Abstract
We establish some stability results for the cubic functional equations
and
in the setting of various -fuzzy normed spaces that in turn generalize a Hyers-Ulam stability result in the framework of classical normed spaces. First, we shall prove the stability of cubic functional equations in the -fuzzy normed space under arbitrary t-norm which generalizes previous studies. Then, we prove the stability of cubic functional equations in the non-Archimedean -fuzzy normed space. We therefore provide a link among different disciplines: fuzzy set theory, lattice theory, non-Archimedean spaces, and mathematical analysis.
Mathematics Subject Classification (2000): Primary 54E40; Secondary 39B82, 46S50, 46S40.
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1. Introduction
The study of stability problems for functional equations is related to a question of Ulam [1] concerning the stability of group homomorphisms and it was affirmatively answered for Banach spaces by Hyers [2]. Subsequently, the result of Hyers was generalized by Aoki [3] for additive mappings and by Rassias [4] for linear mappings by considering an unbounded Cauchy difference. The article [4] of Rassias has provided a lot of influence in the development of what we now call Hyers-Ulam-Rassias stability of functional equations. For more informations on such problems, refer to the papers [5–15].
The functional equations
and
are called the cubic functional equations, since the function f(x) = cx3 is their solution. Every solution of the cubic functional equations is said to be a cubic mapping. The stability problem for the cubic functional equations was studied by Jun and Kim [16] for mappings f : X → Y, where X is a real normed space and Y is a Banach space. Later a number of mathematicians worked on the stability of some types of cubic equations [4, 17–19]. Furthermore, Mirmostafaee and Moslehian [20], Mirmostafaee et al. [21], Alsina [22], Miheţ and Radu [23] and others [24–28] investigated the stability in the settings of fuzzy, probabilistic, and random normed spaces.
2. Preliminaries
In this section, we recall some definitions and results which are needed to prove our main results.
A triangular norm (shorter t-norm) is a binary operation on the unit interval [0,1], i.e., a function T : [0,1] × [0,1] → [0,1] such that for all a, b, c ∈ [0,1] the following four axioms are satisfied:
-
(i)
T (a, b) = T (b, a) (: commutativity);
-
(ii)
T (a, (T (b, c))) = T (T (a, b), c) (: associativity);
-
(iii)
T (a, 1) = a (: boundary condition);
-
(iv)
T (a, b) ≤ T (a, c) whenever b ≤ c (: monotonicity).
Basic examples are the Lukasiewicz t-norm T L , T L (a, b) = max(a + b - 1, 0) ∀a, b ∈ [0,1] and the t-norms T P , T M , T D , where T P (a, b) := ab, T M (a, b) := min{a, b},
If T is a t-norm then is defined for every x ∈ [0,1] and n ∈ N ∪ {0} by 1, if n = 0 and if n ≥ 1. A t-norm T is said to be of Hadžić-type (we denote by T ∈ ) if the family is equicontinuous at x = 1 (cf. [29]).
Other important triangular norms are (see [30]):
-
the Sugeno-Weber family is defined by , and
if λ ∈ (-1, ∞).
-
the Domby family defined by TD, if λ = 0, TM, if λ = ∞ and
if λ ∈ (0, ∞).
-
the Aczel-Alsina family defined by TD, if λ = 0, TM, if λ = ∞ and
if λ ∈ (0, ∞).
A t-norm T can be extended (by associativity) in a unique way to an n-array operation taking for (x1, . . . , x n ) ∈ [0,1]n the value T (x1, . . . , x n ) defined by
T can also be extended to a countable operation taking for any sequence (x n ) n∈N in [0,1] the value
The limit on the right side of (2.1) exists, since the sequence is non-increasing and bounded from below.
Proposition 2.1. [30] (1) For T ≥ T L the following implication holds:
-
(2)
If T is of Hadžić-type then
for every sequence {x n }n∈Nin [0, 1] such that .
