# Nonlinear -Fuzzy stability of cubic functional equations

## 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.

## 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 [515].

The functional equations

(1.1)
(1.2)

and

(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 : XY, 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, 1719]. Furthermore, Mirmostafaee and Moslehian [20], Mirmostafaee et al. [21], Alsina [22], Miheţ and Radu [23] and others [2428] 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:

1. (i)

T (a, b) = T (b, a) (: commutativity);

2. (ii)

T (a, (T (b, c))) = T (T (a, b), c) (: associativity);

3. (iii)

T (a, 1) = a (: boundary condition);

4. (iv)

T (a, b) ≤ T (a, c) whenever bc (: 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},

${T}_{D}\phantom{\rule{0.3em}{0ex}}\left(a,\phantom{\rule{0.3em}{0ex}}b\right)\phantom{\rule{0.3em}{0ex}}:=\phantom{\rule{0.3em}{0ex}}\left\{\begin{array}{c}\text{min}\left(a,\phantom{\rule{0.3em}{0ex}}b\right),\phantom{\rule{0.3em}{0ex}}\mathsf{\text{if}}\phantom{\rule{0.3em}{0ex}}\text{max}\left(a,b\right)\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}1;\\ 0,\phantom{\rule{1em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\mathsf{\text{otherwise}}\mathsf{\text{.}}\end{array}\right\$

If T is a t-norm then ${x}_{T}^{\left(n\right)}$ is defined for every x [0,1] and n N {0} by 1, if n = 0 and $T\left({x}_{T}^{\left(n-1\right)}\phantom{\rule{0.3em}{0ex}},\phantom{\rule{0.3em}{0ex}}x\right),$ if n ≥ 1. A t-norm T is said to be of Hadžić-type (we denote by T ) if the family ${\left({x}_{T}^{\left(n\right)}\right)}_{n\in N}$ is equicontinuous at x = 1 (cf. [29]).

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

• the Sugeno-Weber family ${\left\{{T}_{\lambda }^{\mathsf{\text{SW}}}\right\}}_{\lambda \in \left[-1,\phantom{\rule{0.3em}{0ex}}\infty \right]}$ is defined by ${T}_{-1}^{\mathsf{\text{SW}}}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}{T}_{D}$, ${T}_{\infty }^{\mathsf{\text{SW}}}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}{T}_{P}$ and

${T}_{\lambda }^{\mathsf{\text{SW}}}\phantom{\rule{0.3em}{0ex}}\left(x,\phantom{\rule{0.3em}{0ex}}y\right)\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}\text{max}\phantom{\rule{0.3em}{0ex}}\left(0,\phantom{\rule{0.3em}{0ex}}\frac{x\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}y\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}1\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}\lambda xy}{1\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}\lambda }\right)$

if λ (-1, ∞).

• the Domby family ${\left\{{T}_{\lambda }^{\mathsf{\text{D}}}\right\}}_{\lambda \in \left[0,\phantom{\rule{0.3em}{0ex}}\infty \right],}$defined by TD, if λ = 0, TM, if λ = ∞ and

${T}_{\lambda }^{\mathsf{\text{D}}}\left(x,y\right)=\frac{1}{1+{\left({\left(\frac{1-x}{x}\right)}^{\lambda }+{\left(\frac{1-y}{y}\right)}^{\lambda }\right)}^{1/\lambda }}$

if λ (0, ∞).

• the Aczel-Alsina family ${\left\{{T}_{\lambda }^{\mathsf{\text{AA}}}\right\}}_{\lambda \in \left[0,\phantom{\rule{0.3em}{0ex}}\infty \right],}$ defined by TD, if λ = 0, TM, if λ = ∞ and

${T}_{\lambda }^{\mathsf{\text{AA}}}\left(x,\phantom{\rule{0.3em}{0ex}}y\right)\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}{\mathsf{\text{e}}}^{-{\left({\left|\text{log}\phantom{\rule{0.3em}{0ex}}x\right|}^{\lambda }\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{\left|\text{log}\phantom{\rule{0.3em}{0ex}}y\right|}^{\lambda }\right)}^{1/\lambda }}$

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

${\mathsf{\text{T}}}_{i\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}1}^{0}{x}_{i}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}1,{\mathsf{\text{T}}}_{i=1}^{n}{x}_{i}=\mathsf{\text{T}}\left({\mathsf{\text{T}}}_{i=1}^{n-1}{x}_{i},\phantom{\rule{0.3em}{0ex}}{x}_{n}\right)\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}T\left({x}_{1},\phantom{\rule{0.3em}{0ex}}.\phantom{\rule{0.3em}{0ex}}.\phantom{\rule{0.3em}{0ex}}.\phantom{\rule{0.3em}{0ex}},\phantom{\rule{0.3em}{0ex}}{x}_{n}\right).$

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

${\mathsf{\text{T}}}_{i=1}^{\infty }{x}_{i}=\underset{n\to \infty }{\text{lim}}\phantom{\rule{0.3em}{0ex}}{\mathsf{\text{T}}}_{i=1}^{n}\phantom{\rule{0.3em}{0ex}}{x}_{i}.$
(2.1)

The limit on the right side of (2.1) exists, since the sequence ${\left\{{\mathsf{\text{T}}}_{i=1}^{n}{x}_{i}\right\}}_{n\in ℕ}$ is non-increasing and bounded from below.

Proposition 2.1. [30] (1) For TT L the following implication holds:

$\underset{n\to \infty }{\text{lim}}{\mathsf{\text{T}}}_{i=1}^{\infty }{x}_{n+i}=1⇔\sum _{n=1}^{\infty }\left(1-{x}_{n}\right)<\infty .$
1. (2)

If T is of Hadžić-type then

$\underset{n\to \infty }{\text{lim}}{\mathsf{\text{T}}}_{i=1}^{\infty }{x}_{n+i}=1$

for every sequence {x n }nNin [0, 1] such that $\mathsf{\text{li}}{\mathsf{\text{m}}}_{n\to \infty }{x}_{n}=1$.

1. (3)

If $T\in {\left\{{T}_{\lambda }^{\mathsf{\text{AA}}}\right\}}_{\lambda \in \left(0,\infty \right)}\cup {\left\{{T}_{\lambda }^{\mathsf{\text{D}}}\right\}}_{\lambda \in \left(0,\infty \right)}$, then

$\underset{n\to \infty }{\text{lim}}{\mathsf{\text{T}}}_{i=1}^{\infty }{x}_{n+i}=1⇔\sum _{n=1}^{\infty }{\left(1-{x}_{n}\right)}^{\alpha }<\infty .$
2. (4)

If $T\in {\left\{{T}_{\lambda }^{\mathsf{\text{sw}}}\right\}}_{\lambda \in \left[-1,\infty \right)}$, then

$\underset{n\to \infty }{\text{lim}}{\mathsf{\text{T}}}_{i=1}^{\infty }{x}_{n+i}=1⇔\sum _{n=1}^{\infty }\left(1-{x}_{n}\right)<\infty .$

## 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 [3240, 4350]. 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, 5155]). 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 $\mathcal{L}=\left(L,{\le }_{L}\right)$ be a complete lattice and U be a non-empty set called universe. A -fuzzy set on U is defined as a mapping $\mathcal{A}:U\to L$. For any u U, $\mathcal{A}\left(u\right)$ represents the degree (in L) to which u satisfies .

Lemma 3.2 ([44]). Consider the set L* and operation ${\le }_{L*}$ defined by:

${L}^{*}=\left\{\left({x}_{1},\phantom{\rule{2.77695pt}{0ex}}{x}_{2}\right)\phantom{\rule{2.77695pt}{0ex}}:\phantom{\rule{2.77695pt}{0ex}}\left({x}_{1},\phantom{\rule{2.77695pt}{0ex}}{x}_{2}\right)\in {\left[0,1\right]}^{2}\phantom{\rule{2.77695pt}{0ex}}and\phantom{\rule{2.77695pt}{0ex}}{x}_{1}+{x}_{2}\le 1\right\},$

$\left({x}_{1},\phantom{\rule{2.77695pt}{0ex}}{x}_{2}\right){\le }_{{L}^{*}}\left({y}_{1},\phantom{\rule{2.77695pt}{0ex}}{y}_{2}\right)⇔{x}_{1}\le {y}_{1}$and ${x}_{2}\ge {y}_{2}$for all (x1, x2), (y1, y2) L*. Then (L*, ≤ L* ) is a complete lattice.

Definition 3.3 ([45]). An intuitionistic fuzzy set ${\mathcal{A}}_{\zeta ,\eta }$ on a universe U is an object ${\mathcal{A}}_{\zeta ,\eta }=\left\{\left({\zeta }_{\mathcal{A}}\left(u\right),\phantom{\rule{2.77695pt}{0ex}}{\eta }_{\mathcal{A}}\left(u\right)\right)\phantom{\rule{2.77695pt}{0ex}}:\phantom{\rule{2.77695pt}{0ex}}u\in U\right\}$, where, for all u U, ${\zeta }_{\mathcal{A}}\left(u\right)\in \left[0,1\right]$ and ${\eta }_{\mathcal{A}}\left(u\right)\in \left[0,1\right]$ are called the membership degree and the non-membership degree, respectively, of u in ${\mathcal{A}}_{\zeta ,\eta }$ and, furthermore, satisfy${\zeta }_{\mathcal{A}}\left(u\right)+{\eta }_{\mathcal{A}}\left(u\right)\le 1$.

In Section 2, we presented the classical definition of t-norm, which can be easily extended to any lattice $\mathcal{L}=\left(L,\phantom{\rule{2.77695pt}{0ex}}{\le }_{L}\right)$. Define first ${0}_{\mathcal{L}}=\text{inf}L$ and ${1}_{\mathcal{L}}=\text{sup}L$.

Definition 3.4. A triangular norm (t-norm) on is a mapping $\mathcal{T}:{L}^{2}\to L$ satisfying the following conditions:

1. (i)

for any $x\in L,\mathcal{T}\left(x,\phantom{\rule{2.77695pt}{0ex}}{1}_{\mathcal{L}}\right)=x$ (: boundary condition);

2. (ii)

for any $\left(x,\phantom{\rule{2.77695pt}{0ex}}y\right)\in {L}^{2},\mathcal{T}\left(x,\phantom{\rule{2.77695pt}{0ex}}y\right)=\mathcal{T}\left(y,\phantom{\rule{2.77695pt}{0ex}}x\right)$ (: commutativity);

3. (iii)

for any $\left(x,\phantom{\rule{2.77695pt}{0ex}}y,\phantom{\rule{2.77695pt}{0ex}}z\right)\in {L}^{3},\mathcal{T}\left(x,\phantom{\rule{2.77695pt}{0ex}}\mathcal{T}\left(y,\phantom{\rule{2.77695pt}{0ex}}z\right)\right)=\mathcal{T}\left(\mathcal{T}\left(x,\phantom{\rule{2.77695pt}{0ex}}y\right),\phantom{\rule{2.77695pt}{0ex}}z\right)$ (: associativity);

4. (iv)

for any $\left(x,\phantom{\rule{2.77695pt}{0ex}}{x}^{\prime },\phantom{\rule{2.77695pt}{0ex}}y,\phantom{\rule{2.77695pt}{0ex}}{y}^{\prime }\right)\in {L}^{4},\phantom{\rule{0.3em}{0ex}}x{\le }_{L}{x}^{\prime }$and $y{\le }_{L}{y}^{\prime }⇒\mathcal{T}\left(x,\phantom{\rule{2.77695pt}{0ex}}y\right){\le }_{L}\mathcal{T}\left({x}^{\prime },\phantom{\rule{2.77695pt}{0ex}}{y}^{\prime }\right)$(: monotonicity).

