Abstract
In this article, we adopt the fixed point and direct methods to prove the Hyers-Ulam stability for an additive-quadratic functional equation in fuzzy Banach spaces.
Mathematics Subject Classification 2010: 39B52, 46S40; 26E50.
Journal of Inequalities and Applications volume 2012, Article number: 64 (2012)
In this article, we adopt the fixed point and direct methods to prove the Hyers-Ulam stability for an additive-quadratic functional equation in fuzzy Banach spaces.
Mathematics Subject Classification 2010: 39B52, 46S40; 26E50.
A classical question in the theory of functional equations is the following: "When is it true that a function which approximately satisfies a functional equation must be close to an exact solution of the equation?". If the problem accepts a solution, we say that the equation is stable. The first stability problem concerning group homomorphisms was raised by Ulam [1]. In the next year, Hyers [2] gave a positive answer to the above question for additive groups under the assumption that the groups are Banach spaces. In 1978, Rassias [3] extended the theorem of Hyers by considering the unbounded Cauchy difference ∥f(x + y)- f(x) - f(y)∥ ≤ ε(∥x∥p+ ∥y∥p), (ε > 0, p ∈[0,1)). Furthermore, in 1994, a generalization of Rassias' theorem was obtained by Găvruta [4] by replacing the bound ϵ(∥x∥p+ ∥y∥p) by a general control function φ(x, y).
In 1983, a Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [5] for mappings f : X → Y, where X is a normed space and Y is a Banach space. In 1984, Cholewa [6] noticed that the theorem of Skof is still true if the relevant domain X is replaced by an Abelian group and, in 2002, Czerwik [7] proved the Hyers-Ulam stability of the quadratic functional equation.
Katsaras [8] defined a fuzzy norm on a vector space to construct a fuzzy vector topological structure on the space. Some mathematicians have defined fuzzy norms of a vector space from various points of view (see [9–17]).
In particular, Bag and Samanta [18], following Cheng and Mordeson [19], gave an idea of fuzzy norm in such a manner that the corresponding fuzzy metric is of Karmosil and Michalek type [20]. They established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy normed spaces [21].
In this article, we prove the Hyers-Ulam stability of the following additive-quadratic functional equation
where r is a positive real number, in fuzzy Banach spaces.
Lee and Jun [22] proved that an even (odd) mapping f : X → Y satisfies the functional Equation (1.1) if and only if the mapping f : X → Y is quadratic (additive). Moreover, they proved the Hyers-Ulam stability of the functional Equation (1.1) in non-Archimedean normed spaces. The stability problems of several functional equations have been extensively investigated by a number of authors, and there are many interesting results concerning this problem (see [23–33]).
Definition 1.1. [18] Let X be a real vector space. A function N : X × ℝ → [0,1] is called a fuzzy norm on X if for all x, y ∈ X and all s,t ∈ ℝ,
(N 1) N(x, t) = 0 for t ≤ 0;
(N 2) x = 0 if and only if N(x, t) = 1 for all t > 0;
(N 3) if c ≠ 0;
(N 4) N(x + y, c + t) ≥ min{N(x, s), N(y, t)};
(N 5) N(x,.) is a non-decreasing function of ℝ and limt→∞N(x, t) = 1;
(N 6) 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. [18] 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 limt→∞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 we denote it by N - limt→∞x n = x.
Definition 1.4. [18] Let (X, N) be a fuzzy normed vector space. A sequence {x n } in X is called Cauchy if for each ϵ > 0 and each t > 0 there exists an n0 ∈ ℕ such that for all n ≥ n0 and all p > 0, we have N(xn+p- x n , t) > 1 - ϵ.
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.
