Solution and Hyers-Ulam-Rassias Stability of Generalized Mixed Type Additive-Quadratic Functional Equations in Fuzzy Banach Spaces

and Applied Analysis 3 with f 0 0 in a non-Archimedean space. It is easy to see that the function f x ax bx2 is a solution of the functional equation 1.8 , which explains why it is called additive-quadratic functional equation. For more detailed definitions of mixed type functional equations, we can refer to 26–47 . Definition 1.1 see 48 . Let X be a real vector space. A functionN : X × R → 0, 1 is called a fuzzy norm on X if for all x, y ∈ X and all s, t ∈ R, (N1) N x, t 0 for t ≤ 0; (N2) x 0 if and only ifN x, t 1 for all t > 0; (N3) N cx, 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 a nondecreasing function of R and limt→∞N x, t 1; (N6) for x / 0,N x, · is continuous on R. The pair X,N is called a fuzzy normed vector space. Example 1.2. Let X, ‖ · ‖ be a normed linear space and α, β > 0. Then


Introduction and Preliminaries
The stability problem of functional equations was originated from a question of Ulam 1 in 1940, concerning the stability of group homomorphisms. Let G 1 , · be a group and let G 2 , * , d be a metric group with the metric d ·, · . Given > 0, does there exist a δ > 0, such that if a mapping h : G 1 → G 2 satisfies the inequality d h x · y , h x * h y < δ for all x, y ∈ G 1 , then there exists a homomorphism H : G 1 → G 2 with d h x , H x < for all x ∈ G 1 ? In other words, under what condition does there exist 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. In 1941, 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 with f 0 0 in a non-Archimedean space. It is easy to see that the function f x ax bx 2 is a solution of the functional equation 1.8 , which explains why it is called additive-quadratic functional equation. For more detailed definitions of mixed type functional equations, we can refer to 26-47 . (N5) N x, · is a nondecreasing function of R and lim t → ∞ N x, t 1; (N6) for x / 0, N x, · is continuous on R.
The pair X, N is called a fuzzy normed vector space.
Example 1.2. Let X, · be a normed linear space and α, β > 0. Then is a fuzzy norm on X. Definition 1.3. Let X, N be a fuzzy normed vector space. A sequence {x n } in X is said to be convergent or converge if there exists an x ∈ X such that lim n → ∞ N x n − x, t 1 for all t > 0. In this case, x is called the limit of the sequence {x n } in X and one denotes it by N − lim n → ∞ x n x. Definition 1.4. Let X, N be a fuzzy normed vector space. A sequence {x n } in X is called Cauchy if for each > 0 and each t > 0 there exists an n 0 ∈ N such that for all n ≥ n 0 and all p > 0, one has N x n p − x n , t > 1 − .
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.

4 Abstract and Applied Analysis
The R, N is a fuzzy Banach space. Let {x n } be a Cauchy sequence in R, δ > 0, and δ/ 1 δ . Then there exist m ∈ N such that for all n ≥ m and all p > 0, one has So |x n p − x n | < δ for all n ≥ m and all p > 0. Therefore {x n } is a Cauchy sequence in R, | · | . Let x n → x 0 ∈ R as n → ∞. Then lim n → ∞ N x n − x 0 , t 1 for all t > 0.
We say that a mapping f : X → Y between fuzzy normed vector spaces X and Y is continuous at a point x ∈ X if for each sequence {x n } converging to x 0 ∈ X, the sequence {f x n } converges to f x 0 . If f : X → Y is continuous at each x ∈ X, then f : X → Y is said to be continuous on X 49 . We have the following theorem from 42 , which investigates the solution of 1.8 .
where the function Q is symmetric biadditive and A is additive.

