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Fuzzy Hyers-Ulam stability of an additive functional equation
Journal of Inequalities and Applications volume 2011, Article number: 140 (2011)
Abstract
In this paper, using the fixed point and direct methods, we prove the Hyers-Ulam stability of the following additive functional equation
in fuzzy normed spaces.
Mathematics Subject Classification (2010): 39B22; 39B52; 39B82; 46S10; 47S10; 46S40.
1. Introduction
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 1940. 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] proved a generalization of the Hyers' theorem for additive mappings.
Theorem 1.1. (Th.M. Rassias) Let f : X → Y be a mapping from a normed vector space X into a Banach space Y subject to the inequality
for all x, y ∈ X, where ε and p are constants with ε > 0 and 0 ≤ p < 1. Then the limit
exists for all x ∈ X and L : X → Y is the unique additive mapping which satisfies
for all x ∈ X. Also, if for each x ∈ X, the function f(tx) is continuous in t ∈ ℝ, then L is ℝ-linear.
Furthermore, in 1994, a generalization of Rassias' theorem was obtained by Gǎvruta [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. The reader is referred to ([8–20]) and references therein for detailed information on stability of functional equations.
Katsaras [21] defined a fuzzy norm on a vector space to construct a fuzzy vector topological structure on the space. Some mathematicians have defined fuzzy norms on a vector space from various points of view (see [22, 23]). In particular, Bag and Samanta [24], following Cheng and Mordeson [25], gave an idea of fuzzy norm in such a manner that the corresponding fuzzy metric is of Karmosil and Michalek type [26]. They established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy normed spaces [27].
Definition 1.2. 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 1.3. Let (X, ||.||) be a normed linear space and α, β > 0. Then
is a fuzzy norm on X.
Definition 1.4. 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 1.5. 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 ≥ n0and 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.
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.
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:
(a) d(x, y) = 0 if and only if x = y for all x, y ∈ X;
(b) d(x, y) = d(y, x) for all x, y ∈ X;
(c) d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z ∈ X.
Theorem 1.7. ([28, 29]) 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 d(Jnx, Jn+1x) = ∞ for all nonnegative integers n or there exists a positive integer n0such that
(a) d(Jnx, Jn+1x) < ∞ for all n0 ≥ n0;
(b) the sequence {Jnx} converges to a fixed point y* of J;
(c) y* is the unique fixed point of J in the set;
(d)for all y ∈ Y.
2. Fuzzy stability of the functional Eq. (0.1)
Throughout this section, using the fixed point and direct methods, we prove the Hyers-Ulam stability of functional Eq. (0.1) in fuzzy normed spaces.
2.1. Fixed point alternative approach
Throughout this subsection, using the fixed point alternative approach, we prove the Hyers-Ulam stability of functional Eq. (0.1) in fuzzy Banach spaces.
In this subsection, assume that X is a vector space and that (Y, N) is a fuzzy Banach space.
Theorem 2.1. Let φ : X3 → [0, ∞) be a function such that there exists an L < 1 with
for all x, y, z ∈ X. Let f : X → Y be a mapping satisfying
for all x, y, z ∈ X and all t > 0. Then the limit
exists for each x ∈ X and defines a unique additive mapping A : X → Y such that
Proof. Putting y = 2x and z = x in (2.1) and replacing x by , we have
for all x ∈ X and t > 0. Consider the set
and the generalized metric d in S defined by
where inf ∅ = +∞. It is easy to show that (S, d) is complete (see [30, Lemma 2.1]). Now, we consider a linear mapping J : S → S such that
for all x ∈ X. Let g, h ∈ S be such that d(g, h) = ε. Then
for all x ∈ X and t > 0. Hence,
for all x ∈ X and t > 0. Thus, d(g, h) = ε implies that d(Jg, Jh) ≤ Lε. This means that
for all g, h ∈ S. It follows from (2.3) that
Therefore,
This means 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,
(2.6)
for all x ∈ X. The mapping A is a unique fixed point of J in the set
This implies that A is a unique mapping satisfying (2.6) such that there exists μ ∈ (0, ∞) satisfying
for all x ∈ X and t > 0.
-
(2)
d(Jnf, A) → 0 as n → ∞. This implies the equality
for all x ∈ X.
-
(3)
with f ∈ Ω, which implies the inequality
This implies that the inequality (2.2) holds. Furthermore, since
for all x, y, z ∈ X, t > 0. So for all x, y, z ∈ X and all t > 0. Thus the mapping A : X → Y is additive, as desired. □
Corollary 2.2. Let θ ≥ 0 and let p be a real number with p > 1. Let X be a normed vector space with norm ||.||. Let f : X → Y be a mapping satisfying
for all x, y, z ∈ X and all t > 0. Then the limit
exists for each x ∈ X and defines a unique additive mapping A : X → Y such that
for all x ∈ X.
Proof. The proof follows from Theorem 2.1 by taking φ(x, y, z): = θ(||x|| p + ||y|| p + ||z|| p ) for all x, y, z ∈ X. Then we can choose L = 2-pand we get the desired result. □
Theorem 2.3. Let φ : X3 → [0, ∞) be a function such that there exists an L < 1 with
for all x, y, z ∈ X. Let f : X → Y be a mapping satisfying (2.1). Then
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. Let (S, d) be the generalized metric space defined as in the proof of Theorem 2.1.