Example 1.5. Let N : X × ℝ → [0,1] be a fuzzy norm on ℝ defined by
Then (ℝ, N) is a fuzzy Banach space. Let {x n } be a Cauchy sequence in ℝ, δ > 0 and. Then there exist m ∈ ℕ such that for all n ≥ m and all p > 0, we have
So |xn+p-x n | < δ for all n ≥ m and all p > 0. Therefore {x n } is a Cauchy sequence in (ℝ, |·|). Let x n → x0 ∈ ℝ as n → ∞. Then limn→∞N (x n - x0, 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 x0 ∈ X, then the sequence {f(x n )} converges to f(x0). If f: X → Y is continuous at each x ∈ X, then f : X → Y is said to be continuous on X ([21]).
Definition 1.6. Let X be a set. A function d : X × X → [0, ∞] is called a generalized metric on X if d satisfies the following conditions:
d(x, y) = 0 if and only if x = y for all x, y ∈ X;
d(x, y) = d(y, x) for all x, y ∈ X;
d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z ∈ X.
Theorem 1.7. Let (X, d) be a complete generalized metric space and J : X → X be a strictly contractive mapping with Lipschitz constant L < 1. Then, for all x ∈ X, either
for all nonnegative integers n or there exists a positive integer n 0 such that
d(J n x, J n+ 1 x) < ∞ for all n 0 ≥ n 0;
the sequence {J n x} converges to a fixed point y* of J;
y* is the unique fixed point of J in the set ;
for all y ∈ Y.
In this section, using the fixed point method, we prove the Hyers-Ulam stability of the additive-quadratic functional equation (1.1) in fuzzy Banach spaces.
Remark 2.1. If f be an odd mapping satisfying (1.1) and r = 2, then an additive mapping f is nonzero in general. But if r is a rational number and r ≠ 2 in (1.1), then f ≡ 0. So, to avoid the trivial case, let r = 2 or r is an positive irrational real number.
Theorem 2.2. Let φ: X3 → [0, ∞) be a function such that there exists an α < 1 with
for all x, y, z ∈ X. Let f : X → Y be an odd mapping satisfying
for all x, y, z ∈ X and all t > 0. Then
exists for each x ∈ X and defines a unique additive mapping A:X → Y such that
Proof. Note that f (0) = 0 and f (-x) = -f(x) for all x ∈ X since f is an odd function. Putting y = z = 0 in (2.2) and replacing x by 2x, we get
for all x ∈ X and all t > 0. Putting y = x and z = 0 in (2.2), we have
for all x ∈ X and all t > 0. By (2.4), (2.5), (N 3) and (N 4), we get
for all x ∈ X and all t > 0. Consider the set S := {h : X → Y} and introduce the generalized metric on S:
where, as usual, inf ϕ = + ∞. It is easy to show that (S, d) is complete (see [11]). Now we consider the linear mapping J : (S, d) → (S, d) such that
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 (2.6) that
By Theorem 1.7, there exists a mapping A : X → Y satisfying the following:
A is a fixed point of J, i.e.,
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 (2.7) such that there exists a μ ∈ (0, ∞) satisfying
for all x ∈ X;
d(J n f, A) → 0 as n → ∞. This implies the equality for all x ∈ X;
, which implies the inequality . This implies that the inequalities (2.3) holds.
It follows from (2.1) and (2.2) that
for all x, y, z ∈ X, all t > 0 and all n ∈ ℕ. So
for all x, y, z ∈ X, all t > 0 and all n ∈ ℕ. Since
for all x, y, z ∈ X and all t > 0, we obtain that
for all x, y, z ∈ X and all t > 0. Hence the mapping A : X → Y is additive, as desired.
Corollary 2.3. Let θ ≥ 0 and 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
for all x, y, z ∈ X and all t > 0. Then the limitexists 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 2.2 by taking φ(x, y, z) := θ(∥x∥r+ ∥y∥r+ ∥z∥r) for all x, y, z ∈ X. Then we can choose α = 2r-1and we get the desired result.
