Nonlinear -Fuzzy stability of cubic functional equations

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.

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.

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 x T n is defined for every x ∈ [0,1] and n ∈ N ∪ {0} by 1, if n = 0 and T x T n - 1 , x , if n ≥ 1. A t-norm T is said to be of Hadžić-type (we denote by T ∈ ) if the family x T n n ∈ N is equicontinuous at x = 1 (cf. [29]).

Other important triangular norms are (see [30]):

-the Sugeno-Weber family T λ SW λ ∈ - 1 , ∞ is defined by T - 1 SW = T D , T ∞ SW = T P and

if λ ∈ (-1, ∞).

-the Domby family T λ D λ ∈ 0 , ∞ , defined by T_D, if λ = 0, T_M, if λ = ∞ and

if λ ∈ (0, ∞).

-the Aczel-Alsina family T λ AA λ ∈ 0 , ∞ , defined by T_D, if λ = 0, T_M, 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 T i = 1 n x i n ∈ ℕ 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∈N in [0, 1] such that li m n → ∞ x n = 1 .

(3) If T ∈ { T λ AA } λ ∈ ( 0 , ∞ ) ∪ { T λ D } λ ∈ ( 0 , ∞ ) , then

(4) If T ∈ { T λ sw } λ ∈ [ - 1 , ∞ ) , then

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 L = ( L , ≤ L ) be a complete lattice and U be a non-empty set called universe. A -fuzzy set on U is defined as a mapping A : U → L . For any u ∈ U, A u represents the degree (in L) to which u satisfies .

Lemma 3.2 ([44]). Consider the set L* and operation ≤ L * defined by:

( x 1 , x 2 ) ≤ L * ( y 1 , y 2 ) ⇔ x 1 ≤ y 1 and x 2 ≥ y 2 for all (x₁, x₂), (y₁, y₂) ∈ L*. Then (L*, ≤_L*) is a complete lattice.

Definition 3.3 ([45]). An intuitionistic fuzzy set A ζ , η on a universe U is an object A ζ , η = { ( ζ A ( u ) , η A ( u ) ) : u ∈ U } , where, for all u ∈ U, ζ A ( u ) ∈ [ 0 , 1 ] and η A ( u ) ∈ [ 0 , 1 ] are called the membership degree and the non-membership degree, respectively, of u in A ζ , η and, furthermore, satisfy ζ A ( u ) + η A ( u ) ≤ 1 .

In Section 2, we presented the classical definition of t-norm, which can be easily extended to any lattice L = ( L , ≤ L ) . Define first 0 L = inf L and 1 L = sup L .

Definition 3.4. A triangular norm (t-norm) on is a mapping T : L 2 → L satisfying the following conditions:

(i) for any x ∈ L , T ( x , 1 L ) = x (: boundary condition);

(ii) for any ( x , y ) ∈ L 2 , T ( x , y ) = T ( y , x ) (: commutativity);

(iii) for any ( x , y , z ) ∈ L 3 , T ( x , T ( y , z ) ) = T ( T ( x , y ) , z ) (: associativity);

(iv) for any ( x , x ′ , y , y ′ ) ∈ L 4 , x ≤ L x ′ and y ≤ L y ′ ⇒ T ( x , y ) ≤ L T ( x ′ , y ′ ) (: monotonicity).

A t-norm can also be defined recursively as an (n + 1)-array operation (n ∈ N \ {0}) by T 1 = T and

The t-norm M defined by

is a continuous t-norm.

Definition 3.5. A t-norm T 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 N : L → L satisfying N ( 0 L ) = 1 L and N ( 1 L ) = 0 L . If N ( N ( x ) ) = x for all x ∈ L, then is called an involutive negation.

In this article, let N : L → L 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 ( V , P , T ) is said to be a -fuzzy normed space if V is a vector space, T 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) 0 L < L P ( x , t ) ;

(ii) P ( x , t ) = 1 L if and only if x = 0;

(iii) P ( α x , t ) = P ( x , t | α | ) for all α ≠ 0;

(iv) T ( P ( x , t ) , P ( y , s ) ) ≤ L P ( x + y , t + s ) ;

(v) P ( x ,   · ) : ] 0 , ∞ [ → L is continuous;

(vi) lim t → 0 P ( x , t ) = 0 L and lim t → ∞ P ( x , t ) = 1 L .