Consider the linear mapping J : S → S such that
for all x ∈ X. Let g, h ∈ S be such that d(g, h) = ε. Then
for all x ∈ X and t > 0. Hence,
for all x ∈ X and t > 0. Thus, d(g, h) = ε implies that d(Jg, Jh) ≤ Lε. This means that
for all g, h ∈ S. It follows from (2.3) that
Therefore,
By Theorem 1.7, there exists a mapping A : X → Y satisfying the following:
-
(1)
A is a fixed point of J, that is,
(2.8)
for all x ∈ X. The mapping A is a unique fixed point of J in the set
This implies that A is a unique mapping satisfying (2.8) such that there exists μ ∈ (0, ∞) satisfying
for all x ∈ X and t > 0.
-
(2)
d(Jn f, A) → 0 as n → ∞. This implies the equality
for all x ∈ X.
-
(3)
with f ∈ Ω which implies the inequality
This implies that the inequality (2.7) holds.
The rest of the proof is similar to that of the proof of Theorem 2.1. □
Corollary 2.4. Let θ ≥ 0 and let p be a real number with. Let X be a normed vector space with norm || . ||. Let f : X → Y be a 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
for all x ∈ X.
Proof. The proof follows from Theorem 2.3 by taking φ(x, y, z): = θ(||x|| p · ||y|| p · ||z|| p ) for all x, y, z ∈ X. Then we can choose L = 2-3pand we get the desired result. □
2.2. Direct method. In this subsection, using direct method, we prove the Hyers-Ulam stability of the functional Eq. (0.1) in fuzzy Banach spaces.
Throughout this subsection, 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 2.5. Assume that a mapping f : X → Y satisfies the inequality
for all x, y, z ∈ X, t > 0 and φ : X3 → Z is a mapping for which there is a constant r ∈ ℝ satisfyingand
for all x, y, z ∈ X and all t > 0. Then there exist a unique additive mapping A : X → Y satisfying (0.1) and the inequality
for all x ∈ X and all t > 0.
Proof. It follows from (2.10) that
So
for all x, y, z ∈ X and all t > 0. Substituting y = 2x and z = x in (2.9), we obtain
So
for all x ∈ X and all t > 0. Replacing x by in (2.14), we have
for all x ∈ X, all t > 0 and any integer j ≥ 0. So
Replacing x by in the above inequality, we find that
for all x ∈ X, t > 0 and all integers n ≥ 0, p ≥ 0. So
for all x ∈ X, t > 0 and all integers n > 0, p ≥ 0. Hence, one obtains
for all x ∈ X, t > 0 and all integers n > 0, p ≥ 0. Since the series is convergent, by taking the limit p → ∞ in the last inequality, we know that a 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 (2.17) that
for all x ∈ X and all t > 0. So
for sufficiently large n and for all x ∈ X, t > 0 and N with 0 < N < 1. Since N is arbitrary and N' is left continuous, we obtain
for all x ∈ X and t > 0. It follows from (2.9) that
for all x, y, z ∈ X, t > 0 and all n ∈ ℕ. Since
and so
for all x, y, z ∈ X and all t > 0. Therefore, we obtain in view of (2.18)
which implies
for all x, y, z ∈ X. Thus, A: X → Y is a mapping satisfying the Eq. (0.1) and the inequality (2.11).
To prove the uniqueness, assume that there is another mapping L : X → Y which satisfies the inequality (2.11). Since for all x ∈ X, we have
for all t > 0. Therefore, A(x) = L(x) for all x ∈ X, this completes the proof. □
Corollary 2.6. 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 a mapping f : X → Y satisfies the following inequality
for all x, y, z ∈ X and t > 0. Then there is a unique additive mapping A : X → Y satisfying (0.1) and the inequality
Proof. Let φ(x, y, z): = θ(||x|| p + ||y|| p + ||z|| p ) and . Applying Theorem 2.5, we get the desired result. □
Theorem 2.7. Assume that a mapping f : X → Y satisfies (2.9) and φ : X2 → Z is a mapping for which there is a constant r ∈ ℝ satisfying 0 < |r| < 2 and
for all x, y, z ∈ X and all t > 0. Then there is a unique additive mapping A : X → Y satisfying (0.1) and the following inequality
for all x ∈ X and all t > 0.
Proof. It follows from (2.13) that
for all x ∈ X and all t > 0. Replacing x by 2 nx in (2.21), we obtain
So
for all x ∈ X and all t > 0. Proceeding as in the proof of Theorem 2.5, we obtain that
for all x ∈ X, all t > 0 and all integers n > 0. So
The rest of the proof is similar to the proof of Theorem 2.5. □
Corollary 2.8. Let X be a normed spaces and (ℝ, N') a fuzzy Banach space. Assume that there exist real numbers θ ≥ 0 andsuch that a mapping f : X → Y satisfies the following inequality
for all x, y, z ∈ X and t > 0. Then there is a unique additive mapping A : X → Y satisfying (0.1) and the inequality
Proof. Let and |r| = 1. Applying Theorem 2.7, we get the desired result. □
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Kenary, H.A., Rezaei, H., Ghaffaripour, A. et al. Fuzzy Hyers-Ulam stability of an additive functional equation. J Inequal Appl 2011, 140 (2011). https://doi.org/10.1186/1029-242X-2011-140
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DOI: https://doi.org/10.1186/1029-242X-2011-140