Theorem 2.4. Let φ : X3 → [0, ∞) be a function such that there exists an α < 1 with
for all x, y, z ∈ X. Let f : X → Y be an odd mapping satisfying (2.2). Then the limitexists for each x ∈ X and defines a unique additive mapping A : X → Y such that
Proof. Let (S, d) be the generalized metric space defined as in the proof of Theorem 2.2. Consider the linear mapping J : S → S such that
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 (2.6) that
for all x ∈ X and t > 0. Therefore
So . By Theorem 1.7, there exists a mapping A : X → Y satisfying the following:
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 Ω = {h ∈ S : d(g, h) < ∞}. This implies that A is a unique mapping satisfying (2.12) such that there exists μ ∈ (0, ∞) satisfying
for all x ∈ X and t > 0.
d(J n f, A) → 0 as n → ∞. This implies the equality for all x ∈ X.
with f ∈ Ω, which implies the inequality . This implies that the inequality (2.10) holds.
The rest of the proof is similar to that of the proof of Theorem 2.2.
Corollary 2.5. 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 (2.8). Then, the limitexists 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 2.4 by taking φ(x, y, z) := θ(∥x∥r+ ∥y∥r+ ∥z∥r) for all x, y, z ∈ X. Then we can choose α = 21-rand we get the desired result.
Theorem 2.6. Let φ : X3 → [0, ∞) be a function such that there exists an α < 1 with
for all x, y, z ∈ X. Let f : X → Y be an even mapping with f(0) = 0 and satisfying (2.2). Thenexists for each x ∈ X and defines a unique quadratic mapping Q : X → Y such that
Proof. Putting y = x and z = 0 in (2.2), we get
for all x ∈ X and all t > 0. Putting y = z = 0 and replacing x by 2x in (2.2), we have
for all x ∈ X and all t > 0. By (2.15), (2.16), (N 3) and (N 4), we get
for all x ∈ X and all t > 0. So
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 [11]). Now we consider the linear mapping J : (S*, d) → (S*, d) such that for all x ∈ X. Proceeding as in the proof of Theorem 2.2, we obtain that 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 (2.17) that
By Theorem 1.7, there exists a mapping Q : X → Y satisfying the following:
Q is a fixed point of J, i.e.,
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 (2.18) such that there exists a μ ∈ (0, ∞) satisfying for all x ∈ X;
d(J n f, Q) → 0 as n → ∞. This implies the equality for all x ∈ X;
, which implies the inequality . This implies that the inequalities (2.14) holds.
The rest of the proof is similar to the proof of Theorem 2.2.
Corollary 2.7. Let θ ≥ 0 and 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 satisfying (2.8) and f(0) = 0. Then the limitexists 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 2.6 by taking φ(x, y, z) := θ(∥x∥r+ ∥y∥r+ ∥z∥r) for all x, y, z ∈ X. Then we can choose α = 4r-1and we get the desired result.
Theorem 2.8. Let φ : X3 → [0, ∞) be a function such that there exists an α < 1 with
for all x, y, z ∈ X. Let f : X → Y be an even mapping satisfying (2.2) and f(0) = 0. Then the limitexists for each x ∈ X and defines a unique quadratic mapping Q : X → Y such that
Proof. Let (S*, d) be the generalized metric space defined as in the proof of Theorem 2.6.
It follows from (2.17) that
for all x ∈ X and t > 0. So
The rest of the proof is similar to the proof of Theorems 2.2 and 2.4.
Corollary 2.9. Let θ ≥ 0 and 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 satisfying f(0) = 0 and (2.8). Thenexists 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 2.8 by taking φ(x, y, z) := θ(∥x∥r+ ∥y∥r+ ∥z∥r) for all x, y, z ∈ X. Then we can choose α = 41-rand we get the desired result.
Let f : X → Y be a mapping satisfying f(0) = 0 and (1.1). Let and . Then f e is an even mapping satisfying (1.1) and f o is an odd mapping satisfying (1.1) such that f(x) = f e (x) + f o (x). So we obtain the following.
Theorem 2.10. Let φ : X3 → [0, ∞) be a function such that there exists an α < 1 with
for all x, y, z ∈ X. Let f : X → Y be a mapping satisfying (2.2) and f(0) = 0. Then there exist a unique additive mapping A: X → Y and a unique quadratic mapping Q : X → Y such that
Corollary 2.11. 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 a mapping with f(0) = 0 and satisfying (2.8). Then there exist a unique additive mapping A : X → Y and a unique quadratic mapping Q : X → Y such that
for all x ∈ X and all t > 0.