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

${\mathcal{T}}^{n}\left({x}_{\left(1\right)},\phantom{\rule{2.77695pt}{0ex}}\dots ,\phantom{\rule{2.77695pt}{0ex}}{x}_{\left(n+1\right)}\right)=\mathcal{T}\left({\mathcal{T}}^{n-1}\left({x}_{\left(1\right)},\phantom{\rule{2.77695pt}{0ex}}\dots ,\phantom{\rule{2.77695pt}{0ex}}{x}_{\left(n\right)}\right),\phantom{\rule{2.77695pt}{0ex}}{x}_{\left(n+1\right)}\right),\phantom{\rule{1em}{0ex}}\forall n\ge 2,\phantom{\rule{2.77695pt}{0ex}}{x}_{\left(i\right)}\in L.$

The t-norm M defined by

is a continuous t-norm.

Definition 3.5. A t-norm $\mathcal{T}$ 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

$\mathcal{T}\left(x,\phantom{\rule{2.77695pt}{0ex}}y\right)=\left(T\left({x}_{1},\phantom{\rule{2.77695pt}{0ex}}{y}_{1}\right),\phantom{\rule{2.77695pt}{0ex}}S\left({x}_{2},\phantom{\rule{2.77695pt}{0ex}}{y}_{2}\right)\right),\phantom{\rule{1em}{0ex}}\forall x=\left({x}_{1},\phantom{\rule{2.77695pt}{0ex}}{x}_{2}\right),\phantom{\rule{2.77695pt}{0ex}}y=\left({y}_{1},\phantom{\rule{2.77695pt}{0ex}}{y}_{2}\right)\in {L}^{*}.$

Definition 3.6. A negation on is any strictly decreasing mapping $\mathcal{N}:L\to L$satisfying $\mathcal{N}\left({0}_{\mathcal{L}}\right)={1}_{\mathcal{L}}$and $\mathcal{N}\left({1}_{\mathcal{L}}\right)={0}_{\mathcal{L}}$. If $\mathcal{N}\left(\mathcal{N}\left(x\right)\right)=x$for all x L, then is called an involutive negation.

In this article, let $\mathcal{N}:L\to L$ 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 $\left(V,\phantom{\rule{2.77695pt}{0ex}}\mathcal{P},\phantom{\rule{2.77695pt}{0ex}}\mathcal{T}\right)$ is said to be a -fuzzy normed space if V is a vector space, $\mathcal{T}$ is a continuous t-norm on and is a -fuzzy set on $V×\right]0,+\infty \left[$ satisfying the following conditions: for all x, y V and t, s ]0, +∞[,

1. (i)

${0}_{\mathcal{L}}{<}_{L}\mathcal{P}\left(x,\phantom{\rule{2.77695pt}{0ex}}t\right);$

2. (ii)

$\mathcal{P}\left(x,\phantom{\rule{2.77695pt}{0ex}}t\right)={1}_{\mathcal{L}}$ if and only if x = 0;

3. (iii)

$\mathcal{P}\left(\alpha x,\phantom{\rule{2.77695pt}{0ex}}t\right)=\mathcal{P}\left(x,\phantom{\rule{2.77695pt}{0ex}}\frac{t}{|\alpha |}\right)$ for all α ≠ 0;

4. (iv)

$\mathcal{T}\left(\mathcal{P}\left(x,\phantom{\rule{2.77695pt}{0ex}}t\right),\phantom{\rule{2.77695pt}{0ex}}\mathcal{P}\left(y,\phantom{\rule{2.77695pt}{0ex}}s\right)\right){\le }_{L}\mathcal{P}\left(x+y,\phantom{\rule{2.77695pt}{0ex}}t+s\right)$;

5. (v)

$\mathcal{P}\left(x,·\right):\right]0,\infty \left[\to L$ is continuous;

6. (vi)

$\underset{t\to 0}{\text{lim}}\mathcal{P}\left(x,\phantom{\rule{2.77695pt}{0ex}}t\right)={0}_{\mathcal{L}}$ and ${\mathrm{lim}}_{t\to \infty }\mathcal{P}\left(x,\phantom{\rule{2.77695pt}{0ex}}t\right)={1}_{\mathcal{L}}$.

In this case, is called a -fuzzy norm. If $\mathcal{P}={\mathcal{P}}_{\mu ,\nu }$is an intuitionistic fuzzy set and the t-norm $\mathcal{T}$ is t-representable, then the 3-tuple $\left(V,{\mathcal{P}}_{\mu ,v,}\mathcal{T}\right)$ 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 $\epsilon \in L\\left\{{0}_{\mathcal{L}}\right\}$ and t > 0, there exists a positive integer n0 such that

$\mathcal{N}\left(\epsilon \right){<}_{L}\mathcal{P}\left({x}_{n+p}-{x}_{n},\phantom{\rule{2.77695pt}{0ex}}t\right),\phantom{\rule{1em}{0ex}}\forall n\ge {n}_{0},\phantom{\rule{2.77695pt}{0ex}}p>0.$
1. (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.

2. (3)

The sequence {x n } is said to be convergent to x V in the -fuzzy normed space $\left(V,\phantom{\rule{2.77695pt}{0ex}}\mathcal{P},\phantom{\rule{2.77695pt}{0ex}}\mathcal{T}\right)$ (denoted by ${x}_{n}\underset{\mathcal{P}}{\overset{}{\to }}x\right)$if $\mathcal{P}\left({x}_{n}-x,\phantom{\rule{2.77695pt}{0ex}}t\right)\to {1}_{\mathcal{L}}$, whenever n → + ∞ for all t > 0.

Lemma 3.9 ([46]). Let$\mathcal{P}$ be a -fuzzy norm on V. Then

1. (1)

For all × V, $\mathcal{P}\left(x,t\right)$ is nondecreasing with respect to t.

2. (2)

$\mathcal{P}\left(x-y,\phantom{\rule{2.77695pt}{0ex}}t\right)=\mathcal{P}\left(y-x,\phantom{\rule{2.77695pt}{0ex}}t\right)$ for all x, y V and t ]0, +∞ [.

Definition 3.10. Let $\left(V,\phantom{\rule{2.77695pt}{0ex}}\mathcal{P},\phantom{\rule{2.77695pt}{0ex}}\mathcal{T}\right)$ be a -fuzzy normed space. For any t ]0, +∞[, we define the open ball B(x, r, t) with center x V and radius $r\in L\\left\{{0}_{\mathcal{L}},\phantom{\rule{2.77695pt}{0ex}}{1}_{\mathcal{L}}\right\}$ as

$B\left(x,\phantom{\rule{2.77695pt}{0ex}}r,\phantom{\rule{2.77695pt}{0ex}}t\right)=\left\{y\in V\phantom{\rule{2.77695pt}{0ex}}:\phantom{\rule{2.77695pt}{0ex}}\mathcal{N}\left(r\right){<}_{L}\mathcal{P}\left(x-y,\phantom{\rule{2.77695pt}{0ex}}t\right)\right\}.$

## 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$\left(Y,\phantom{\rule{2.77695pt}{0ex}}\mathcal{P},\phantom{\rule{2.77695pt}{0ex}}\mathcal{T}\right)$ 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:

$\begin{array}{ll}\hfill \mathcal{P}\left(3f\left(x+3y\right)& +f\left(3x-y\right)-15f\left(x+y\right)-15f\left(x-y\right)-80f\left(y\right),t\right)\phantom{\rule{2em}{0ex}}\\ {\ge }_{L}Q\left(x,y,t\right),\phantom{\rule{1em}{0ex}}\forall x,y\in X,\phantom{\rule{2.77695pt}{0ex}}t>0.\phantom{\rule{2em}{0ex}}\end{array}$
(4.1)

If

${\mathcal{T}}_{i=1}^{\infty }\left(Q\left({3}^{n+i-1}x,0,{3}^{3n+2i+1}t\right)\right)={1}_{\mathcal{L}},\phantom{\rule{1em}{0ex}}\forall x\in X,\phantom{\rule{2.77695pt}{0ex}}t>0,$

and

$\underset{n\to \infty }{\text{lim}}Q\left({3}^{n}x,{3}^{n}y,{3}^{3n}t\right)={1}_{\mathcal{L}},\phantom{\rule{1em}{0ex}}\forall x,y\in X,\phantom{\rule{2.77695pt}{0ex}}t>0,$

then there exists a unique cubic mapping C : XY such that

$\mathcal{P}\left(f\left(x\right)-C\left(x\right),t\right){\ge }_{L}{\mathcal{T}}_{i=1}^{\infty }\left(Q\left({3}^{i-1}x,0,{3}^{2i+2}t\right)\right),\phantom{\rule{1em}{0ex}}\forall x\in X,\phantom{\rule{2.77695pt}{0ex}}t>0.$
(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

$\mathcal{P}\left(\frac{f\left(3x\right)}{27}-f\left(x\right),t\right){\ge }_{{L}^{*}}Q\left(x,0,{3}^{3}t\right),\phantom{\rule{1em}{0ex}}\forall x\in X,\phantom{\rule{2.77695pt}{0ex}}t>0.$

Therefore, it follows that

$\mathcal{P}\left(\frac{f\left({3}^{k+1}x\right)}{{3}^{3\left(k+1\right)}}-\frac{f\left({3}^{k}x\right)}{{3}^{3k}},\frac{t}{{3}^{k+1}}\right){\ge }_{L}Q\left({3}^{k}x,0,{3}^{2\left(k+1\right)}t\right).\phantom{\rule{1em}{0ex}}\forall k\ge 1,\phantom{\rule{2.77695pt}{0ex}}t>0.$

By the triangle inequality, it follows that

$\mathcal{P}\left(\frac{f\left({3}^{n}x\right)}{{27}^{n}}-f\left(x\right),t\right){\ge }_{L}{\mathcal{T}}_{i=1}^{n}\left(Q\left({3}^{i-1}x,0,{3}^{2i+2}t\right)\right),\phantom{\rule{1em}{0ex}}\forall x\in X,\phantom{\rule{2.77695pt}{0ex}}t>0.$
(4.3)

In order to prove the convergence of the sequence $\left\{\frac{f\left({3}^{n}x\right)}{{27}^{n}}\right\}$, we replace x with 3mx in (4.3) to find that, for all m, n > 0,

$\mathcal{P}\left(\frac{f\left({3}^{n+m}x\right)}{{27}^{\left(n+m\right)}}-\frac{f\left({3}^{m}x\right)}{{27}^{m}},t\right){\ge }_{L}{\mathcal{T}}_{i=1}^{n}\left(Q\left({3}^{i+m-1}x,0,{3}^{2i+3m+2}t\right)\right),\phantom{\rule{1em}{0ex}}\forall x\in X,\phantom{\rule{2.77695pt}{0ex}}t>0.$

Since the right-hand side of the inequality tends to ${1}_{\mathcal{L}}$ as m tends to infinity, the sequence $\left\{\frac{f\left({3}^{n}x\right)}{{3}^{3n}}\right\}$ is a Cauchy sequence. Thus, we may define $C\left(x\right)=\underset{n\to \infty }{\text{lim}}\frac{f\left({3}^{n}x\right)}{{3}^{3n}}$ 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 . Hence it follows from (4.2) that

$\begin{array}{c}\mathcal{P}\left(C\left(x\right)-{C}^{\prime }\left(x\right),t\right)\\ {\ge }_{L}\mathcal{P}\left(C\left({3}^{n}x\right)-{C}^{\prime }\left({3}^{n}x\right),{3}^{3n}t\right)\\ {\ge }_{L}\mathcal{T}\left(\mathcal{P}\left(C\left({3}^{n}x\right)-f\left({3}^{n}x\right),{3}^{3n-1}t\right),\mathcal{P}\left(f\left({3}^{n}x\right)-{C}^{\prime }\left({3}^{n}x\right),{2}^{3n-1}t\right)\right)\\ {\ge }_{L}\mathcal{T}\left({\mathcal{T}}_{i=1}^{\infty }\left(Q\left({3}^{n+i-1}x,0,{3}^{3n+2i+1}t\right)\right),{\mathcal{T}}_{i=1}^{\infty }\left(Q\left({3}^{n+i-1}x,0,{3}^{3n+2i+1}t\right)\right)\\ =\mathcal{T}\left({1}_{\mathcal{L}},{1}_{\mathcal{L}}\right)={1}_{\mathcal{L}},\phantom{\rule{1em}{0ex}}\forall x\in X,\phantom{\rule{2.77695pt}{0ex}}t>0,\end{array}$

which proves the uniqueness of C. This completes the proof.

