Research

# Nonlinear -Fuzzy stability of cubic functional equations

Ravi P Agarwal1,2*, Yeol J Cho3, Reza Saadati4 and Shenghua Wang5

Author Affiliations

1 Department of Mathematics, Texas A&M University - Kingsville, Kingsville, TX 78363, USA

2 Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia

3 Department of Mathematics Education and the RINS, Gyeongsang National University, Chinju 660-701, Korea

4 Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, I.R. Iran

5 Department of Mathematics and Physics, North China Electric Power University, Baoding 071003, China

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Journal of Inequalities and Applications 2012, 2012:77 doi:10.1186/1029-242X-2012-77

 Received: 2 December 2011 Accepted: 2 April 2012 Published: 2 April 2012

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

We establish some stability results for the cubic functional equations

3 f x +  3 y   + f 3 x - y   =  15 f x + y   +  15 f x - y   +  8 0 f y ,

f 2 x + y   + f 2 x   - y   =  2 f x + y   +  2 f x   - y   +  12 f x

and

f 3 x + y   + f 3 x   - y   =  3 f x + y   +  3 f x   - y   +  48 f x

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.

##### Keywords:
stability; cubic functional equation; fuzzy normed space; -fuzzy set

### 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

3 f x +  3 y   + f 3 x - y   =  15 f x + y   +  15 f x - y   +  8 0 f y , (1.1)

f 2 x + y   + f 2 x   - y   =  2 f x + y   +  2 f x   - y   +  12 f x (1.2)

and

f 3 x + y   + f 3 x   - y   =  3 f x + y   +  3 f x   - y   +  48 f x (1.3)

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 TL, TL(a, b) = max(a + b - 1, 0) ∀a, b ∈ [0,1] and the t-norms TP, TM, TD, where TP (a, b) := ab, TM (a, b) := min{a, b},

T D a , b : = min a , b , if max a , b = 1 ; 0 , otherwise .

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

T λ SW x , y = max 0 , x + y - 1 + λ x y 1 + λ

if λ ∈ (-1, ∞).

-the Domby family T λ D λ 0 , , defined by TD, if λ = 0, TM, if λ = ∞ and

T λ D ( x , y ) = 1 1 + 1 - x x λ + 1 - y y λ 1 / λ

if λ ∈ (0, ∞).

-the Aczel-Alsina family T λ AA λ 0 , , defined by TD, if λ = 0, TM, if λ = ∞ and

T λ AA x , y = e - log x λ + log y λ 1 / λ

if λ ∈ (0, ∞).

A t-norm T can be extended (by associativity) in a unique way to an n-array operation taking for (x1, . . . , xn) ∈ [0,1] n the value T (x1, . . . , xn) defined by

T i = 1 0 x i = 1 , T i = 1 n x i = T T i = 1 n - 1 x i , x n = T x 1 , . . . , x n .

T can also be extended to a countable operation taking for any sequence (xn)n∈N in [0,1] the value

T i = 1 x i = lim n T i = 1 n x i . (2.1)

The limit on the right side of (2.1) exists, since the sequence T i = 1 n x i n is non-increasing and bounded from below.

Proposition 2.1. [30] (1) For T TL the following implication holds:

lim n T i = 1 x n + i = 1 n = 1 ( 1 - x n ) < .

(2) If T is of Hadžić-type then

lim n T i = 1 x n + i = 1

for every sequence {xn}nN 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

lim n T i = 1 x n + i = 1 n = 1 ( 1 - x n ) α < .

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

lim n T i = 1 x n + i = 1 n = 1 ( 1 - x n ) < .

### 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 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:

L * = { ( x 1 , x 2 ) : ( x 1 , x 2 ) [ 0 , 1 ] 2 a n d x 1 + x 2 1 } ,

( 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

T n ( x ( 1 ) , , x ( n + 1 ) ) = T ( T n - 1 ( x ( 1 ) , , x ( n ) ) , x ( n + 1 ) ) , n 2 , x ( i ) L .