Proof. The proof follows from Theorem 2.10 by taking φ(x, y,z) := θ(∥x∥r+ ∥y∥r+ ∥z∥r) for all x, y, z ∈ X. Then we can choose α = 21-rand we get the desired result.
In this section, using direct method, we prove the Hyers-Ulam stability of functional equation (1.1) 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 spaces. 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 satisfying the inequality
for all x, y, z ∈ X, t > 0 and that φ : X3 → Z is a mapping for which there is a constant r ∈ ℝ satisfyingsuch that
for all x, y, z ∈ X and all t > 0. Then there exists a unique additive mapping A : X → Y satisfying (1.1) and the inequality
for all x ∈ X and all t > 0.
Proof. It follows from (3.2) that
for all x, y, z ∈ X and all t > 0. Putting y = z = 0 in (3.1) and replacing x by 2x, we get
for all x ∈ X and all t > 0. Putting y = x and z = 0 in (3.1), we have
for all x ∈ X and all t > 0. By (3.4), (3.5), (N 3) and (N 4), we get
for all x ∈ X and all t > 0. Replacing x by in (3.6), we have
Replacing x by in (3.7), we have
for all x ∈ X, all t > 0 and any integer j ≥ 0. So
which yields
for all x ∈ X, t > 0 and any integers n > 0, p ≥ 0. So
for all x ∈ X, t > 0 and any integers n > 0, p ≥ 0. Hence one obtains
for all x ∈ X, t > 0 and any integers n > 0, p ≥ 0. Since the series is a convergent series, we see by taking the limit p → ∞ in the last inequality that the sequence 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 is well-defined for all x ∈ X. It 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
for sufficiently large n and for all x ∈ X, t > 0 and ϵ with 0 < ϵ < 1. Since ϵ is arbitrary and N' is left continuous, we obtain
for all x ∈ X and t > 0. It follows from (3.1) that
for all x, y, z ∈ X, t > 0. So
for all x, y, z ∈ X and all t > 0. Therefore, we obtain in view of (3.11)
which implies
for all x, y, z ∈ X. Thus A : X → Y is a mapping satisfying the Equation (1.1) and the inequality (3.3). Thus the mapping A : X → Y is additive, as desired.
To prove the uniqueness, assume that there is another mapping L : X → Y which satisfies the inequality (3.3). Since L(2nx) = 2nL(x) for all x ∈ X, we have
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 spaces and (ℝ, N') a fuzzy Banach space. Assume that there exist real numbers θ ≥ 0 and p > 1 such that an odd mapping f : X → Y satisfies the inequality
for all x, y, z ∈ X and t > 0. Then there is a unique additive mapping A : X → Y satisfying (1.1) and the inequality
Proof. Let φ(x, y, z) := θ (∥x∥p+ ∥y∥p+ ∥z∥p) and |r| = 2-p. Applying Theorem 3.1, we get the desired result.
Theorem 3.3. Assume that an odd mapping f : X → Y satisfies the inequality (3.1) and that φ:X3 → Z is a mapping for which there is a constant r ∈ ℝ satisfying 0 < |r| < 2 such that
for all x, y, z ∈ X and all t > 0. Then there exists a unique additive mapping A : X → Y satisfying (1.1) and the inequality
for all x ∈ X and all t > 0.
Proof. It follows from (3.6) that
for all x ∈ X and all t > 0. Replacing x by 2nx in (3.15), we obtain
So
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 spaces and that (ℝ, N') a fuzzy Banach space. Assume that there exist real numbers θ ≥ 0 and 0 < p < 1 such that an odd mapping f : X → Y satisfies (3.12). Then there is a unique additive mapping A : X → Y satisfying (1.1) and the inequality
Proof. Let φ(x, y, z) := θ(∥x∥p+ ∥y∥p+ ∥z∥p) and |r| = 4p. Applying Theorem 3.3, we get the desired results.