A Fixed Point Method
Using the fixed point methods, we prove the Hyers-Ulam stability of the additive-quadratic functional equation 1.8 in fuzzy Banach spaces. Throughout this paper, assume that X is a vector space and that Y, N is a fuzzy Banach space. for all x, y ∈ X. Let f : X → Y be an odd function satisfying f 0 0 and for all x, y ∈ X and all t > 0. Then A x : N − lim n → ∞ f k n x /k n exists for all x ∈ X and defines a unique additive mapping A : X → Y such that for all x ∈ X and t > 0.
Proof. Note that f 0 0 and f −x −f x for all x ∈ X since f is an odd function. Putting x 0 in 2.2 , we get for all y ∈ X and all t > 0. Replacing y by x in 2.4 , we have for all x ∈ X and all t > 0. Consider the set S : {h : X → Y ; h 0 0} and introduce the generalized metric on S: where, as usual, inf φ ∞. It is easy to show that S, d is complete see 50 . We consider the mapping J : S, d → S, d as follows: for all x ∈ X. Let g, h ∈ S be given such that d g, h β. Then for all x ∈ X and all t > 0. Hence for all x ∈ X and all t > 0. So d g, h β implies that d Jg, Jh ≤ αβ. This means that d Jg, Jh ≤ αd g, h for all g, h ∈ S. It follows from 2.5 that By Theorem 1.7, there exists a mapping A : X → Y satisfying the following. 1 A is a fixed point of J, that is, for all x ∈ X. The mapping A is a unique fixed point of J in the set M {g ∈ S : d h, g < ∞}.
This implies that A is a unique mapping satisfying 2.11 such that there exists a μ ∈ 0, ∞ satisfying This implies that the inequality 2.3 holds.

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It follows from 2.1 and 2.2 that 2.14 for all x, y ∈ X, all t > 0, and all n ∈ N. So for all x, y ∈ X, all t > 0, and all n ∈ N. Since lim n → ∞ |k| n t/ |k| n t |k| n α n ϕ x, y 1 for all x, y ∈ X and all t > 0, we obtain that

2.16
for all x, y, z ∈ X and all t > 0. Hence the mapping A : X → Y is additive, as desired.

Corollary 2.2.
Let θ ≥ 0 and let r be a real positive number with r < 1. Let X be a normed vector space with norm · . Let f : X → Y be an odd mapping satisfying

2.17
for all x, y ∈ X and all t > 0. Then the limit A x : N − lim n → ∞ f k n x /k n exists for each x ∈ X and defines a unique additive mapping A : for all x ∈ X and all t > 0.

Abstract and Applied Analysis
Proof. The proof follows from Theorem 2.1 by taking ϕ x, y : θ x r y r for all x, y ∈ X.
Then we can choose α |k| r−1 and we get the desired result.
for all x, y ∈ X. Let f : X → Y be an odd mapping satisfying f 0 0 and 2.2 . Then the limit A x : N − lim n → ∞ k n f x/k n exists for all x ∈ X and defines a unique additive mapping for all x ∈ X and all t > 0.
Proof. Let S, d be the generalized metric space defined as in the proof of Theorem 2.1. Consider the mapping J : S → S by for all g ∈ S. Let g, h ∈ S be given such that d g, h β. Then for all x ∈ X and all t > 0. Hence for all x ∈ X and all t > 0. So d g, h β implies that d Jg, Jh ≤ αβ. This means that d Jg, Jh ≤ αd g, h for all g, h ∈ S. It follows from 2.5 that Abstract and Applied Analysis 9 for all x ∈ X and all t > 0. Therefore So d f, Jf ≤ α. By Theorem 1.7, there exists a mapping A : X → Y satisfying the following. 1 A is a fixed point of J, that is, This implies that A is a unique mapping satisfying 2.26 such that there exists μ ∈ 0, ∞ satisfying for all x ∈ X and t > 0. .