Theorem 4.2. Let X be a linear space and$\left(Y,\mathcal{P},\mathcal{T}\right)$ be a complete -fuzzy normed space. If f : XY is a mapping with f (0) = 0 and Q is a -fuzzy set on X2 × (0, ∞) with the following property:

$\mathcal{P}\left(f\left(2x+y\right)+f\left(2x-y\right)-2f\left(x+y\right)-2f\left(x-y\right)-12f\left(x\right),t\right)$
(4.4)
${\ge }_{L}Q\left(x,y,t\right),\phantom{\rule{1em}{0ex}}\forall x,y\in X,\phantom{\rule{2.77695pt}{0ex}}t>0.$

If

${\mathcal{T}}_{i=1}^{\infty }\left(Q\left({2}^{n+i-1}x,0,{2}^{3n+2i+1}t\right)\right)={1}_{\mathcal{L}},\phantom{\rule{1em}{0ex}}\forall x\in X,\phantom{\rule{2.77695pt}{0ex}}t>0,$

and

$\underset{n\to \infty }{\text{lim}}Q\left({2}^{n}x,{2}^{n}y,{2}^{3n}t\right)={1}_{\mathcal{L}},\phantom{\rule{1em}{0ex}}\forall x,y\in X,\phantom{\rule{2.77695pt}{0ex}}t>0,$

then there exists a unique cubic mapping C : XY such that

$\mathcal{P}\left(f\left(x\right)-C\left(x\right),t\right){\ge }_{L}{\mathcal{T}}_{i=1}^{\infty }\left(Q\left({2}^{i-1}x,0,{2}^{2i+1}t\right)\right),\phantom{\rule{1em}{0ex}}\forall x\in X,\phantom{\rule{2.77695pt}{0ex}}t>0.$
(4.5)

Proof. We omit the proof because it is similar as the last theorem and see [28].

Corollary 4.3. Let$\left(X,{\mathcal{P}}^{\prime },\mathcal{T}\right)$be -fuzzy normed space and$\left(Y,\mathcal{P},\mathcal{T}\right)$ be a complete -fuzzy normed space. If f : XY is a mapping such that

$\mathcal{P}\left(f\left(2x+y\right)+f\left(2x-y\right)-2f\left(x+y\right)-2f\left(x-y\right)-12f\left(x\right),t\right)$
${\ge }_{L}{\mathcal{P}}^{\prime }\left(x+y,t\right),\phantom{\rule{1em}{0ex}}\forall x,y\in X,\phantom{\rule{2.77695pt}{0ex}}t>0,$

and

$\underset{n\to \infty }{\text{lim}}{\mathcal{T}}_{i=1}^{\infty }\left({\mathcal{P}}^{\prime }\left(x,{2}^{2n+i+2}t\right)\right)={1}_{\mathcal{L}},\phantom{\rule{1em}{0ex}}\forall x\in X,\phantom{\rule{2.77695pt}{0ex}}t>0,$

then there exists a unique cubic mapping C : XY such that

$\mathcal{P}\left(f\left(x\right)-C\left(x\right),t\right){\ge }_{L}{\mathcal{T}}_{i=1}^{\infty }\left({\mathcal{P}}^{\prime }\left(x,{2}^{i+2}t\right)\right),\phantom{\rule{1em}{0ex}}\forall x\in X,\phantom{\rule{2.77695pt}{0ex}}t>0.$

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, $\left(X,{\mathcal{P}}_{\mu ,\nu },{\mathcal{T}}_{M}\right)$ be an intuitionistic fuzzy normed space in which ${\mathcal{T}}_{\mathbf{M}}\left(a,b\right)=\left(\text{min}\left\{{a}_{1},{b}_{1}\right\},\text{max}\left\{{a}_{2},{b}_{2}\right\}\right)$ and

${\mathcal{P}}_{\mu ,\nu }\left(x,t\right)=\left(\frac{t}{t+||x||},\frac{||x||}{t+||x||}\right),\phantom{\rule{1em}{0ex}}\forall x\in X,\phantom{\rule{2.77695pt}{0ex}}t>0,$

also $\left(Y,{\mathcal{P}}_{\mu ,\nu },{\mathcal{T}}_{M}\right)$ be a complete intuitionistic fuzzy normed space. Define a mapping f : XY by f (x) = x3 + x0 for all x X, where x0 is a unit vector in X. A straightforward computation shows that

$\begin{array}{c}{\mathcal{P}}_{\mu ,\nu }\left(f\left(2x+y\right)+f\left(2x-y\right)-2f\left(x+y\right)-2f\left(x-y\right)-12f\left(x\right),t\right)\\ \phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{\ge }_{{L}^{*}}{\mathcal{P}}_{\mu ,\nu }\left(x+y,t\right),\phantom{\rule{1em}{0ex}}\forall x,y\in X,\phantom{\rule{2.77695pt}{0ex}}t>0.\end{array}$

Also, we have

$\begin{array}{ll}\hfill \underset{n\to \infty }{\text{lim}}{\mathcal{T}}_{M,i=1}^{\infty }\left({\mathcal{P}}_{\mu ,\nu }\left(x,{2}^{2n+i+1}t\right)\right)& =\underset{n\to \infty }{\text{lim}}\underset{m\to \infty }{\text{lim}}{\mathcal{T}}_{M,i=1}^{m}\left({\mathcal{P}}_{\mu ,\nu }\left(x,{2}^{2n+i+1}t\right)\right)\phantom{\rule{2em}{0ex}}\\ =\underset{n\to \infty }{\text{lim}}\underset{m\to \infty }{\text{lim}}{\mathcal{P}}_{\mu ,\nu }\left(x,{2}^{2n+2}t\right)\phantom{\rule{2em}{0ex}}\\ =\underset{n\to \infty }{\text{lim}}{\mathcal{P}}_{\mu ,\nu }\left(x,{2}^{2n+2}t\right)\phantom{\rule{2em}{0ex}}\\ ={1}_{{L}^{*}}.\phantom{\rule{2em}{0ex}}\end{array}$

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

${\mathcal{P}}_{\mu ,\nu }\left(f\left(x\right)-C\left(x\right),t\right){\ge }_{{L}^{*}}{\mathcal{P}}_{\mu ,\nu }\left(x,{2}^{2}t\right),\phantom{\rule{1em}{0ex}}\forall x\in X,\phantom{\rule{2.77695pt}{0ex}}t>0.$

## 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 $|\cdot |:\mathcal{K}\to \left[0,+\infty \left[$such that, for any a, b ,

1. (i)

|a| ≥ 0 and equality holds if and only if a = 0;

2. (ii)

|ab| = |a| |b|;

3. (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 $\left(V,\mathcal{P},\mathcal{T}\right)$, 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, +∞[,

1. (i)

${0}_{\mathcal{L}}{<}_{L}\mathcal{P}\left(x,t\right)$;

2. (ii)

$\mathcal{P}\left(x,t\right)={1}_{\mathcal{L}}$ if and only if x = 0;

3. (iii)

$\mathcal{P}\left(\alpha x,t\right)=\mathcal{P}\left(x,\frac{t}{|\alpha |}\right)$ for all α ≠ 0;

4. (vi)

$\mathcal{T}\left(\mathcal{P}\left(x,t\right),\mathcal{P}\left(y,s\right)\right){\le }_{L}\mathcal{P}\left(x+y,\text{max}\left\{t,s\right\}\right)$;

5. (v)

$\mathcal{P}\left(x,\cdot \right):\right]0,\infty \left[\to L$ is continuous;

6. (vi)

$\underset{t\to 0}{\text{lim}}\mathcal{P}\left(x,t\right)={0}_{\mathcal{L}}$ and $\underset{t\to \infty }{\text{lim}}\mathcal{P}\left(x,t\right)={1}_{\mathcal{L}}$.

Example 5.3. Let (X, || · ||) be a non-Archimedean normed linear space. Then the triple $\left(X,\mathcal{P},\text{min}\right)$, where

$\mathcal{P}\left(x,\phantom{\rule{2.77695pt}{0ex}}t\right)=\left\{\begin{array}{cc}\hfill 0,\hfill & \hfill \mathsf{\text{if}}\phantom{\rule{2.77695pt}{0ex}}t\le ||x||;\hfill \\ \hfill 1,\hfill & \hfill \mathsf{\text{if}}\phantom{\rule{2.77695pt}{0ex}}t>||x||,\hfill \end{array}\right\$

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 ${\mathcal{T}}_{M}\left(a,\phantom{\rule{2.77695pt}{0ex}}b\right)=\left(\text{min}\left\{{a}_{1},\phantom{\rule{2.77695pt}{0ex}}{b}_{1}\right\},\phantom{\rule{2.77695pt}{0ex}}\text{max}\left\{{a}_{2},\phantom{\rule{2.77695pt}{0ex}}{b}_{2}\right\}\right)$ for all a = (a1, a2), b = (b1, b2) L* and ${\mathcal{P}}_{\mu ,\nu }$ be the intuitionistic fuzzy set on X × ]0, +∞[ defined as follows:

${\mathcal{P}}_{\mu ,\nu }\left(x,\phantom{\rule{2.77695pt}{0ex}}t\right)=\left(\frac{t}{t+||x||},\phantom{\rule{2.77695pt}{0ex}}\frac{||x||}{t+||x||}\right),\phantom{\rule{1em}{0ex}}\forall x\in X,\phantom{\rule{2.77695pt}{0ex}}t\in {ℝ}^{+}.$

Then $\left(X,\phantom{\rule{2.77695pt}{0ex}}{\mathcal{P}}_{\mu ,\nu },\phantom{\rule{2.77695pt}{0ex}}{\mathcal{T}}_{M}\right)$ 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 $\left(Y,\phantom{\rule{2.77695pt}{0ex}}\mathcal{P},\phantom{\rule{2.77695pt}{0ex}}\mathcal{T}\right)$ 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,

$\mathrm{\Psi }\left(cx,\phantom{\rule{2.77695pt}{0ex}}cx,\phantom{\rule{2.77695pt}{0ex}}t\right){\ge }_{L}\mathrm{\Psi }\left(x,\phantom{\rule{2.77695pt}{0ex}}x,\phantom{\rule{2.77695pt}{0ex}}\frac{t}{|c|}\right),\phantom{\rule{1em}{0ex}}\forall x\in X,\phantom{\rule{2.77695pt}{0ex}}c\ne 0$

and

$\underset{t\to \infty }{\text{lim}}\mathrm{\Psi }\left(x,\phantom{\rule{2.77695pt}{0ex}}y,\phantom{\rule{2.77695pt}{0ex}}t\right)={1}_{\mathcal{L}},\phantom{\rule{1em}{0ex}}\forall x,\phantom{\rule{2.77695pt}{0ex}}y\in X,\phantom{\rule{2.77695pt}{0ex}}t>0.$