-
(3)
If , then
-
(4)
If , then
3. -Fuzzy normed spaces
The theory of fuzzy sets was introduced by Zadeh [31]. After the pioneering study of Zadeh, there has been a great effort to obtain fuzzy analogs of classical theories. Among other fields, a progressive development is made in the field of fuzzy topology [32–40, 43–50]. One of the problems in -fuzzy topology is to obtain an appropriate concept of -fuzzy metric spaces and -fuzzy normed spaces. Saadati and Park [40], respectively, introduced and studied a notion of intuitionistic fuzzy metric (normed) spaces and then Deschrijver et al. [41] generalized the concept of intuitionistic fuzzy metric (normed) spaces and studied a notion of -fuzzy metric spaces and -fuzzy normed spaces (also, see [41, 42, 51–55]). In this section, we give some definitions and related lemmas for our main results.
In this section, we give some definitions and related lemmas which are needed later.
Definition 3.1 ([43]). Let be a complete lattice and U be a non-empty set called universe. A -fuzzy set on U is defined as a mapping . For any u ∈ U, represents the degree (in L) to which u satisfies .
Lemma 3.2 ([44]). Consider the set L* and operation defined by:
and for all (x1, x2), (y1, y2) ∈ L*. Then (L*, ≤ L* ) is a complete lattice.
Definition 3.3 ([45]). An intuitionistic fuzzy set on a universe U is an object , where, for all u ∈ U, and are called the membership degree and the non-membership degree, respectively, of u in and, furthermore, satisfy.
In Section 2, we presented the classical definition of t-norm, which can be easily extended to any lattice . Define first and .
Definition 3.4. A triangular norm (t-norm) on is a mapping satisfying the following conditions:
-
(i)
for any (: boundary condition);
-
(ii)
for any (: commutativity);
-
(iii)
for any (: associativity);
-
(iv)
for any and (: monotonicity).
A t-norm can also be defined recursively as an (n + 1)-array operation (n ∈ N \ {0}) by and
The t-norm M defined by
is a continuous t-norm.
Definition 3.5. A t-norm on L* is said to be t-representable if there exist a t-norm T and a t-conorm S on [0,1] such that
Definition 3.6. A negation on is any strictly decreasing mapping satisfying and . If for all x ∈ L, then is called an involutive negation.
In this article, let be a given mapping. The negation N s on ([0,1], ≤) defined as N s (x) = 1 - x for all x ∈ [0, 1] is called the standard negation on ([0,1], ≤).
Definition 3.7. The 3-tuple is said to be a -fuzzy normed space if V is a vector space, is a continuous t-norm on and is a -fuzzy set on satisfying the following conditions: for all x, y ∈ V and t, s ∈]0, +∞[,
-
(i)
-
(ii)
if and only if x = 0;
-
(iii)
for all α ≠ 0;
-
(iv)
;
-
(v)
is continuous;
-
(vi)
and .
In this case, is called a -fuzzy norm. If is an intuitionistic fuzzy set and the t-norm is t-representable, then the 3-tuple is said to be an intuitionistic fuzzy normed space.
Definition 3.8. (1) A sequence {x n } in X is called a Cauchy sequence if, for any and t > 0, there exists a positive integer n0 such that
-
(2)
If every Cauchy sequence is convergent, then the -fuzzy norm is said to be complete and the -fuzzy normed space is called a -fuzzy Banach space, where is an involutive negation.
-
(3)
The sequence {x n } is said to be convergent to x ∈ V in the -fuzzy normed space (denoted by if , whenever n → + ∞ for all t > 0.
Lemma 3.9 ([46]). Let be a -fuzzy norm on V. Then
-
(1)
For all × ∈ V, is nondecreasing with respect to t.
-
(2)
for all x, y ∈ V and t ∈ ]0, +∞ [.
Definition 3.10. Let be a -fuzzy normed space. For any t ∈ ]0, +∞[, we define the open ball B(x, r, t) with center x ∈ V and radius as
4. Stability result in -fuzzy normed spaces
In this section, we study the stability of functional equations in -fuzzy normed spaces.
