M ( x , t ) = min { N ′ ( φ ( x , x ) , k 2 ( k 2 − 1 ) 9 k 2 + 4 t ) , N ′ ( φ ( 2 x , x ) , k 2 ( k 2 − 1 ) 9 k 2 + 4 t ) , N ′ ( φ ( x ,2 x ) , k 2 ( k 2 − 1 ) 9 k 2 + 4 t ) , N ′ ( φ ( ( k + 1 ) x , x ) , k 2 ( k 2 − 1 ) 9 k 2 + 4 t ) , N ′ ( φ ( ( k − 1 ) x , x ) , k 2 ( k 2 − 1 ) 9 k 2 + 4 t ) ) , N ′ ( φ ( 2 x ,2 x ) , k 2 ( k 2 − 1 ) 9 k 2 + 4 t ) , N ′ ( φ ( x ,3 x ) , k 2 ( k 2 − 1 ) 9 k 2 + 4 t ) , N ′ ( φ ( ( 2 k + 1 ) x , x ) , k 2 ( k 2 − 1 ) 9 k 2 + 4 t ) , N ′ ( φ ( ( 2 k − 1 ) x , x ) , k 2 ( k 2 − 1 ) 9 k 2 + 4