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

$\begin{array}{c}\mathcal{P}\left(3f\left(x+3y\right)+f\left(3x-y\right)-15f\left(x+y\right)-15f\left(x-y\right)-80f\left(y\right),\phantom{\rule{2.77695pt}{0ex}}t\right)\\ \phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{\ge }_{L}\mathrm{\Psi }\left(x,\phantom{\rule{2.77695pt}{0ex}}y,\phantom{\rule{2.77695pt}{0ex}}t\right),\phantom{\rule{1em}{0ex}}\forall x,\phantom{\rule{2.77695pt}{0ex}}y\in X,\phantom{\rule{2.77695pt}{0ex}}t>0.\end{array}$
(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 $\left(Y,\phantom{\rule{2.77695pt}{0ex}}\mathcal{P},\phantom{\rule{2.77695pt}{0ex}}\mathcal{T}\right)$ be a non-Archimedean -fuzzy Banach space over : Let f: XY be a Ψ-approximately cubic mapping. If there exist a α (α > 0) and an integer k, k ≥ 2 with | 3k| < α and | 3| ≠ 1 such that

$\mathrm{\Psi }\left({3}^{-k}x,\phantom{\rule{2.77695pt}{0ex}}{3}^{-k}y,\phantom{\rule{2.77695pt}{0ex}}t\right){\ge }_{L}\mathrm{\Psi }\left(x,\phantom{\rule{2.77695pt}{0ex}}y,\phantom{\rule{2.77695pt}{0ex}}\alpha t\right),\phantom{\rule{1em}{0ex}}\forall x,y\in X,\phantom{\rule{2.77695pt}{0ex}}t>0,$
(6.2)

and

$\underset{n\to \infty }{\text{lim}}{\mathcal{T}}_{j=n}^{\infty }\mathcal{M}\left(x,\phantom{\rule{2.77695pt}{0ex}}\frac{{\alpha }^{j}t}{|3{|}^{kj}}\right)={1}_{\mathcal{L}},\phantom{\rule{1em}{0ex}}\forall x\in X,\phantom{\rule{2.77695pt}{0ex}}t>0,$

then there exists a unique cubic mapping C : XY such that

$\mathcal{P}\left(f\left(x\right)-C\left(x\right),\phantom{\rule{2.77695pt}{0ex}}t\right)\ge {\mathcal{T}}_{i=1}^{\infty }\mathcal{M}\left(x,\frac{{\alpha }^{i+1}t}{|3{|}^{ki}}\right),\phantom{\rule{1em}{0ex}}\forall x\in X,\phantom{\rule{2.77695pt}{0ex}}t>0,$
(6.3)

where

$\mathcal{M}\left(x,\phantom{\rule{2.77695pt}{0ex}}t\right)\phantom{\rule{2.77695pt}{0ex}}:=\mathcal{T}\left(\mathrm{\Psi }\left(x,\phantom{\rule{2.77695pt}{0ex}}0,\phantom{\rule{2.77695pt}{0ex}}t\right),\phantom{\rule{2.77695pt}{0ex}}\mathrm{\Psi }\left(3x,\phantom{\rule{2.77695pt}{0ex}}0,\phantom{\rule{2.77695pt}{0ex}}t\right),\phantom{\rule{2.77695pt}{0ex}}\dots ,\phantom{\rule{2.77695pt}{0ex}}\mathrm{\Psi }\left({3}^{k-1}x,\phantom{\rule{2.77695pt}{0ex}}0,\phantom{\rule{2.77695pt}{0ex}}t\right)\right),\phantom{\rule{1em}{0ex}}\forall x\in X,\phantom{\rule{2.77695pt}{0ex}}t>0.$

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

$\mathcal{P}\left(f\left({3}^{j}x\right)-2{7}^{j}f\left(x\right),\phantom{\rule{2.77695pt}{0ex}}t\right){\ge }_{L}{\mathcal{M}}_{j}\left(x,\phantom{\rule{2.77695pt}{0ex}}t\right):=\mathcal{T}\left(\mathrm{\Psi }\left(x,\phantom{\rule{2.77695pt}{0ex}}0,\phantom{\rule{2.77695pt}{0ex}}t\right),\phantom{\rule{2.77695pt}{0ex}}\dots ,\phantom{\rule{2.77695pt}{0ex}}\mathrm{\Psi }\left({3}^{j-1}x,\phantom{\rule{2.77695pt}{0ex}}0,\phantom{\rule{2.77695pt}{0ex}}t\right)\right).$
(6.4)

Putting y = 0 in (6.1), we obtain

$\mathcal{P}\left(f\left(3x\right)-27f\left(x\right),\phantom{\rule{2.77695pt}{0ex}}t\right){\ge }_{L}\mathrm{\Psi }\left(x,\phantom{\rule{2.77695pt}{0ex}}0,\phantom{\rule{2.77695pt}{0ex}}t\right),\phantom{\rule{1em}{0ex}}\forall x\in X,\phantom{\rule{2.77695pt}{0ex}}t>0.$

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

$\mathcal{P}\left(f\left({3}^{j+1}x\right)-27f\left({3}^{j}x\right),\phantom{\rule{2.77695pt}{0ex}}t\right){\ge }_{L}\mathrm{\Psi }\left({3}^{j}x,\phantom{\rule{2.77695pt}{0ex}}0,\phantom{\rule{2.77695pt}{0ex}}t\right),\phantom{\rule{1em}{0ex}}\forall x\in X,\phantom{\rule{2.77695pt}{0ex}}t>0.$

Since | 27| ≤ 1, it follows that

$\begin{array}{c}\mathcal{P}\left(f\left({3}^{j+1}x\right)-2{7}^{j+1}f\left(x\right),\phantom{\rule{2.77695pt}{0ex}}t\right)\\ {\ge }_{L}\mathcal{T}\left(\mathcal{P}\left(f\left({3}^{j+1}x\right)-27f\left({3}^{j}x\right),\phantom{\rule{2.77695pt}{0ex}}t\right),\phantom{\rule{2.77695pt}{0ex}}\mathcal{P}\left(8f\left({3}^{j}x\right)-{27}^{j+1}f\left(x\right),\phantom{\rule{2.77695pt}{0ex}}t\right)\right)\\ =\mathcal{T}\left(\mathcal{P}\left(f\left({2}^{j+1}x\right)-8f\left({2}^{j}x\right),\phantom{\rule{2.77695pt}{0ex}}t\right),\phantom{\rule{2.77695pt}{0ex}}\mathcal{P}\left(f\left({3}^{j}x\right)-{27}^{j}f\left(x\right),\phantom{\rule{2.77695pt}{0ex}}\frac{t}{|27|}\right)\right)\\ {\ge }_{L}\mathcal{T}\left(\mathcal{P}\left(f\left({3}^{j+1}x\right)-27f\left({3}^{j}x\right),\phantom{\rule{2.77695pt}{0ex}}t\right),\phantom{\rule{2.77695pt}{0ex}}\mathcal{P}\left(f\left({3}^{j}x\right)-{27}^{j}f\left(x\right),\phantom{\rule{2.77695pt}{0ex}}t\right)\right)\\ {\ge }_{L}\mathcal{T}\left(\mathrm{\Psi }\left({3}^{j}x,\phantom{\rule{2.77695pt}{0ex}}0,\phantom{\rule{2.77695pt}{0ex}}t\right),\phantom{\rule{2.77695pt}{0ex}}{\mathcal{M}}_{j}\left(x,\phantom{\rule{2.77695pt}{0ex}}t\right)\right)\\ ={\mathcal{M}}_{j+1}\left(x,\phantom{\rule{2.77695pt}{0ex}}t\right),\phantom{\rule{1em}{0ex}}\forall x\in X,\phantom{\rule{2.77695pt}{0ex}}t>0.\end{array}$

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

$\mathcal{P}\left(f\left({3}^{k}x\right)-2{7}^{k}f\left(x\right),\phantom{\rule{2.77695pt}{0ex}}t\right){\ge }_{L}\mathcal{M}\left(x,\phantom{\rule{2.77695pt}{0ex}}t\right),\phantom{\rule{1em}{0ex}}\forall x\in X,\phantom{\rule{2.77695pt}{0ex}}t>0.$
(6.5)

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

$\begin{array}{ll}\hfill \mathcal{P}\left(f\left(\frac{x}{{3}^{kn}}\right)-{27}^{k}f\left(\frac{x}{{3}^{kn+k}}\right),\phantom{\rule{2.77695pt}{0ex}}t\right)& {\ge }_{L}\mathcal{M}\left(\frac{x}{{3}^{kn+k}},\phantom{\rule{2.77695pt}{0ex}}t\right)\phantom{\rule{2em}{0ex}}\\ {\ge }_{L}\mathcal{M}\left(x,\phantom{\rule{2.77695pt}{0ex}}{\alpha }^{n+1}t\right)\phantom{\rule{1em}{0ex}}\forall x\in X,\phantom{\rule{2.77695pt}{0ex}}t>0,\phantom{\rule{2.77695pt}{0ex}}n\ge 0\phantom{\rule{2em}{0ex}}\end{array}$

and so

$\begin{array}{c}\mathcal{P}\left({\left({3}^{3k}\right)}^{n}f\left(\frac{x}{{\left({3}^{k}\right)}^{n}}\right)-{\left({3}^{3k}\right)}^{n+1}f\left(\frac{x}{{\left({3}^{k}\right)}^{n+1}}\right),\phantom{\rule{2.77695pt}{0ex}}t\right)\\ {\ge }_{L}\mathcal{M}\left(x,\phantom{\rule{2.77695pt}{0ex}}\frac{{\alpha }^{n+1}}{|{\left({3}^{3k}\right)}^{n}|}t\right)\\ {\ge }_{L}\mathcal{M}\left(x,\phantom{\rule{2.77695pt}{0ex}}\frac{{\alpha }^{n+1}}{|{\left({3}^{k}\right)}^{n}|}t\right),\phantom{\rule{1em}{0ex}}\forall x\in X,\phantom{\rule{2.77695pt}{0ex}}t>0,\phantom{\rule{2.77695pt}{0ex}}n\ge 0.\end{array}$

$\begin{array}{c}\mathcal{P}\left({\left({3}^{3k}\right)}^{n}f\left(\frac{x}{{\left({3}^{k}\right)}^{n}}\right)-{\left({3}^{3k}\right)}^{n+p}f\left(\frac{x}{{\left({3}^{k}\right)}^{n+p}}\right),\phantom{\rule{2.77695pt}{0ex}}t\right)\\ {\ge }_{L}{\mathcal{T}}_{j=n}^{n+p}\left(\mathcal{P}\left({\left({3}^{3k}\right)}^{j}f\left(\frac{x}{{\left({3}^{k}\right)}^{j}}\right)-{\left({3}^{3k}\right)}^{j+p}f\left(\frac{x}{{\left({3}^{k}\right)}^{j+p}}\right),\phantom{\rule{2.77695pt}{0ex}}t\right)\right)\\ {\ge }_{L}{\mathcal{T}}_{j=n}^{n+p}\mathcal{M}\left(x,\phantom{\rule{2.77695pt}{0ex}}\frac{{\alpha }^{j+1}}{|{\left({3}^{k}\right)}^{j}|}t\right),\phantom{\rule{1em}{0ex}}\forall x\in X,\phantom{\rule{2.77695pt}{0ex}}t>0,\phantom{\rule{2.77695pt}{0ex}}n\ge 0.\end{array}$