Theorem 4.1. Let X be a linear space and be a complete -fuzzy normed space. If f : × → Y is a mapping with f (0) = 0 and Q is a -fuzzy set on X2 × (0, ∞) with the following property:
If
and
then there exists a unique cubic mapping C : X → Y such that
Proof. We brief the proof because it is similar as the random case [47, 27]. Putting y = 0 in (4.1), we have
Therefore, it follows that
By the triangle inequality, it follows that
In order to prove the convergence of the sequence , we replace x with 3mx in (4.3) to find that, for all m, n > 0,
Since the right-hand side of the inequality tends to as m tends to infinity, the sequence is a Cauchy sequence. Thus, we may define for all x ∈ X. Replacing x, y with 3nx and 3ny, respectively, in (4.1), it follows that C is a cubic mapping. To prove (4.2), take the limit as n → ∞ in (4.3). To prove the uniqueness of the cubic mapping C subject to (4.2), let us assume that there exists another cubic mapping C' which satisfies (4.2). Obviously, we have C(3nx) = 33nC(x) and C'(3nx) = 33nC'(x) for all x ∈ X and n ∈ ℕ. Hence it follows from (4.2) that
which proves the uniqueness of C. This completes the proof.
Theorem 4.2. Let X be a linear space and be a complete -fuzzy normed space. If f : X → Y is a mapping with f (0) = 0 and Q is a -fuzzy set on X2 × (0, ∞) with the following property:
If
and
then there exists a unique cubic mapping C : X → Y such that
Proof. We omit the proof because it is similar as the last theorem and see [28].
Corollary 4.3. Letbe -fuzzy normed space and be a complete -fuzzy normed space. If f : X → Y is a mapping such that
and
then there exists a unique cubic mapping C : X → Y such that
Proof. See [28].
Now, we give an example to validate the main result as follows:
Example 4.4 ([28]). Let (X, || · ||) be a Banach space, be an intuitionistic fuzzy normed space in which and
also be a complete intuitionistic fuzzy normed space. Define a mapping f : X → Y by f (x) = x3 + x0 for all x ∈ X, where x0 is a unit vector in X. A straightforward computation shows that
Also, we have
Therefore, all the conditions of Theorem 4.2 hold and so there exists a unique cubic mapping C : X → Y such that
5. Non-Archimedean L-fuzzy normed spaces
In 1897, Hensel [?] introduced a field with a valuation in which does not have the Archimedean property.
Definition 5.1. Let be a field. A non-Archimedean absolute value on is a function such that, for any a, b ∈ ,
-
(i)
|a| ≥ 0 and equality holds if and only if a = 0;
-
(ii)
|ab| = |a| |b|;
-
(iii)
|a + b| ≤ max {|a|, |b|} (: the strict triangle inequality).
Note that |n| ≤ 1 for each integer n ≥ 1. We always assume, in addition, that | · | is non-trivial, i.e., there exists a0 ∈ such that |a0| ≠ 0, 1.
Definition 5.2. A non-Archimedean -fuzzy normed space is a triple , where V is a vector space, is a continuous t-norm on and is a -fuzzy set on V × ]0, +∞[ satisfying the following conditions: for all x, y ∈ V and t, s ∈ ]0, +∞[,
-
(i)
;
-
(ii)
if and only if x = 0;
-
(iii)
for all α ≠ 0;
-
(vi)
;
-
(v)
is continuous;
-
(vi)
and .
Example 5.3. Let (X, || · ||) be a non-Archimedean normed linear space. Then the triple , where
is a non-Archimedean -fuzzy normed space in which L = [0,1].
Example 5.4. Let (X, ||·||) be a non-Archimedean normed linear space. Denote for all a = (a1, a2), b = (b1, b2) ∈ L* and be the intuitionistic fuzzy set on X × ]0, +∞[ defined as follows:
Then is a non-Archimedean intuitionistic fuzzy normed space.
6. -fuzzy Hyers-Ulam-Rassias stability for cubic functional equations in non-Archimedean-fuzzy normed space
Let be a non-Archimedean field, X be a vector space over and be a non-Archimedean -fuzzy Banach space over . In this section, we investigate the stability of the cubic functional equation (1.1).
Next, we define a -fuzzy approximately cubic mapping. Let Ψ be a -fuzzy set on X × X × [0, ∞) such that Ψ (x, y, ·) is nondecreasing,
and
Definition 6.1. A mapping f : X → Y is said to be Ψ-approximately cubic if
Here, we assume that 3 ≠ 0 in (i.e., characteristic of is not 3).