The t-norm M defined by

M ( x , y ) = x if  x L y y if  y L x

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

T ( x , y ) = ( T ( x 1 , y 1 ) , S ( x 2 , y 2 ) ) , x = ( x 1 , x 2 ) , y = ( y 1 , y 2 ) L * .

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 Ns on ([0,1], ≤) defined as Ns(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 {xn} 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

N ( ε ) < L P ( x n + p - x n , t ) , n n 0 , p > 0 .

(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 {xn} 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

B ( x , r , t ) = { y V : N ( r ) < L P ( x - y , t ) } .

### 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 ( 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:

P ( 3 f ( x + 3 y ) + f ( 3 x - y ) - 15 f ( x + y ) - 15 f ( x - y ) - 80 f ( y ) , t ) L Q ( x , y , t ) , x , y X , t > 0 . (4.1)

If

T i = 1 ( Q ( 3 n + i - 1 x , 0 , 3 3 n + 2 i + 1 t ) ) = 1 L , x X , t > 0 ,

and

lim n Q ( 3 n x , 3 n y , 3 3 n t ) = 1 L , x , y X , t > 0 ,

then there exists a unique cubic mapping C : X Y such that

P ( f ( x ) - C ( x ) , t ) L T i = 1 ( Q ( 3 i - 1 x , 0 , 3 2 i + 2 t ) ) , x X , t > 0 . (4.2)

Proof. We brief the proof because it is similar as the random case [47,27]. Putting y = 0 in (4.1), we have

P f ( 3 x ) 27 - f ( x ) , t L * Q ( x , 0 , 3 3 t ) , x X , t > 0 .

Therefore, it follows that

P f ( 3 k + 1 x ) 3 3 ( k + 1 ) - f ( 3 k x ) 3 3 k , t 3 k + 1 L Q ( 3 k x , 0 , 3 2 ( k + 1 ) t ) . k 1 , t > 0 .

By the triangle inequality, it follows that

P f ( 3 n x ) 27 n - f ( x ) , t L T i = 1 n ( Q ( 3 i - 1 x , 0 , 3 2 i + 2 t ) ) , x X , t > 0 . (4.3)

In order to prove the convergence of the sequence f ( 3 n x ) 27 n , we replace x with 3mx in (4.3) to find that, for all m, n > 0,

P f ( 3 n + m x ) 27 ( n + m ) - f ( 3 m x ) 27 m , t L T i = 1 n ( Q ( 3 i + m - 1 x , 0 , 3 2 i + 3 m + 2 t ) ) , x X , t > 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

P C ( x ) - C ( x ) , t L P C ( 3 n x ) - C ( 3 n x ) , 3 3 n t L T P C ( 3 n x ) - f ( 3 n x ) , 3 3 n - 1 t , P f ( 3 n x ) - C ( 3 n x ) , 2 3 n - 1 t L T T i = 1 ( Q ( 3 n + i - 1 x , 0 , 3 3 n + 2 i + 1 t ) ) , T i = 1 ( Q ( 3 n + i - 1 x , 0 , 3 3 n + 2 i + 1 t ) = T ( 1 L , 1 L ) = 1 L , x X , t > 0 ,

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:

P ( f ( 2 x + y ) + f ( 2 x - y ) - 2 f ( x + y ) - 2 f ( x - y ) - 12 f ( x ) , t ) (4.4)

L Q ( x , y , t ) , x , y X , t > 0 .

If

T i = 1 ( Q ( 2 n + i - 1 x , 0 , 2 3 n + 2 i + 1 t ) ) = 1 L , x X , t > 0 ,

and

lim n Q ( 2 n x , 2 n y , 2 3 n t ) = 1 L , x , y X , t > 0 ,

then there exists a unique cubic mapping C : X Y such that

P ( f ( x ) - C ( x ) , t ) L T i = 1 ( Q ( 2 i - 1 x , 0 , 2 2 i + 1 t ) ) , x X , t > 0 . (4.5)

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

P ( f ( 2 x + y ) + f ( 2 x - y ) - 2 f ( x + y ) - 2 f ( x - y ) - 12 f ( x ) , t )

L P ( x + y , t ) , x , y X , t > 0 ,

and

lim n T i = 1 ( P ( x , 2 2 n + i + 2 t ) ) = 1 L , x X , t > 0 ,

then there exists a unique cubic mapping C : X Y such that

P ( f ( x ) - C ( x ) , t ) L T i = 1 ( P ( x , 2 i + 2 t ) ) , x X , t > 0 .