Since $\underset{}{\text{lim}\text{n}\to \infty }{\mathcal{T}}_{j=n}^{\infty }\mathcal{M}\left(x,\frac{{\alpha }^{j+1}}{|{\left({3}^{k}\right)}^{j}|}t\right)={1}_{\mathcal{L}}$ for all x X and t > 0, ${\left\{{\left({3}^{3k}\right)}^{n}f\left(\frac{x}{{\left({3}^{k}\right)}^{n}}\right)\right\}}_{n\in ℕ}$ is a Cauchy sequence in the non-Archimedean -fuzzy Banach space $\left(Y,\mathcal{P},\mathcal{T}\right)$. Hence we can define a mapping C : XY such that

$\underset{n\to \infty }{\text{lim}}\mathcal{P}\left({\left({3}^{3k}\right)}^{n}f\left(\frac{x}{{\left({3}^{k}\right)}^{n}}\right)-C\left(x\right),t\right)={1}_{\mathcal{L}},\phantom{\rule{1em}{0ex}}\forall x\in X,\phantom{\rule{2.77695pt}{0ex}}t>0.$
(6.6)

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

$\begin{array}{l}\mathcal{P}\left(f\left(x\right)-{\left({3}^{3k}\right)}^{n}f\left(\frac{x}{{\left({3}^{k}\right)}^{n}}\right),t\right)\\ =\left(\sum _{i=0}^{n-1}{\left({3}^{3k}\right)}^{i}f\left(\frac{x}{{\left({3}^{k}\right)}^{i}}\right)-{\left({3}^{3k}\right)}^{i+1}f\left(\frac{x}{{\left({3}^{k}\right)}^{i+1}}\right),t\right)\\ {\ge }_{L}{\mathcal{T}}_{i=0}^{n-1}\left(\left({\left({3}^{3k}\right)}^{i}f\left(\frac{x}{{\left({3}^{k}\right)}^{i}}\right)-{\left({3}^{3k}\right)}^{i+1}f\left(\frac{x}{{\left({3}^{k}\right)}^{i+1}}\right),t\right)\right)\\ {\ge }_{L}{\mathcal{T}}_{i=0}^{n-1}ℳ\left(x,\frac{{\alpha }^{i+1}t}{|{3}^{k}{|}^{i}}\right)\end{array}$

and so

$\begin{array}{c}\mathcal{P}\left(f\left(x\right)-C\left(x\right),\phantom{\rule{2.77695pt}{0ex}}t\right)\\ \phantom{\rule{1em}{0ex}}{\ge }_{L}\mathcal{T}\left(\mathcal{P}\left(f\left(x\right)-{\left({3}^{3k}\right)}^{n}f\left(\frac{x}{{\left({3}^{k}\right)}^{n}}\right),\phantom{\rule{2.77695pt}{0ex}}t\right),\phantom{\rule{2.77695pt}{0ex}}\mathcal{P}\left({\left({3}^{3k}\right)}^{n}f\left(\frac{x}{{\left({3}^{k}\right)}^{n}}\right)-C\left(x\right),\phantom{\rule{2.77695pt}{0ex}}t\right)\right)\\ \phantom{\rule{1em}{0ex}}{\ge }_{L}\mathcal{P}\left({\mathcal{T}}_{i=0}^{n-1}\mathcal{M}\left(x,\phantom{\rule{2.77695pt}{0ex}}\frac{{\alpha }^{i+1}t}{|{3}^{k}{|}^{i}}\right),\phantom{\rule{2.77695pt}{0ex}}\mathcal{P}\left({\left({3}^{3k}\right)}^{n}f\left(\frac{x}{{\left({3}^{k}\right)}^{n}}\right)-C\left(x\right),\phantom{\rule{2.77695pt}{0ex}}t\right)\right).\end{array}$
(6.7)

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

$\mathcal{P}\left(f\left(x\right)-C\left(x\right),\phantom{\rule{2.77695pt}{0ex}}t\right){\ge }_{L}{\mathcal{T}}_{i=1}^{\infty }\mathcal{M}\left(x,\phantom{\rule{2.77695pt}{0ex}}\frac{{\alpha }^{i+1}t}{|{3}^{k}{|}^{i}}\right),$

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

$\begin{array}{c}\underset{n\to \infty }{\text{lim}}\mathcal{P}\left({\left(2{7}^{k}\right)}^{n}f\left({3}^{-kn}\left(x+3y\right)\right)+{\left(2{7}^{k}\right)}^{n}f\left({3}^{-kn}\left(3x-y\right)\right)-15{\left(2{7}^{k}\right)}^{n}f\left({3}^{-kn}\left(x+y\right)\right)\\ \phantom{\rule{1em}{0ex}}-15{\left(2{7}^{k}\right)}^{n}f\left({3}^{-kn}\left(x-y\right)\right)-80{\left(2{7}^{k}\right)}^{n}f\left({3}^{-kn}y\right),\phantom{\rule{2.77695pt}{0ex}}t\right)\\ \phantom{\rule{1em}{0ex}}=\mathcal{P}\left(C\left(x+3y\right)+C\left(3x-y\right)-15C\left(x+y\right)-15C\left(x-y\right)-80C\left(y\right),\phantom{\rule{2.77695pt}{0ex}}t\right),\phantom{\rule{1em}{0ex}}\forall t>0.\end{array}$

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

$\begin{array}{c}\mathcal{P}\left({\left(2{7}^{k}\right)}^{n}f\left({3}^{-kn}\left(x+3y\right)\right)+{\left(2{7}^{k}\right)}^{n}f\left({3}^{-kn}\left(3x-y\right)\right)-15{\left(2{7}^{k}\right)}^{n}f\left({3}^{-kn}\left(x+y\right)\right)\\ \phantom{\rule{1em}{0ex}}-15{\left(2{7}^{k}\right)}^{n}f\left({3}^{-kn}\left(x-y\right)\right)-80{\left(2{7}^{k}\right)}^{n}f\left({3}^{-kn}y\right),\phantom{\rule{2.77695pt}{0ex}}t\right)\\ \phantom{\rule{1em}{0ex}}{\ge }_{L}\mathrm{\Psi }\left({3}^{-kn}x,\phantom{\rule{2.77695pt}{0ex}}{3}^{-kn}y,\phantom{\rule{2.77695pt}{0ex}}\frac{t}{|{3}^{3k}{|}^{n}}\right)\\ \phantom{\rule{1em}{0ex}}{\ge }_{L}\mathrm{\Psi }\left(x,\phantom{\rule{2.77695pt}{0ex}}y,\phantom{\rule{2.77695pt}{0ex}}\frac{{\alpha }^{n}t}{|{3}^{k}{|}^{n}}\right),\phantom{\rule{1em}{0ex}}\forall x,\phantom{\rule{2.77695pt}{0ex}}y\in X,\phantom{\rule{2.77695pt}{0ex}}t>0.\end{array}$

Since $\underset{n\to \infty }{\text{lim}}\mathrm{\Psi }\left(x,y,\frac{{\alpha }^{n}t}{|{3}^{k}{|}^{n}}\right)={1}_{\mathcal{L}}$, we infer that C is a cubic mapping.

For the uniqueness of C, let C' : XY be another cubic mapping such that

$\mathcal{P}\left({C}^{\prime }\left(x\right)-f\left(x\right),\phantom{\rule{2.77695pt}{0ex}}t\right){\ge }_{L}\mathcal{M}\left(x,\phantom{\rule{2.77695pt}{0ex}}t\right),\phantom{\rule{1em}{0ex}}\forall x\in X,\phantom{\rule{2.77695pt}{0ex}}t>0.$

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

$\begin{array}{c}\mathcal{P}\left(C\left(x\right)-{C}^{\prime }\left(x\right),\phantom{\rule{2.77695pt}{0ex}}t\right)\\ \phantom{\rule{1em}{0ex}}{\ge }_{L}\mathcal{T}\left(\mathcal{P}\left(C\left(x\right)-{\left({3}^{3k}\right)}^{n}f\left(\frac{x}{{\left({3}^{k}\right)}^{n}}\right),\phantom{\rule{2.77695pt}{0ex}}t\right),\phantom{\rule{2.77695pt}{0ex}}\mathcal{P}\left({\left({3}^{3k}\right)}^{n}f\left(\frac{x}{{\left({3}^{k}\right)}^{n}}\right)-{C}^{\prime }\left(x\right),\phantom{\rule{2.77695pt}{0ex}}t\right),\phantom{\rule{2.77695pt}{0ex}}t\right)\right).\end{array}$

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$\left(Y,\mathcal{P},\mathcal{T}\right)$ be a non-Archimedean -fuzzy Banach space over under a t-norm. Let f: XY be a Ψ-approximately cubic mapping. If there exist α (α > 0),| 3| ≠ 1 and an integer k, k ≥ 3 with | 3k| < α such that

$\mathrm{\Psi }\left({3}^{-k}x,\phantom{\rule{2.77695pt}{0ex}}{3}^{-k}y,\phantom{\rule{2.77695pt}{0ex}}t\right){\ge }_{L}\mathrm{\Psi }\left(x,\phantom{\rule{2.77695pt}{0ex}}y,\phantom{\rule{2.77695pt}{0ex}}\alpha t\right),\phantom{\rule{1em}{0ex}}\forall x,\phantom{\rule{2.77695pt}{0ex}}y\in X,\phantom{\rule{2.77695pt}{0ex}}t>0,$

then there exists a unique cubic mapping C : XY such that

$\mathcal{P}\left(f\left(x\right)-C\left(x\right),\phantom{\rule{2.77695pt}{0ex}}t\right){\ge }_{L}{\mathcal{T}}_{i=1}^{\infty }\mathcal{M}\left(x,\phantom{\rule{2.77695pt}{0ex}}\frac{{\alpha }^{i+1}t}{|3{|}^{ki}}\right),\phantom{\rule{1em}{0ex}}\forall x\in X,\phantom{\rule{2.77695pt}{0ex}}t>0,$

where

$\mathcal{M}\left(x,\phantom{\rule{2.77695pt}{0ex}}t\right)\phantom{\rule{2.77695pt}{0ex}}:=\mathcal{T}\left(\mathrm{\Psi }\left(x,\phantom{\rule{2.77695pt}{0ex}}0,\phantom{\rule{2.77695pt}{0ex}}t\right),\phantom{\rule{2.77695pt}{0ex}}\mathrm{\Psi }\left(3x,\phantom{\rule{2.77695pt}{0ex}}0,\phantom{\rule{2.77695pt}{0ex}}t\right),\phantom{\rule{2.77695pt}{0ex}}\dots ,\phantom{\rule{2.77695pt}{0ex}}\mathrm{\Psi }\left({3}^{k-1}x,\phantom{\rule{2.77695pt}{0ex}}0,\phantom{\rule{2.77695pt}{0ex}}t\right)\right),\phantom{\rule{1em}{0ex}}\forall x\in X,\phantom{\rule{2.77695pt}{0ex}}t>0.$

Proof. Since

$\underset{n\to \infty }{\text{lim}}\mathcal{M}\left(x,\phantom{\rule{2.77695pt}{0ex}}\frac{{\alpha }^{j}t}{|3{|}^{kj}}\right)={1}_{\mathcal{L}},\phantom{\rule{1em}{0ex}}\forall x\in X,\phantom{\rule{2.77695pt}{0ex}}t>0,$