Theorem 6.2. Let be a non-Archimedean field, X be a vector space over and be a non-Archimedean -fuzzy Banach space over : Let f: X → Y be a Ψ-approximately cubic mapping. If there exist a α ∈ ℝ (α > 0) and an integer k, k ≥ 2 with | 3k| < α and | 3| ≠ 1 such that
and
then there exists a unique cubic mapping C : X → Y such that
where
Proof. First, we show, by induction on j, that, for all x ∈ X, t > 0 and j ≥ 1,
Putting y = 0 in (6.1), we obtain
This proves (6.4) for j = 1. Let (6.4) hold for some j > 1. Replacing y by 0 and x by 3jx in (6.1), we get
Since | 27| ≤ 1, it follows that
Thus (6.4) holds for all j ≥ 1. In particular, we have
Replacing x by 3-(kn+k)x in (6.5) and using the inequality (6.2), we obtain
and so
Hence, it follow that
Since for all x ∈ X and t > 0, is a Cauchy sequence in the non-Archimedean -fuzzy Banach space . Hence we can define a mapping C : X → Y such that
Next, for all n ≥ 1, x ∈ X and t > 0, we have
and so
Taking the limit as n → ∞ in (6.7), we obtain
which proves (6.3). As is continuous, from a well known result in -fuzzy (probabilistic) normed space (see, [51, Chap. 12]), it follows that
On the other hand, replacing x, y by 3-knx, 3-kny in (6.1) and (6.2), we get
Since , we infer that C is a cubic mapping.
For the uniqueness of C, let C' : X → Y be another cubic mapping such that
Then we have, for all x, y ∈ X and t > 0,
Therefore, from (6.6), we conclude that C = C'. This completes the proof.
Corollary 6.3. Let be a non-Archimedean field, X be a vector space over and be a non-Archimedean -fuzzy Banach space over under a t-norm. Let f: X → Y be a Ψ-approximately cubic mapping. If there exist α ∈ ℝ (α > 0),| 3| ≠ 1 and an integer k, k ≥ 3 with | 3k| < α such that
then there exists a unique cubic mapping C : X → Y such that
where
Proof. Since
and is of Hadžić type, it follows from Proposition 2.1 that
Now, if we apply Theorem 6.2, we get the conclusion.
Now, we give an example to validate the main result as follows:
Example 6.4. Let (X, || · ||) be a non-Archimedean Banach space, be non-Archimedean -fuzzy normed space (intuitionistic fuzzy normed space) in which
and be a complete non-Archimedean -fuzzy normed space (intuitionistic fuzzy normed space) (see, Example 5.4). Define
It is easy to see that (6.2) holds for α = 1. Also, since
we have
Let f : X → Y be a Ψ-approximately cubic mapping. Therefore, all the conditions of Theorem 6.2 hold and so there exists a unique cubic mapping C : X → Y such that
Definition 6.5. A mapping f : X → Y is said to be Ψ-approximately cubic I if
In this section, we assume that 2 ≠ 0 in (i.e., the characteristic of is not 2).
Theorem 6.6. Let be a non-Archimedean field, X be a vector space over and be a non-Archimedean -fuzzy Banach space over. Let f : X → Y be a Ψ-approximately cubic I mapping. If | 2| ≠ 1 and for some α ∈ ℝ, α > 0, and some integer k, k ≥ 2 with | 2k| < α,
and
then there exists a unique cubic mapping C : X → Y such that
where
Proof. We omit the proof because it is similar as the random case (see, [28]).
Corollary 6.7. Let be a non-Archimedean field, X be a vector space over and be a non-Archimedean -fuzzy Banach space over under a t-norm . Let f : X → Y be a Ψ-approximately cubic I mapping. If there exist a α ∈ ℝ (α > 0) and an integer k, k ≥ 2 with |2k| < α such that
then there exists a unique cubic mapping C : X → Y such that
where
Proof. Since
and is of Hadžić type, it follows from Proposition 2.1 that
Now, if we apply Theorem 6.2, we get the conclusion.