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

P μ , ν ( x , t ) = t t + | | x | | , | | x | | t + | | x | | , x X , t > 0 ,

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

P μ , ν ( f ( 2 x + y ) + f ( 2 x - y ) - 2 f ( x + y ) - 2 f ( x - y ) - 12 f ( x ) , t ) L * P μ , ν ( x + y , t ) , x , y X , t > 0 .

Also, we have

lim n T M , i = 1 ( P μ , ν ( x , 2 2 n + i + 1 t ) ) = lim n lim m T M , i = 1 m ( P μ , ν ( x , 2 2 n + i + 1 t ) ) = lim n lim m P μ , ν ( x , 2 2 n + 2 t ) = lim n P μ , ν ( x , 2 2 n + 2 t ) = 1 L * .

Therefore, all the conditions of Theorem 4.2 hold and so there exists a unique cubic mapping C : X Y such that

P μ , ν ( f ( x ) - C ( x ) , t ) L * P μ , ν ( x , 2 2 t ) , x X , t > 0 .

### 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 | | : 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

P ( x , t ) = 0 , if t | | x | | ; 1 , if t > | | x | | ,

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:

P μ , ν ( x , t ) = t t + | | x | | , | | x | | t + | | x | | , x X , t + .

Then ( X , P μ , ν , T M ) 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 ( 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,

Ψ ( c x , c x , t ) L Ψ x , x , t | c | , x X , c 0

and

lim t Ψ ( x , y , t ) = 1 L , x , y X , t > 0 .

Definition 6.1. A mapping f : X Y is said to be Ψ-approximately cubic if

P ( 3 f ( x + 3 y ) + f ( 3 x - y ) - 15 f ( x + y ) - 15 f ( x - y ) - 80 f ( y ) , t ) L Ψ ( x , y , t ) , x , y X , t > 0 . (6.1)

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 |3k| < α and |3| ≠ 1 such that

Ψ ( 3 - k x , 3 - k y , t ) L Ψ ( x , y , α t ) , x , y X , t > 0 , (6.2)

and

lim n T j = n M x , α j t | 3 | k j = 1 L , x X , t > 0 ,

then there exists a unique cubic mapping C : X Y such that

P ( f ( x ) - C ( x ) , t ) T i = 1 M x , α i + 1 t | 3 | k i , x X , t > 0 , (6.3)

where

M ( x , t ) : = T ( Ψ ( x , 0 , t ) , Ψ ( 3 x , 0 , t ) , , Ψ ( 3 k - 1 x , 0 , t ) ) , x X , t > 0 .

Proof. First, we show, by induction on j, that, for all x X, t > 0 and j ≥ 1,

P ( f ( 3 j x ) - 2 7 j f ( x ) , t ) L M j ( x , t ) : = T ( Ψ ( x , 0 , t ) , , Ψ ( 3 j - 1 x , 0 , t ) ) . (6.4)

Putting y = 0 in (6.1), we obtain

P ( f ( 3 x ) - 27 f ( x ) , t ) L Ψ ( x , 0 , t ) , x X , t > 0 .

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

P ( f ( 3 j + 1 x ) - 27 f ( 3 j x ) , t ) L Ψ ( 3 j x , 0 , t ) , x X , t > 0 .