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

$\underset{n\to \infty }{\text{lim}}{\mathcal{T}}_{j=n}^{\infty }\mathcal{M}\left(x,\phantom{\rule{2.77695pt}{0ex}}\frac{{\alpha }^{j}t}{|3{|}^{kj}}\right)={1}_{\mathcal{L}},\phantom{\rule{1em}{0ex}}\forall x\in X,\phantom{\rule{2.77695pt}{0ex}}t>0.$

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, $\left(X,\phantom{\rule{2.77695pt}{0ex}}{\mathcal{P}}_{\mu ,\nu },\phantom{\rule{2.77695pt}{0ex}}{\mathcal{T}}_{M}\right)$ be non-Archimedean -fuzzy normed space (intuitionistic fuzzy normed space) in which

${\mathcal{P}}_{\mu ,\nu }\left(x,\phantom{\rule{2.77695pt}{0ex}}t\right)=\left(\frac{t}{t+||x||},\phantom{\rule{2.77695pt}{0ex}}\frac{||x||}{t+||x||}\right),\phantom{\rule{1em}{0ex}}\forall x\in X,\phantom{\rule{2.77695pt}{0ex}}t>0,$

and $\left(Y,\phantom{\rule{2.77695pt}{0ex}}{\mathcal{P}}_{\mu ,\nu },\phantom{\rule{2.77695pt}{0ex}}{\mathcal{T}}_{M}\right)$ be a complete non-Archimedean -fuzzy normed space (intuitionistic fuzzy normed space) (see, Example 5.4). Define

$\mathrm{\Psi }\left(x,\phantom{\rule{2.77695pt}{0ex}}y,\phantom{\rule{2.77695pt}{0ex}}t\right)=\left(\frac{t}{1+t},\phantom{\rule{2.77695pt}{0ex}}\frac{1}{1+t}\right),\phantom{\rule{1em}{0ex}}\forall x,\phantom{\rule{2.77695pt}{0ex}}y\in X,\phantom{\rule{2.77695pt}{0ex}}t>0.$

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

$\mathcal{M}\left(x,\phantom{\rule{2.77695pt}{0ex}}t\right)=\left(\frac{t}{1+t},\phantom{\rule{2.77695pt}{0ex}}\frac{1}{1+t}\right),\phantom{\rule{1em}{0ex}}\forall x\in X,\phantom{\rule{2.77695pt}{0ex}}t>0,$

we have

$\begin{array}{ll}\hfill \underset{n\to \infty }{\text{lim}}{\mathcal{T}}_{M,j=n}^{\infty }\mathcal{M}\left(x,\phantom{\rule{2.77695pt}{0ex}}\frac{{\alpha }^{j}t}{|3{|}^{kj}}\right)& =\underset{n\to \infty }{\text{lim}}\left(\underset{m\to \infty }{\text{lim}}{\mathcal{T}}_{M,j=n}^{m}\mathcal{M}\left(x,\phantom{\rule{2.77695pt}{0ex}}\frac{t}{|3{|}^{kj}}\right)\right)\phantom{\rule{2em}{0ex}}\\ =\underset{n\to \infty }{\text{lim}}\underset{m\to \infty }{\text{lim}}\left(\frac{t}{t+|{3}^{k}{|}^{n}},\phantom{\rule{2.77695pt}{0ex}}\frac{|{2}^{k}{|}^{n}}{t+|{3}^{k}{|}^{n}}\right)\phantom{\rule{2em}{0ex}}\\ =\left(1,0\right)={1}_{{L}^{*}},\phantom{\rule{1em}{0ex}}\forall x\in X,\phantom{\rule{1em}{0ex}}t>0.\phantom{\rule{2em}{0ex}}\end{array}$

Let f : XY be a Ψ-approximately cubic mapping. Therefore, all the conditions of Theorem 6.2 hold and so there exists a unique cubic mapping C : XY such that

${\mathcal{P}}_{\mu ,\nu }\left(f\left(x\right)-C\left(x\right),\phantom{\rule{2.77695pt}{0ex}}t\right){\ge }_{{L}^{*}}\left(\frac{t}{t+|{3}^{k}|},\phantom{\rule{2.77695pt}{0ex}}\frac{|{3}^{k}|}{t+|{3}^{k}|}\right),\phantom{\rule{1em}{0ex}}\forall x\in X,\phantom{\rule{2.77695pt}{0ex}}t>0.$

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

$\mathcal{P}\left(f\left(2x+y\right)+f\left(2x-y\right)-2f\left(x+y\right)-2f\left(x-y\right)-12f\left(x\right),\phantom{\rule{2.77695pt}{0ex}}t\right)$
(6.8)
${\ge }_{L}\mathrm{\Psi }\left(x,\phantom{\rule{2.77695pt}{0ex}}y,\phantom{\rule{2.77695pt}{0ex}}t\right),\phantom{\rule{1em}{0ex}}\forall x,\phantom{\rule{2.77695pt}{0ex}}y\in X,\phantom{\rule{2.77695pt}{0ex}}t>0.$

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$\left(Y,\mathcal{P},\mathcal{T}\right)$ be a non-Archimedean -fuzzy Banach space over. Let f : XY be a Ψ-approximately cubic I mapping. If | 2| ≠ 1 and for some α , α > 0, and some integer k, k ≥ 2 with | 2k| < α,

$\mathrm{\Psi }\left({2}^{-k}x,\phantom{\rule{2.77695pt}{0ex}}{2}^{-k}y,\phantom{\rule{2.77695pt}{0ex}}t\right){\ge }_{L}\mathrm{\Psi }\left(x,\phantom{\rule{2.77695pt}{0ex}}y,\phantom{\rule{2.77695pt}{0ex}}\alpha t\right),\phantom{\rule{1em}{0ex}}\forall x,y\in X,\phantom{\rule{2.77695pt}{0ex}}t>0,$
(6.9)

and

$\underset{n\to \infty }{\text{lim}}{\mathcal{T}}_{j=n}^{\infty }\mathcal{M}\left(x,\phantom{\rule{2.77695pt}{0ex}}\frac{{\alpha }^{j}t}{|2{|}^{kj}}\right)={1}_{\mathcal{L}},\phantom{\rule{1em}{0ex}}\forall x\in X,\phantom{\rule{2.77695pt}{0ex}}t>0,$

then there exists a unique cubic mapping C : XY such that

$\mathcal{P}\left(f\left(x\right)-C\left(x\right),\phantom{\rule{2.77695pt}{0ex}}t\right)\ge {\mathcal{T}}_{i=1}^{\infty }\mathcal{M}\left(x,\frac{{\alpha }^{i+1}t}{|2{|}^{ki}}\right),\phantom{\rule{1em}{0ex}}\forall x\in X,\phantom{\rule{2.77695pt}{0ex}}t>0,$
(6.10)

where

$\mathcal{M}\left(x,\phantom{\rule{2.77695pt}{0ex}}t\right)\phantom{\rule{2.77695pt}{0ex}}:=\mathcal{T}\left(\mathrm{\Psi }\left(x,\phantom{\rule{2.77695pt}{0ex}}0,\phantom{\rule{2.77695pt}{0ex}}t\right),\phantom{\rule{2.77695pt}{0ex}}\mathrm{\Psi }\left(2x,\phantom{\rule{2.77695pt}{0ex}}0,\phantom{\rule{2.77695pt}{0ex}}t\right),\phantom{\rule{2.77695pt}{0ex}}\dots ,\phantom{\rule{2.77695pt}{0ex}}\mathrm{\Psi }\left({2}^{k-1}x,\phantom{\rule{2.77695pt}{0ex}}0,\phantom{\rule{2.77695pt}{0ex}}t\right)\right),\phantom{\rule{1em}{0ex}}\forall x\in X,\phantom{\rule{2.77695pt}{0ex}}t>0.$

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 $\left(Y,\mathcal{P},\mathcal{T}\right)$ be a non-Archimedean -fuzzy Banach space over under a t-norm . Let f : XY be a Ψ-approximately cubic I mapping. If there exist a α (α > 0) and an integer k, k ≥ 2 with |2k| < α such that

$\mathrm{\Psi }\left({2}^{-k}x,\phantom{\rule{2.77695pt}{0ex}}{2}^{-k}y,\phantom{\rule{2.77695pt}{0ex}}t\right){\ge }_{L}\mathrm{\Psi }\left(x,\phantom{\rule{2.77695pt}{0ex}}y,\phantom{\rule{2.77695pt}{0ex}}\alpha t\right),\phantom{\rule{1em}{0ex}}\forall x,\phantom{\rule{2.77695pt}{0ex}}y\in X,\phantom{\rule{2.77695pt}{0ex}}t>0,$

then there exists a unique cubic mapping C : XY such that

$\mathcal{P}\left(f\left(x\right)-C\left(x\right),\phantom{\rule{2.77695pt}{0ex}}t\right){\ge }_{L}{\mathcal{T}}_{i=1}^{\infty }\mathcal{M}\left(x,\phantom{\rule{2.77695pt}{0ex}}\frac{{\alpha }^{i+1}t}{|2{|}^{ki}}\right),\phantom{\rule{1em}{0ex}}\forall x\in X,\phantom{\rule{2.77695pt}{0ex}}t>0,$

where

$\mathcal{M}\left(x,\phantom{\rule{2.77695pt}{0ex}}t\right)\phantom{\rule{2.77695pt}{0ex}}:=\mathcal{T}\left(\mathrm{\Psi }\left(x,\phantom{\rule{2.77695pt}{0ex}}0,\phantom{\rule{2.77695pt}{0ex}}t\right),\phantom{\rule{2.77695pt}{0ex}}\mathrm{\Psi }\left(2x,\phantom{\rule{2.77695pt}{0ex}}0,\phantom{\rule{2.77695pt}{0ex}}t\right),\phantom{\rule{2.77695pt}{0ex}}\dots ,\phantom{\rule{2.77695pt}{0ex}}\mathrm{\Psi }\left({2}^{k-1}x,\phantom{\rule{2.77695pt}{0ex}}0,\phantom{\rule{2.77695pt}{0ex}}t\right)\right),\phantom{\rule{1em}{0ex}}\forall x\in X,\phantom{\rule{2.77695pt}{0ex}}t>0.$

Proof. Since

$\underset{n\to \infty }{\text{lim}}\mathcal{M}\left(x,\phantom{\rule{2.77695pt}{0ex}}\frac{{\alpha }^{j}t}{|2{|}^{kj}}\right)={1}_{\mathcal{L}},\phantom{\rule{1em}{0ex}}\forall x\in X,\phantom{\rule{2.77695pt}{0ex}}t>0,$

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

$\underset{n\to \infty }{\text{lim}}{\mathcal{T}}_{j=n}^{\infty }\mathcal{M}\left(x,\phantom{\rule{2.77695pt}{0ex}}\frac{{\alpha }^{j}t}{|2{|}^{kj}}\right)={1}_{\mathcal{L}},\phantom{\rule{1em}{0ex}}\forall x\in X,\phantom{\rule{2.77695pt}{0ex}}t>0.$

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, $\left(X,\phantom{\rule{2.77695pt}{0ex}}{\mathcal{P}}_{\mu ,\nu },\phantom{\rule{2.77695pt}{0ex}}{\mathcal{T}}_{M}\right)$ be non-Archimedean -fuzzy normed space (intuitionistic fuzzy normed space) in which

${\mathcal{P}}_{\mu ,\nu }\left(x,\phantom{\rule{2.77695pt}{0ex}}t\right)=\left(\frac{t}{t+||x||},\phantom{\rule{2.77695pt}{0ex}}\frac{||x||}{t+||x||}\right),\phantom{\rule{1em}{0ex}}\forall x\in X,\phantom{\rule{2.77695pt}{0ex}}t>0,$

and $\left(Y,\phantom{\rule{2.77695pt}{0ex}}{\mathcal{P}}_{\mu ,\nu },\phantom{\rule{2.77695pt}{0ex}}{\mathcal{T}}_{M}\right)$ be a complete non-Archimedean -fuzzy normed space (intuitionistic fuzzy normed space) (see, Example 5.4). Define