Now, we give an example to validate the main result as follows:
Example 6.8. Let (X, || · || be a non-Archimedean Banach space, be non-Archimedean -fuzzy normed space (intuitionistic fuzzy normed space) in which
and be a complete non-Archimedean -fuzzy normed space (intuitionistic fuzzy normed space) (see, Example 5.4). Define
It is easy to see that (6.9) holds for α = 1. Also, since
we have
Let f : X → Y be a Ψ-approximately cubic I mapping. Therefore, all the conditions of Theorem 6.6 hold and so there exists a unique cubic mapping C : X → Y such that
Definition 6.9. A mapping f : X → Y is said to be Ψ-approximately cubic II if
Here, we assume that 3 ≠ 0 in (i.e., the characteristic of is not 3).
Theorem 6.10. Let be a non-Archimedean field, X be a vector space over and be a non-Archimedean -fuzzy Banach space over . Let f : X → Y be a Ψ-approximately cubic II function. If |3| ≠ 1 and, for some α ∈ ℝ, α > 0, and some integer k, k ≥ 3, with |3k| < α,
and
then there exists a unique cubic mapping C : X → Y such tha
for all × ∈ X and t > 0, where
Proof. First, we show, by induction on j, that, for all x ∈ X, t > 0 and j ≥ 1,
Put y = 0 in (6.11) to obtain
This proves (6.15) for j = 1. Let (6.15) hold for some j > 1. Replacing y by 0 and x by 3jx in (6.16), we get
Since |27| ≤ 1, then we have
Thus (6.15) holds for all j ≥ 1. In particular, it follows that
Replacing x by 3-(kn+k)x in (6.17) and using inequality (6.12) we obtain
Then we have
Hence we have
Since for all x ∈ X and t > 0, {kn f (k--nx)} n∈N is a Cauchy sequence in the non-Archimedean -fuzzy Banach space Hence we can define a mapping C : X → Y such that
Next, for all n ≥ 1, x ∈ X and t > 0,
Therefore, we have
By letting n → ∞ in the above inequality, we obtain
This proves (6.14). Since is continuous, from the well known result in -fuzzy (probabilistic) normed space (see, [51, Chap. 12]), it follows that
On the other hand, replace x, y by 3-knx, 3-kny in (6.11) and (6.12) to get
Since we infer that C is a cubic mapping.
If is another cubic mapping such that for all x ∈ X and t > 0, then, for all n ≥ 1, x ∈ X and t > 0,
Thus, from (6.20), we conclude that C = . This completes the proof.
Corollary 6.11. Let be a non-Archimedean field, X be a vector space over and be a non-Archimedean -fuzzy Banach space over under a t-norm T∈ . Let f : X → Y be a Ψ-approximately cubic II function. If, for some α ∈ ℝ, α > 0 and an integer k, k ≥ 3, with | 3k| < α,
then there exists a unique cubic mapping C : X → Y such that, for all × ∈ X and t > 0,
Where
Proof. Since
and T is of Hadžić type, from Proposition 2.1, it follows that
Thus, if we apply Theorem 6.10, then we can get the conclusion. This completes the proof.
7. Conclusion
We established the Hyers-Ulam-Rassias stability of the cubic functional equations (1.1), (1.2), and (1.3) in various fuzzy spaces. In Section 4, we proved the stability of functional equations (1.1), (1.2), and (1.3) in a -fuzzy normed space under arbitrary t-norm which is a generalization of [26]. In Section 6, we proved the stability of functional equations (1.1), (1.2), and (1.3) in a non-Archimedean -fuzzy normed space. We therefore provided a link among three various discipline: fuzzy set theory, lattice theory, and mathematical analysis.
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Acknowledgment
This study was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant Number: 2011-0021821).
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Agarwal, R.P., Cho, Y.J., Saadati, R. et al. Nonlinear -Fuzzy stability of cubic functional equations. J Inequal Appl 2012, 77 (2012). https://doi.org/10.1186/1029-242X-2012-77
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DOI: https://doi.org/10.1186/1029-242X-2012-77