2.28
This implies that the inequality 2.20 holds. The rest of proof is similar to the proof of Theorem 2.1.

Corollary 2.4.
Let θ ≥ 0 and let r be a real number with r > 1. Let X be a normed vector space with norm · . Let f : X → Y be an odd mapping satisfying 2.17 . Then A x : N − lim n → ∞ k n f x/k n exists for each x ∈ X and defines a unique additive mapping A : X → Y such that for all x ∈ X and all t > 0.
Proof. The proof follows from Theorem 2.3 by taking ϕ x, y : θ x r y r for all x, y ∈ X.
Then we can choose α |k| 1−r and we get the desired result. for all x, y ∈ X. Let f : X → Y be an even mapping with f 0 0 and satisfying 2.2 . Then Q x : N − lim n → ∞ f k n x /k 2n exists for all x ∈ X and defines a unique quadratic mapping for all x ∈ X and all t > 0.
Proof. Replacing x by kx in 2.2 , we get for all x, y ∈ X and all t > 0. Putting x 0 and replacing y by x in 2.32 , we have for all x ∈ X and all t > 0. By 2.33 , N3 , and N4 , we get for all x ∈ X and all t > 0. Consider the set S * : {h : X → Y ; h 0 0} and introduce the generalized metric on S * : where, as usual, inf φ ∞. It is easy to show that S * , d is complete see 50 . Now we consider the linear mapping J : S * , d → S * , d such that for all x ∈ X. The mapping Q is a unique fixed point of J in the set M {g ∈ S * : d h, g < ∞}.
This implies that Q is a unique mapping satisfying 2.38 such that there exists a μ ∈ 0, ∞ satisfying N f x − Q x , μt ≥ t/ t ϕ 0, x for all x ∈ X.
2 d J n f, Q → 0 as n → ∞. This implies the equality lim n → ∞ f k n x /k 2n Q x for all x ∈ X.
The rest of the proof is similar to the proof of Theorem 2.1.
Corollary 2.6. Let θ ≥ 0 and let r be a real positive number with r < 1. Let X be a normed vector space with norm · . Let f : X → Y be an even mapping with f 0 0 and satisfying 2.17 . Then the limit Q x : N − lim n → ∞ f k n x /k 2n exists for each x ∈ X and defines a unique quadratic mapping Q : X → Y such that for all x ∈ X and all t > 0.
Proof. The proof follows from Theorem 2.5 by taking ϕ x, y : θ x r y r for all x, y ∈ X. Then we can choose α k 2r−2 and we get the desired result.
Theorem 2.7. Let ϕ : X 2 → 0, ∞ be a function such that there exists an α < 1 with for all x, y ∈ X. Let f : X → Y be an even mapping with f 0 0 and satisfying 2.2 . Then the limit Q x : N − lim n → ∞ k 2n f x/k n exists for all x ∈ X and defines a unique quadratic mapping for all x ∈ X and t > 0.
Proof. Let S * , d be the generalized metric space defined as in the proof of Theorem 2.5. It follows from 2.34 that for all x ∈ X and t > 0. So The rest of the proof is similar to the proofs of Theorems 2.1 and 2.3.
Corollary 2.8. Let θ ≥ 0 and let r be a real number with r > 1. Let X be a normed vector space with norm · . Let f : X → Y be an even mapping with f 0 0 and satisfying 2.17 . Then Q x : N − lim n → ∞ k 2n f x/k n exists for each x ∈ X and defines a unique quadratic mapping

2.44
for all x ∈ X and all t > 0.
Proof. It follows from Theorem 2.7 by taking ϕ x, y : θ x r y r for all x, y ∈ X. Then we can choose α k 2−2r and we get the desired result.

Direct Method
In this section, using direct method, we prove the Hyers-Ulam stability of functional equation 1.8 in fuzzy Banach spaces. Throughout this section, we assume that X is a linear space, Y, N is a fuzzy Banach space, and Z, N is a fuzzy normed space. Moreover, we assume that N x, · is a left continuous function on R.
Theorem 3.1. Assume that a mapping f : X → Y is an odd mapping with f 0 0 satisfying the inequality for all x, y ∈ X, t > 0, and ϕ : X 2 → Z is a mapping for which there is a constant r ∈ R satisfying 0 < |r| < 1/|k| such that for all x, y ∈ X and all t > 0. Then there exists a unique additive mapping A : X → Y satisfying 1.8 and the inequality for all x ∈ X and all t > 0.
Proof. It follows from 3.2 that for all x, y ∈ X and all t > 0. Putting x 0 in 3.1 and then replacing y by x/k, we get for all x ∈ X and all t > 0. Replacing x by x/k j in 3.5 , we have for all x ∈ X, all t > 0, and all integer j ≥ 0. So

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Abstract and Applied Analysis for all x ∈ X, t > 0, and all integers n > 0, p ≥ 0. So for all x ∈ X, t > 0, and any integers n > 0, p ≥ 0. Hence one can obtain for all x ∈ X, t > 0, and any integers n > 0, p ≥ 0. Since the series ∞ j 0 k j |r| j is a convergent series, we see by taking the limit p → ∞ in the last inequality that the sequence {k n f x/k n } is a Cauchy sequence in the fuzzy Banach space Y, N and so it converges in Y . Therefore a mapping A : X → Y defined by A x : N − lim n → ∞ k n f x/k n is well defined for all x ∈ X. This means that for all x ∈ X and all t > 0. In addition, it follows from 3.10 that for all x ∈ X and all t > 0. So

3.13
for sufficiently large n and for all x ∈ X, t > 0, and with 0 < < 1. Since ε is arbitrary and N is left continuous, we obtain To prove the uniqueness, let there be another mapping L : X → Y which satisfies the inequality 3.3 . Since L k n x k n L x for all x ∈ X, we have

3.18
for all t > 0. Therefore A x L x for all x ∈ X. This completes the proof.