$\mathrm{\Psi }\left(x,\phantom{\rule{2.77695pt}{0ex}}y,\phantom{\rule{2.77695pt}{0ex}}t\right)=\left(\frac{t}{1+t},\phantom{\rule{2.77695pt}{0ex}}\frac{1}{1+t}\right),\phantom{\rule{1em}{0ex}}\forall x,\phantom{\rule{2.77695pt}{0ex}}y\in X,\phantom{\rule{2.77695pt}{0ex}}t>0.$

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

$\mathcal{M}\left(x,\phantom{\rule{2.77695pt}{0ex}}t\right)=\left(\frac{t}{1+t},\phantom{\rule{2.77695pt}{0ex}}\frac{1}{1+t}\right),\phantom{\rule{1em}{0ex}}\forall x\in X,\phantom{\rule{2.77695pt}{0ex}}t>0,$

we have

$\begin{array}{ll}\hfill \underset{n\to \infty }{\text{lim}}{\mathcal{T}}_{M,j=n}^{\infty }\mathcal{M}\left(x,\phantom{\rule{2.77695pt}{0ex}}\frac{{\alpha }^{j}t}{|2{|}^{kj}}\right)& =\underset{n\to \infty }{\text{lim}}\left(\underset{m\to \infty }{\text{lim}}{\mathcal{T}}_{M,j=n}^{m}\mathcal{M}\left(x,\phantom{\rule{2.77695pt}{0ex}}\frac{t}{|2{|}^{kj}}\right)\right)\phantom{\rule{2em}{0ex}}\\ =\underset{n\to \infty }{\text{lim}}\underset{m\to \infty }{\text{lim}}\left(\frac{t}{t+|{2}^{k}{|}^{n}},\phantom{\rule{2.77695pt}{0ex}}\frac{|{2}^{k}{|}^{n}}{t+|{2}^{k}{|}^{n}}\right)\phantom{\rule{2em}{0ex}}\\ =\left(1,0\right)={1}_{{L}^{*}},\phantom{\rule{1em}{0ex}}\forall x\in X,\phantom{\rule{2.77695pt}{0ex}}t>0.\phantom{\rule{2em}{0ex}}\end{array}$

Let f : XY 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 : XY such that

${\mathcal{P}}_{\mu ,\nu }\left(f\left(x\right)-C\left(x\right),\phantom{\rule{2.77695pt}{0ex}}t\right){\ge }_{{L}^{*}}\left(\frac{t}{t+|{2}^{k}|},\phantom{\rule{2.77695pt}{0ex}}\frac{|{2}^{k}|}{t+|{2}^{k}|}\right),\phantom{\rule{1em}{0ex}}\forall x\in X,\phantom{\rule{2.77695pt}{0ex}}t>0.$

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

$\begin{array}{c}P\left(f\left(3x\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}y\right)\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}f\left(3x\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}y\right)\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}3f\left(x\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}y\right)\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}3f\left(x\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}y\right)\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}48\phantom{\rule{0.3em}{0ex}}f\left(x\right),\phantom{\rule{0.3em}{0ex}}t\right)\\ \phantom{\rule{1em}{0ex}}{\ge }_{L}\phantom{\rule{0.3em}{0ex}}\mathrm{\Psi }\left(x,\phantom{\rule{0.3em}{0ex}}y,\phantom{\rule{0.3em}{0ex}}t\right),\phantom{\rule{1em}{0ex}}\forall x,\phantom{\rule{0.3em}{0ex}}y\phantom{\rule{0.3em}{0ex}}\in \phantom{\rule{0.3em}{0ex}}X,\phantom{\rule{0.3em}{0ex}}t>0.\end{array}$
(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 $\left(Y,\mathcal{P},T\right)$ be a non-Archimedean -fuzzy Banach space over . Let f : XY be a Ψ-approximately cubic II function. If |3| ≠ 1 and, for some α , α > 0, and some integer k, k ≥ 3, with |3k| < α,

$\mathrm{\Psi }\left({3}^{-k}x,\phantom{\rule{0.3em}{0ex}}{3}^{-k}y,\phantom{\rule{0.3em}{0ex}}t\right)\phantom{\rule{0.3em}{0ex}}{\ge }_{L}\phantom{\rule{0.3em}{0ex}}\mathrm{\Psi }\left(x,\phantom{\rule{0.3em}{0ex}}y,\phantom{\rule{0.3em}{0ex}}\alpha t\right),\phantom{\rule{1em}{0ex}}\forall x,\phantom{\rule{0.3em}{0ex}}y\phantom{\rule{0.3em}{0ex}}\in \phantom{\rule{0.3em}{0ex}}X,\phantom{\rule{2.77695pt}{0ex}}t\phantom{\rule{0.3em}{0ex}}>\phantom{\rule{0.3em}{0ex}}0,$
(6.12)

and

$\underset{n\to \infty }{\text{lim}}\phantom{\rule{0.3em}{0ex}}{\mathcal{T}}_{j=n}^{\infty }\mathcal{M}\phantom{\rule{0.3em}{0ex}}\left(x,\phantom{\rule{0.3em}{0ex}}\frac{{\alpha }^{j}t}{|3{|}^{kj}}\right)\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}{1}_{\mathcal{L}},\phantom{\rule{1em}{0ex}}\forall x\phantom{\rule{0.3em}{0ex}}\in \phantom{\rule{0.3em}{0ex}}X,\phantom{\rule{2.77695pt}{0ex}}t\phantom{\rule{0.3em}{0ex}}>\phantom{\rule{0.3em}{0ex}}0,$
(6.13)

then there exists a unique cubic mapping C : XY such tha

$\mathcal{P}\left(f\left(x\right)\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}C\left(x\right),\phantom{\rule{0.3em}{0ex}}t\right)\phantom{\rule{0.3em}{0ex}}\ge \phantom{\rule{0.3em}{0ex}}{T}_{i=1}^{\infty }\mathcal{M}\phantom{\rule{0.3em}{0ex}}\left(x,\phantom{\rule{0.3em}{0ex}}\frac{{\alpha }^{i+1}\phantom{\rule{0.3em}{0ex}}t}{|3{|}^{ki}}\right)\phantom{\rule{0.3em}{0ex}},$
(6.14)

for all × X and t > 0, where

$\mathcal{M}\left(x,\phantom{\rule{0.3em}{0ex}}t\right)\phantom{\rule{0.3em}{0ex}}:=\phantom{\rule{0.3em}{0ex}}\mathcal{T}\left(\mathrm{\Psi }\left(x,\phantom{\rule{0.3em}{0ex}}0,\phantom{\rule{0.3em}{0ex}}2t\right),\phantom{\rule{0.3em}{0ex}}\mathrm{\Psi }\left(3x,\phantom{\rule{0.3em}{0ex}}0,\phantom{\rule{0.3em}{0ex}}2t\right),\phantom{\rule{0.3em}{0ex}}.\phantom{\rule{0.3em}{0ex}}.\phantom{\rule{0.3em}{0ex}}.\phantom{\rule{0.3em}{0ex}},\phantom{\rule{0.3em}{0ex}}\mathrm{\Psi }\left({3}^{k-1}x,\phantom{\rule{0.3em}{0ex}}0,\phantom{\rule{0.3em}{0ex}}2t\right)\right),\phantom{\rule{1em}{0ex}}\forall x\phantom{\rule{0.3em}{0ex}}\in \phantom{\rule{0.3em}{0ex}}X,\phantom{\rule{0.3em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}t\phantom{\rule{0.3em}{0ex}}>\phantom{\rule{0.3em}{0ex}}0.$

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

$\mathcal{P}\left(f\left({3}^{j}x\right)\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}{27}^{j}\phantom{\rule{0.3em}{0ex}}f\left(x\right),\phantom{\rule{0.3em}{0ex}}t\right)\phantom{\rule{0.3em}{0ex}}{\ge }_{L\phantom{\rule{0.3em}{0ex}}}{\mathcal{M}}_{j}\phantom{\rule{0.3em}{0ex}}\left(x,\phantom{\rule{0.3em}{0ex}}t\right)\phantom{\rule{0.3em}{0ex}}:=\phantom{\rule{0.3em}{0ex}}\mathcal{T}\left(\mathrm{\Psi }\left(x,\phantom{\rule{0.3em}{0ex}}0,\phantom{\rule{0.3em}{0ex}}2t\right),\phantom{\rule{0.3em}{0ex}}.\phantom{\rule{0.3em}{0ex}}.\phantom{\rule{0.3em}{0ex}}.\phantom{\rule{0.3em}{0ex}},\phantom{\rule{0.3em}{0ex}}\mathrm{\Psi }\left({3}^{j-1}\phantom{\rule{0.3em}{0ex}}x,\phantom{\rule{0.3em}{0ex}}0,\phantom{\rule{0.3em}{0ex}}2t\right)\right).$
(6.15)

Put y = 0 in (6.11) to obtain

$\mathcal{P}\left(f\left(3x\right)\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}27f\left(x\right),\phantom{\rule{0.3em}{0ex}}t\right)\phantom{\rule{0.3em}{0ex}}{\ge }_{L}\phantom{\rule{0.3em}{0ex}}\mathrm{\Psi }\left(x,\phantom{\rule{0.3em}{0ex}}0,\phantom{\rule{0.3em}{0ex}}2t\right),\phantom{\rule{1em}{0ex}}\forall x\phantom{\rule{0.3em}{0ex}}\in \phantom{\rule{0.3em}{0ex}}X,\phantom{\rule{0.3em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}t\phantom{\rule{0.3em}{0ex}}>\phantom{\rule{0.3em}{0ex}}0.\phantom{\rule{0.3em}{0ex}}$
(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

$\mathcal{P}\left(f\left({3}^{j+1}\phantom{\rule{0.3em}{0ex}}x\right)\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}27f\left({3}^{j}x\right),\phantom{\rule{0.3em}{0ex}}t\right)\phantom{\rule{0.3em}{0ex}}{\ge }_{L}\phantom{\rule{0.3em}{0ex}}\mathrm{\Psi }\left({3}^{j}\phantom{\rule{0.3em}{0ex}}x,\phantom{\rule{0.3em}{0ex}}0,\phantom{\rule{0.3em}{0ex}}2t\right),\phantom{\rule{1em}{0ex}}\forall x\phantom{\rule{0.3em}{0ex}}\in \phantom{\rule{0.3em}{0ex}}X,\phantom{\rule{0.3em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}t\phantom{\rule{0.3em}{0ex}}>\phantom{\rule{0.3em}{0ex}}0.$