In this case, is called a -fuzzy norm. If P = P μ , ν is an intuitionistic fuzzy set and the t-norm T is t-representable, then the 3-tuple ( V , P μ , v , T ) 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 ε ∈ L \ { 0 L } 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 ( V , P , T ) (denoted by x n → P x ) if P ( x n - x , t ) → 1 L , whenever n → + ∞ for all t > 0.

Lemma 3.9 ([46]). Let P be a -fuzzy norm on V. Then

(1) For all × ∈ V, P ( x , t ) is nondecreasing with respect to t.

(2) P ( x - y , t ) = P ( y - x , t ) for all x, y ∈ V and t ∈ ]0, +∞ [.

Definition 3.10. Let ( V , P , T ) be a -fuzzy normed space. For any t ∈ ]0, +∞[, we define the open ball B(x, r, t) with center x ∈ V and radius r ∈ L \ { 0 L , 1 L } as

In this section, we study the stability of functional equations in -fuzzy normed spaces.

Theorem 4.1. Let X be a linear space and ( Y , P , T ) 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 f ( 3 n x ) 27 n , we replace x with 3^mx in (4.3) to find that, for all m, n > 0,

Since the right-hand side of the inequality tends to 1 L as m tends to infinity, the sequence f ( 3 n x ) 3 3 n is a Cauchy sequence. Thus, we may define C ( x ) = lim n → ∞ f ( 3 n x ) 3 3 n 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 ( Y , P , T ) 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. Let ( X , P ′ , T ) be -fuzzy normed space and ( Y , P , T ) 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, ( X , P μ , ν , T M ) be an intuitionistic fuzzy normed space in which T M ( a , b ) = ( min { a 1 , b 1 } , max { a 2 , b 2 } ) and

also ( Y , P μ , ν , T M ) 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

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 | ⋅ | : K → [ 0 , + ∞ [ 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 ( V , P , T ) , 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) 0 L < L P ( x , t ) ;

(ii) P ( x , t ) = 1 L if and only if x = 0;

(iii) P ( α x , t ) = P ( x , t | α | ) for all α ≠ 0;

(vi) T ( P ( x , t ) , P ( y , s ) ) ≤ L P ( x + y , max { t , s } ) ;

(v) P ( x , ⋅ ) : ] 0 , ∞ [ → L is continuous;

(vi) lim t → 0 P ( x , t ) = 0 L and lim t → ∞ P ( x , t ) = 1 L .

Example 5.3. Let (X, || · ||) be a non-Archimedean normed linear space. Then the triple ( X , P , min ) , 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 T M ( a , b ) = ( min { a 1 , b 1 } , max { a 2 , b 2 } ) for all a = (a₁, a₂), b = (b₁, b₂) ∈ L* and P μ , ν be the intuitionistic fuzzy set on X × ]0, +∞[ defined as follows:

Then ( X , P μ , ν , T M ) is a non-Archimedean intuitionistic fuzzy normed space.

Let be a non-Archimedean field, X be a vector space over and ( Y , P , T ) 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 ( Y , P , T ) 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 |3^k| < 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 3^jx 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 lim n → ∞ T j = n ∞ M x , α j + 1 | ( 3 k ) j | t = 1 L for all x ∈ X and t > 0, ( 3 3 k ) n f x ( 3 k ) n n ∈ ℕ is a Cauchy sequence in the non-Archimedean -fuzzy Banach space ( Y , P , T ) . 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^-kn y in (6.1) and (6.2), we get

Since lim n → ∞ Ψ x , y , α n t | 3 k | n = 1 L , 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 ( Y , P , T ) 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 |3^k| < 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, ( X , P μ , ν , T M ) be non-Archimedean -fuzzy normed space (intuitionistic fuzzy normed space) in which

and ( Y , P μ , ν , T M ) 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 ( Y , P , T ) 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 |2^k| < 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 ( Y , P , T ) 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 |2^k| < 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, ( X , P μ , ν , T M ) be non-Archimedean -fuzzy normed space (intuitionistic fuzzy normed space) in which

and ( Y , P μ , ν , T M ) 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 ( Y , P , T ) 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 |3^k| < 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 3^jx 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