Theorem 3.5. Assume that a mapping f : X → Y is an even mapping satisfies (3.1) and f(0) = 0 and that φ : X3 → Z is a mapping for which there is a constant r ∈ ℝ satisfying 0 < |r| < 4 such that
for all x, y, z ∈ X and all t > 0. Then there exists a unique quadratic mapping Q : X → Y satisfying (1.1) and the inequality
for all x ∈ X and all t > 0.
Proof. Putting y = x and z = 0 in (3.1), we get
for all x ∈ X and all t > 0. Putting y = z = 0 and replacing x by 2 x in (3.1), we have
for all x ∈ X and all t > 0. By (3.20), (3.21), (N 3) and (N 4), we get
for all x ∈ X and all t > 0. Replacing x by 2nx in (3.22), 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 3.5.
Corollary 3.6. Let X be a normed spaces and (ℝ, N') a fuzzy Banach space. Assume that there exist real numbers θ ≥ 0 and 0 < p < 1 such that an even mapping f : X → Y satisfies (3.12). Then there is a unique quadratic mapping Q: X → Y satisfying (1.1) and the inequality
Proof. Let φ(x, y, z) := θ(∥x∥p+ ∥y∥p+ ∥z∥p) and |r| = 4p. Applying Theorem 3.1, we get the desired result.
Theorem 3.7. Assume that an even mapping f : X → Y satisfies the inequality (3.1) and f(0) = 0 and that φ : X3 → Z is a mapping for which there is a constant r ∈ ℝ satisfyingsuch that
for all x, y, z ∈ X and all t > 0. Then there exists a unique quadratic mapping Q : X → Y satisfying (1.1) and the inequality
for all x ∈ X and all t > 0.
Proof. It follows from (3.22) that
for all x ∈ X and all t > 0. Replacing x by in (3.25), we have
for all x ∈ X, all t > 0 and any integer j ≥ 0. By the same technique as in Theorem 3.1, we find that
The rest of the proof is similar to the proof of Theorem 3.1.
Corollary 3.8. Let X be a normed spaces and (ℝ, N') a fuzzy Banach space. Assume that there exists real numbers θ ≥ 0 and p > 1 such that an even mapping f : X → Y satisfies (3.12) and f(0) = 0. Then there is a unique quadratic mapping Q : X → Y satisfying (1.1) and the inequality
Proof. Let φ(x, y, z) := θ (∥x∥p+ ∥y∥p+ ∥z∥p) and |r| = 4-p. Applying Theorem 3.7, we get the desired results.
Let f : X → Y be a mapping satisfying f(0) = 0 and (1.1). Let and . Then f e is an even mapping satisfying (1.1) and f o is an odd mapping satisfying (1.1) such that f(x) = f e (x) + f o (x). So we obtain the following.
Theorem 3.9. Assume that a mapping f : X → Y is a mapping satisfying (3.1) and f(0) = 0 and that φ : X3 → Z is a mapping for which there is a constant r ∈ ℝ satisfyingsuch that
for all x, y, z ∈ X and all t > 0. Then there exist a unique additive mapping A : X → Y and a unique quadratic mapping Q : X → Y satisfying (1.1) and the inequality
for all x ∈ X and all t > 0.
Proof. It follows from Theorems 3.1 and 3.7 that
Corollary 3.10. Let X be a normed spaces and that (ℝ, N') a fuzzy Banach space. Assume that there exists real number θ ≥ 0 and 0 < p < 1 such that a mapping f : X → Y satisfies (3.12) and f(0) = 0. Then there are unique quadratic and additive mappings Q : X → Y and A : X → Y (respectively) satisfying (1.1) and the inequality
Proof. Let φ(x, y, z) := θ (∥x∥p+ ∥y∥p+ ∥z∥p) and |r| = 4p. Applying Theorem 3.7, we get the desired results.
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