Since |27| ≤ 1, it follows that

P ( f ( 3 j + 1 x ) - 2 7 j + 1 f ( x ) , t ) L T P ( f ( 3 j + 1 x ) - 27 f ( 3 j x ) , t ) , P ( 8 f ( 3 j x ) - 27 j + 1 f ( x ) , t ) = T P ( f ( 2 j + 1 x ) - 8 f ( 2 j x ) , t ) , P f ( 3 j x ) - 27 j f ( x ) , t | 27 | L T P ( f ( 3 j + 1 x ) - 27 f ( 3 j x ) , t ) , P ( f ( 3 j x ) - 27 j f ( x ) , t ) L T ( Ψ ( 3 j x , 0 , t ) , M j ( x , t ) ) = M j + 1 ( x , t ) , x X , t > 0 .

Thus (6.4) holds for all j ≥ 1. In particular, we have

P ( f ( 3 k x ) - 2 7 k f ( x ) , t ) L M ( x , t ) , x X , t > 0 . (6.5)

Replacing x by 3-(kn+k)x in (6.5) and using the inequality (6.2), we obtain

P f x 3 k n - 27 k f x 3 k n + k , t L M x 3 k n + k , t L M ( x , α n + 1 t ) x X , t > 0 , n 0

and so

P ( 3 3 k ) n f x ( 3 k ) n - ( 3 3 k ) n + 1 f x ( 3 k ) n + 1 , t L M x , α n + 1 | ( 3 3 k ) n | t L M x , α n + 1 | ( 3 k ) n | t , x X , t > 0 , n 0 .

P ( 3 3 k ) n f x ( 3 k ) n - ( 3 3 k ) n + p f x ( 3 k ) n + p , t L T j = n n + p P 3 3 k j f x ( 3 k ) j - ( 3 3 k ) j + p f x ( 3 k ) j + p , t L T j = n n + p M x , α j + 1 | ( 3 k ) j | t , x X , t > 0 , n 0 .

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 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

lim n P ( 3 3 k ) n f x ( 3 k ) n - C ( x ) , t = 1 L , x X , t > 0 . (6.6)

Next, for all n ≥ 1, x X and t > 0, we have

P ( f ( x ) ( 3 3 k ) n f ( x ( 3 k ) n ) , t ) = ( i = 0 n 1 ( 3 3 k ) i f ( x ( 3 k ) i ) ( 3 3 k ) i + 1 f ( x ( 3 k ) i + 1 ) , t ) L T i = 0 n 1 ( ( ( 3 3 k ) i f ( x ( 3 k ) i ) ( 3 3 k ) i + 1 f ( x ( 3 k ) i + 1 ) , t ) ) L T i = 0 n 1 ( x , α i + 1 t | 3 k | i )

and so

P ( f ( x ) - C ( x ) , t ) L T P f ( x ) - ( 3 3 k ) n f x ( 3 k ) n , t , P ( 3 3 k ) n f x ( 3 k ) n - C ( x ) , t L P T i = 0 n - 1 M x , α i + 1 t | 3 k | i , P ( ( 3 3 k ) n f x ( 3 k ) n - C ( x ) , t ) . (6.7)

Taking the limit as n → ∞ in (6.7), we obtain

P ( f ( x ) - C ( x ) , t ) L T i = 1 M x , α i + 1 t | 3 k | i ,

which proves (6.3). As is continuous, from a well known result in -fuzzy (probabilistic) normed space (see, [51, Chap. 12]), it follows that

lim n P ( ( 2 7 k ) n f ( 3 - k n ( x + 3 y ) ) + ( 2 7 k ) n f ( 3 - k n ( 3 x - y ) ) - 15 ( 2 7 k ) n f ( 3 - k n ( x + y ) ) - 15 ( 2 7 k ) n f ( 3 - k n ( x - y ) ) - 80 ( 2 7 k ) n f ( 3 - k n y ) , t ) = P ( C ( x + 3 y ) + C ( 3 x - y ) - 15 C ( x + y ) - 15 C ( x - y ) - 80 C ( y ) , t ) , t > 0 .