Corollary 3.2.
Let X be a normed space and let R, N be a fuzzy Banach space. Assume that there exist real numbers θ ≥ 0 and p > 1 such that an odd mapping f : X → Y with f 0 0 satisfies the following inequality:

3.19
for all x, y ∈ X and t > 0. Then there is a unique additive mapping A : X → Y satisfying 1.8 and the inequality Proof. Let ϕ x, y : θ x p y p and |r| |k| −p . Applying Theorem 3.1, we get desired results. for all x, y ∈ X and all t > 0. Then there exists a unique additive mapping A : X → Y satisfying 1.8 and the following inequality: for all x ∈ X and all t > 0.
Proof. It follows from 3.5 that for all x ∈ X and all t > 0. Replacing x by k n x in 3.41 , we obtain 3.25 for all x ∈ X and all t > 0. Proceeding as in the proof of Theorem 3.1, we obtain that for all x ∈ X, all t > 0, and any integer n > 0. So The rest of the proof is similar to the proof of Theorem 3.1.

Corollary 3.4.
Let X be a normed space and let R, N be a fuzzy Banach space. Assume that there exist real numbers θ ≥ 0 and 0 < p < 1 such that an odd mapping f : X → Y with f 0 0 satisfies 3.19 . Then there exists a unique additive mapping A : X → Y satisfying 1.8 and the inequality Proof. Let ϕ x, y : θ x p y p and |r| |k| p . Applying Theorem 3.3, we get the desired results.
Theorem 3.5. Let f : X → Y be an even mapping with f 0 0 satisfying the inequality 3.1 and let ϕ : X 2 → Z be a mapping for which there exists a constant r ∈ R such that 0 < |r| < 1/k 2 and that for all x, y ∈ X and all t > 0. Then there exists a unique quadratic mapping Q : X → Y satisfying 1.8 and the inequality for all x ∈ X and all t > 0.
Proof. Replacing x by kx in 3.1 , we get for all x, y ∈ X and all t > 0. Putting x 0 and replacing y by x in 3.31 , we have for all x ∈ X and all t > 0. Replacing x by x/k in 3.32 , we find for all x ∈ X and all t > 0. Also, replacing x by x/k n in 3.33 , we obtain for all x ∈ X and all t > 0. Proceeding as in the proof of Theorem 3.1, we obtain that for all x ∈ X, all t > 0, and any integer n > 0. So The rest of the proof is similar to the proof of Theorem 3.1.
Corollary 3.6. Let X be a normed space and let R, N be a fuzzy Banach space. Assume that there exist real numbers θ ≥ 0 and p > 1 such that an even mapping f : X → Y with f 0 0 satisfies the inequality 3.19 . Then there exists a unique quadratic mapping Q : X → Y satisfying 1.8 and the inequality Proof. Let ϕ x, y : θ x p y p and |r| |k| −2p . Applying Theorem 3.5, we get the desired results.
Theorem 3.7. Assume that an even mapping f : X → Y with f 0 0 satisfies the inequality 3.1 and ϕ : X 2 → Z is a mapping for which there is a constant r ∈ R satisfying 0 < |r| < k 2 such that N ϕ x, y , |r|t ≥ N ϕ x k , y k , t , 3.39 for all x, y ∈ X and all t > 0. Then there exists a unique quadratic mapping Q : X → Y satisfying 1.8 and the following inequality for all x ∈ X and all t > 0.
Proof. It follows from 3.32 that for all x ∈ X and all t > 0. Replacing x by k n x in 3.41 , we obtain N f k n 1 x k 2n 2 − f k n x k 2n , t 2|k| 2n 1 ≥ N ϕ 0, k n x , t ≥ N ϕ 0, x , t |r| n , 3.42 for all x ∈ X and all t > 0. So for all x ∈ X and all t > 0. So The rest of the proof is similar to the proof of Theorem 3.1.
Corollary 3.8. Let X be a normed space and let R, N be a fuzzy Banach space. Assume that there exist real numbers θ ≥ 0 and 0 < p < 1 such that an even mapping f : X → Y with f 0 0 satisfies 3.19 . Then there is a unique quadratic mapping Q : X → Y satisfying 1.8 and the inequality for all x ∈ X, all t > 0.
Proof. Let ϕ x, y : θ x p y p and |r| k 2p . Applying Theorem 3.7, we get the desired results.