On the other hand, replacing x, y by 3-knx, 3-kn y in (6.1) and (6.2), we get

P ( ( 2 7 k ) n f ( 3 - k n ( x + 3 y ) ) + ( 2 7 k ) n f ( 3 - k n ( 3 x - y ) ) - 15 ( 2 7 k ) n f ( 3 - k n ( x + y ) ) - 15 ( 2 7 k ) n f ( 3 - k n ( x - y ) ) - 80 ( 2 7 k ) n f ( 3 - k n y ) , t ) L Ψ 3 - k n x , 3 - k n y , t | 3 3 k | n L Ψ x , y , α n t | 3 k | n , x , y X , t > 0 .

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

P ( C ( x ) - f ( x ) , t ) L M ( x , t ) , x X , t > 0 .

Then we have, for all x, y X and t > 0,

P ( C ( x ) - C ( x ) , t ) L T P C ( x ) - ( 3 3 k ) n f x ( 3 k ) n , t , P ( 3 3 k ) n f x ( 3 k ) n - C ( x ) , t , t ) .

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 |3k| < α such that

Ψ ( 3 - k x , 3 - k y , t ) L Ψ ( x , y , α t ) , x , y X , t > 0 ,

then there exists a unique cubic mapping C : X Y such that

P ( f ( x ) - C ( x ) , t ) L T i = 1 M x , α i + 1 t | 3 | k i , x X , t > 0 ,

where

M ( x , t ) : = T ( Ψ ( x , 0 , t ) , Ψ ( 3 x , 0 , t ) , , Ψ ( 3 k - 1 x , 0 , t ) ) , x X , t > 0 .

Proof. Since

lim n M x , α j t | 3 | k j = 1 L , x X , t > 0 ,

and is of Hadžić type, it follows from Proposition 2.1 that

lim n T j = n M x , α j t | 3 | k j = 1 L , x X , t > 0 .

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

P μ , ν ( x , t ) = t t + | | x | | , | | x | | t + | | x | | , x X , t > 0 ,

and ( Y , P μ , ν , T M ) be a complete non-Archimedean -fuzzy normed space (intuitionistic fuzzy normed space) (see, Example 5.4). Define

Ψ ( x , y , t ) = t 1 + t , 1 1 + t , x , y X , t > 0 .

It is easy to see that (6.2) holds for α = 1. Also, since

M ( x , t ) = t 1 + t , 1 1 + t , x X , t > 0 ,

we have

lim n T M , j = n M x , α j t | 3 | k j = lim n lim m T M , j = n m M x , t | 3 | k j = lim n lim m t t + | 3 k | n , | 2 k | n t + | 3 k | n = ( 1 , 0 ) = 1 L * , x X , t > 0 .

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

P μ , ν ( f ( x ) - C ( x ) , t ) L * t t + | 3 k | , | 3 k | t + | 3 k | , x X , t > 0 .

Definition 6.5. A mapping f : X Y is said to be Ψ-approximately cubic I if

P ( f ( 2 x + y ) + f ( 2 x - y ) - 2 f ( x + y ) - 2 f ( x - y ) - 12 f ( x ) , t ) (6.8)

L Ψ ( x , y , t ) , x , y X , t > 0 .

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 |2k| < α,

Ψ ( 2 - k x , 2 - k y , t ) L Ψ ( x , y , α t ) , x , y X , t > 0 , (6.9)

and

lim n T j = n M x , α j t | 2 | k j = 1 L , x X , t > 0 ,

then there exists a unique cubic mapping C : X Y such that

P ( f ( x ) - C ( x ) , t ) T i = 1 M x , α i + 1 t | 2 | k i , x X , t > 0 , (6.10)

where

M ( x , t ) : = T ( Ψ ( x , 0 , t ) , Ψ ( 2 x , 0 , t ) , , Ψ ( 2 k - 1 x , 0 , t ) ) , x X , t > 0 .

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 |2k| < α such that

Ψ ( 2 - k x , 2 - k y , t ) L Ψ ( x , y , α t ) , x , y X , t > 0 ,

then there exists a unique cubic mapping C : X Y such that

P ( f ( x ) - C ( x ) , t ) L T i = 1 M x , α i + 1 t | 2 | k i , x X , t > 0 ,

where

M ( x , t ) : = T ( Ψ ( x , 0 , t ) , Ψ ( 2 x , 0 , t ) , , Ψ ( 2 k - 1 x , 0 , t ) ) , x X , t > 0 .

Proof. Since

lim n M x , α j t | 2 | k j = 1 L , x X , t > 0 ,

and is of Hadžić type, it follows from Proposition 2.1 that

lim n T j = n M x , α j t | 2 | k j = 1 L , x X , t > 0 .

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

P μ , ν ( x , t ) = t t + | | x | | , | | x | | t + | | x | | , x X , t > 0 ,

and ( Y , P μ , ν , T M ) be a complete non-Archimedean -fuzzy normed space (intuitionistic fuzzy normed space) (see, Example 5.4). Define

Ψ ( x , y , t ) = t 1 + t , 1 1 + t , x , y X , t > 0 .

It is easy to see that (6.9) holds for α = 1. Also, since

M ( x , t ) = t 1 + t , 1 1 + t , x X , t > 0 ,

we have

lim n T M , j = n M x , α j t | 2 | k j = lim n lim m T M , j = n m M x , t | 2 | k j = lim n lim m t t + | 2 k | n , | 2 k | n t + | 2 k | n = ( 1 , 0 ) = 1 L * , x X , t > 0 .

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

P μ , ν ( f ( x ) - C ( x ) , t ) L * t t + | 2 k | , | 2 k | t + | 2 k | , x X , t > 0 .

Definition 6.9. A mapping f : X Y is said to be Ψ-approximately cubic II if

P f 3 x + y + f 3 x - y - 3 f x + y - 3 f x - y - 48 f x , t L Ψ x , y , t , x , y X , t > 0 . (6.11)

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 |3k| < α,

Ψ 3 - k x , 3 - k y , t L Ψ x , y , α t , x , y X , t > 0 , (6.12)

and

lim n T j = n M x , α j t | 3 | k j = 1 L , x X , t > 0 , (6.13)

then there exists a unique cubic mapping C : X Y such tha

P f x - C x , t T i = 1 M x , α i + 1 t | 3 | k i , (6.14)

for all × X and t > 0, where

M x , t : = T Ψ x , 0 , 2 t , Ψ 3 x , 0 , 2 t , . . . , Ψ 3 k - 1 x , 0 , 2 t , x X , t > 0 .

Proof. First, we show, by induction on j, that, for all x X, t > 0 and j ≥ 1,

P f 3 j x - 27 j f x , t L M j x , t : = T Ψ x , 0 , 2 t , . . . , Ψ 3 j - 1 x , 0 , 2 t . (6.15)

Put y = 0 in (6.11) to obtain

P f 3 x - 27 f x , t L Ψ x , 0 , 2 t , x X , t > 0 . (6.16)

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

P f 3 j + 1 x - 27 f 3 j x , t L Ψ 3 j x , 0 , 2 t , x X , t > 0 .

Since |27| ≤ 1, then we have

P f 3 j + 1 x - 27 j + 1 f x , t L T P f 3 j + 1 x - 27 f 3 j x , t , P 27 f 3 j x - 27 j + 1 f x , t = T P f 3 j + 1 x - 27 f 3 j x , t , P f 3 j x - 27 j f x , t | 27 | L T P f 3 j + 1 x - 27 f 3 j x , t , P f 3 j x - 27 j f x , t L T Ψ 3 j x , 0 , 2 t , M j x , t = M j + 1 x , t , x X .

Thus (6.15) holds for all j ≥ 1. In particular, it follows that

P f 3 k x - 27 k f x , t L M x , t , x X , t > 0 . (6.17)

Replacing x by 3-(kn+k)x in (6.17) and using inequality (6.12) we obtain

P f x 3 k n - 27 k f x 3 k n + k , t L M x 3 k n + k , t L M x , α n + 1 t , x X , t > 0 , n 0 . (6.18)

Then we have

P 3 3 k n f x 3 3 k n - 3 3 k n + 1 f x 3 3 k n + 1 , t L M x , α n + 1