Since |27| ≤ 1, then we have

$\begin{array}{c}\mathcal{P}\left(f\left({3}^{j+1}\phantom{\rule{0.3em}{0ex}}x\right)\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}{27}^{j+1}\phantom{\rule{0.3em}{0ex}}f\left(x\right),\phantom{\rule{0.3em}{0ex}}t\right)\\ \phantom{\rule{1em}{0ex}}{\ge }_{L}\phantom{\rule{0.3em}{0ex}}\mathcal{T}\left(\mathcal{P}\left(f\left({3}^{j+1}\phantom{\rule{0.3em}{0ex}}x\right)-\phantom{\rule{0.3em}{0ex}}27f\left({3}^{j}x\right),\phantom{\rule{0.3em}{0ex}}t\right),\phantom{\rule{0.3em}{0ex}}\mathcal{P}\left(27f\left({3}^{j}\phantom{\rule{0.3em}{0ex}}x\right)\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}{27}^{j+1}\phantom{\rule{0.3em}{0ex}}f\left(x\right),\phantom{\rule{0.3em}{0ex}}t\right)\right)\\ \phantom{\rule{1em}{0ex}}=\phantom{\rule{0.3em}{0ex}}\mathcal{T}\left(\mathcal{P}\left(f\left({3}^{j+1}\phantom{\rule{0.3em}{0ex}}x\right)\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}27f\left({3}^{j}\phantom{\rule{0.3em}{0ex}}x\right),\phantom{\rule{0.3em}{0ex}}t\right),\phantom{\rule{0.3em}{0ex}}\mathcal{P}\phantom{\rule{0.3em}{0ex}}\left(f\left({3}^{j}\phantom{\rule{0.3em}{0ex}}x\right)\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}{27}^{j}\phantom{\rule{0.3em}{0ex}}f\left(x\right),\phantom{\rule{0.3em}{0ex}}\frac{t}{|27|}\right)\right)\\ \phantom{\rule{1em}{0ex}}{\ge }_{L}\phantom{\rule{0.3em}{0ex}}\mathcal{T}\left(\mathcal{P}\left(f\left({3}^{j+1}\phantom{\rule{0.3em}{0ex}}x\right)\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}27f\left({3}^{j}\phantom{\rule{0.3em}{0ex}}x\right),\phantom{\rule{0.3em}{0ex}}t\right),\phantom{\rule{0.3em}{0ex}}\mathcal{P}\phantom{\rule{0.3em}{0ex}}\left(f\left({3}^{j}\phantom{\rule{0.3em}{0ex}}x\right)\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}{27}^{j}\phantom{\rule{0.3em}{0ex}}f\left(x\right),\phantom{\rule{0.3em}{0ex}}t\right)\right)\\ \phantom{\rule{1em}{0ex}}{\ge }_{L}\phantom{\rule{0.3em}{0ex}}\mathcal{T}\left(\mathrm{\Psi }\phantom{\rule{0.3em}{0ex}}\left({3}^{j}\phantom{\rule{0.3em}{0ex}}x,\phantom{\rule{0.3em}{0ex}}0,\phantom{\rule{0.3em}{0ex}}2t\right),\phantom{\rule{0.3em}{0ex}}{\mathcal{M}}_{j}\phantom{\rule{0.3em}{0ex}}\left(x,\phantom{\rule{0.3em}{0ex}}t\right)\right)\\ \phantom{\rule{1em}{0ex}}=\phantom{\rule{0.3em}{0ex}}{\mathcal{M}}_{j+1}\phantom{\rule{0.3em}{0ex}}\left(x,\phantom{\rule{0.3em}{0ex}}t\right),\phantom{\rule{1em}{0ex}}\forall x\phantom{\rule{0.3em}{0ex}}\in \phantom{\rule{0.3em}{0ex}}X.\end{array}$

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

$\mathcal{P}\left(f\left({3}^{k}\phantom{\rule{0.3em}{0ex}}x\right)\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}{27}^{k}f\left(x\right),\phantom{\rule{0.3em}{0ex}}t\right)\phantom{\rule{0.3em}{0ex}}{\ge }_{L}\mathcal{M}\left(x,\phantom{\rule{0.3em}{0ex}}t\right),\phantom{\rule{1em}{0ex}}\forall x\phantom{\rule{0.3em}{0ex}}\in X,\phantom{\rule{2.77695pt}{0ex}}t\phantom{\rule{0.3em}{0ex}}>\phantom{\rule{0.3em}{0ex}}0.$
(6.17)

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

$\begin{array}{c}\mathcal{P}\left(f\left(\frac{x}{{3}^{kn}}\right)\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}{27}^{k}\phantom{\rule{0.3em}{0ex}}f\phantom{\rule{0.3em}{0ex}}\left(\frac{x}{{3}^{kn+k}}\right)\phantom{\rule{0.3em}{0ex}},\phantom{\rule{0.3em}{0ex}}t\right)\phantom{\rule{0.3em}{0ex}}{\ge }_{L}\phantom{\rule{0.3em}{0ex}}\mathcal{M}\phantom{\rule{0.3em}{0ex}}\left(\frac{x}{{3}^{kn+k}},\phantom{\rule{0.3em}{0ex}}t\right)\\ {\ge }_{L}\phantom{\rule{0.3em}{0ex}}\mathcal{M}\left(x,\phantom{\rule{0.3em}{0ex}}{\alpha }^{n+1}\phantom{\rule{0.3em}{0ex}}t\right),\phantom{\rule{1em}{0ex}}\forall x\phantom{\rule{0.3em}{0ex}}\in \phantom{\rule{0.3em}{0ex}}X,\phantom{\rule{0.3em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}t\phantom{\rule{0.3em}{0ex}}>\phantom{\rule{0.3em}{0ex}}0,\phantom{\rule{0.3em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}n\phantom{\rule{0.3em}{0ex}}\ge \phantom{\rule{0.3em}{0ex}}0.\end{array}$
(6.18)

Then we have

$\begin{array}{c}\mathcal{P}\phantom{\rule{0.3em}{0ex}}\left({\left({3}^{3k}\right)}^{n}\phantom{\rule{0.3em}{0ex}}f\phantom{\rule{0.3em}{0ex}}\left(\frac{x}{{\left({3}^{3k}\right)}^{n}}\right)\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}{\left({3}^{3k}\right)}^{n+1}f\phantom{\rule{0.3em}{0ex}}\left(\frac{x}{{\left({3}^{3k}\right)}^{n+1}}\right)\phantom{\rule{0.3em}{0ex}},\phantom{\rule{0.3em}{0ex}}t\right)\\ \phantom{\rule{1em}{0ex}}{\ge }_{L}\phantom{\rule{0.3em}{0ex}}\mathcal{M}\phantom{\rule{0.3em}{0ex}}\left(x,\phantom{\rule{0.3em}{0ex}}\frac{{\alpha }^{n+1}}{|{\left({3}^{3k}\right)}^{n}|}\phantom{\rule{0.3em}{0ex}}t\right)\phantom{\rule{0.3em}{0ex}},\phantom{\rule{1em}{0ex}}\forall x\phantom{\rule{0.3em}{0ex}}\in \phantom{\rule{0.3em}{0ex}}X,\phantom{\rule{0.3em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}t\phantom{\rule{0.3em}{0ex}}>\phantom{\rule{0.3em}{0ex}}0,\phantom{\rule{0.3em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}n\phantom{\rule{0.3em}{0ex}}\ge \phantom{\rule{0.3em}{0ex}}0.\end{array}$
(6.19)

Hence we have

$\begin{array}{c}\mathcal{P}\left(f\left(\frac{x}{{3}^{kn}}\right)\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}{27}^{k}\phantom{\rule{0.3em}{0ex}}f\phantom{\rule{0.3em}{0ex}}\left(\frac{x}{{3}^{kn+k}}\right)\phantom{\rule{0.3em}{0ex}},\phantom{\rule{0.3em}{0ex}}t\right)\phantom{\rule{0.3em}{0ex}}{\ge }_{L}\phantom{\rule{0.3em}{0ex}}\mathcal{M}\phantom{\rule{0.3em}{0ex}}\left(\frac{x}{{3}^{kn+k}},\phantom{\rule{0.3em}{0ex}}t\right)\\ {\ge }_{L}\phantom{\rule{0.3em}{0ex}}\mathcal{M}\left(x,\phantom{\rule{0.3em}{0ex}}{\alpha }^{n+1}\phantom{\rule{0.3em}{0ex}}t\right),\phantom{\rule{1em}{0ex}}\forall x\phantom{\rule{0.3em}{0ex}}\in \phantom{\rule{0.3em}{0ex}}X,\phantom{\rule{0.3em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}t\phantom{\rule{0.3em}{0ex}}>\phantom{\rule{0.3em}{0ex}}0,\phantom{\rule{0.3em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}n\phantom{\rule{0.3em}{0ex}}\ge \phantom{\rule{0.3em}{0ex}}0.\end{array}$

Since $\underset{n\to \infty }{\text{lim}}\phantom{\rule{0.3em}{0ex}}{\mathcal{T}}_{j=n}^{\infty }\mathcal{M}\phantom{\rule{0.3em}{0ex}}\left(x,\phantom{\rule{0.3em}{0ex}}\frac{{\alpha }^{j+1}}{\left|{\left({3}^{3k}\right)}^{j}\right|}t\right)=\phantom{\rule{0.3em}{0ex}}{1}_{\mathcal{L}}$ for all x X and t > 0, {kn f (k--nx)} nN is a Cauchy sequence in the non-Archimedean -fuzzy Banach space $\left(Y,\mathcal{P},\mathcal{T}\right).$ Hence we can define a mapping C : XY such that

$\underset{n\to \infty }{\text{lim}}\phantom{\rule{0.3em}{0ex}}\mathcal{P}\phantom{\rule{0.3em}{0ex}}\left(\left({3}^{3k}\right)n\phantom{\rule{0.3em}{0ex}}f\phantom{\rule{0.3em}{0ex}}\left(\frac{x}{{\left({3}^{k}\right)}^{n}}\right)\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}C\left(x\right),\phantom{\rule{0.3em}{0ex}}t\right)\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}{1}_{\mathcal{L}},\phantom{\rule{1em}{0ex}}\forall x\phantom{\rule{0.3em}{0ex}}\in \phantom{\rule{0.3em}{0ex}}X,\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{0.3em}{0ex}}t\phantom{\rule{0.3em}{0ex}}>\phantom{\rule{0.3em}{0ex}}0.$
(6.20)

Next, for all n ≥ 1, x X and t > 0,

$\begin{array}{c}\mathcal{P}\left(f\left(x\right)\phantom{\rule{0.3em}{0ex}}-{\left({3}^{3k}\right)}^{n}\phantom{\rule{0.3em}{0ex}}f\phantom{\rule{0.3em}{0ex}}\left(\frac{x}{{\left({3}^{k}\right)}^{n}}\right)\phantom{\rule{0.3em}{0ex}},\phantom{\rule{0.3em}{0ex}}t\right)\\ \phantom{\rule{1em}{0ex}}=\phantom{\rule{0.3em}{0ex}}\mathcal{P}\phantom{\rule{0.3em}{0ex}}\left(\sum _{i=0}^{n-1}{\left({3}^{3k}\right)}^{i}\phantom{\rule{0.3em}{0ex}}f\phantom{\rule{0.3em}{0ex}}\left(\frac{x}{{\left({3}^{k}\right)}^{i}}\right)\phantom{\rule{0.3em}{0ex}}-{\left({3}^{3k}\right)}^{i+1}\phantom{\rule{0.3em}{0ex}}f\phantom{\rule{0.3em}{0ex}}\left(\frac{x}{{\left({3}^{k}\right)}^{i+1}}\right)\phantom{\rule{0.3em}{0ex}},\phantom{\rule{0.3em}{0ex}}t\right)\\ \phantom{\rule{1em}{0ex}}{\ge }_{L}\phantom{\rule{0.3em}{0ex}}{\mathcal{T}}_{i=0}^{n-1}\phantom{\rule{0.3em}{0ex}}\left(\mathcal{P}\phantom{\rule{0.3em}{0ex}}\left({\left({3}^{3k}\right)}^{i}\phantom{\rule{0.3em}{0ex}}f\phantom{\rule{0.3em}{0ex}}\left(\frac{x}{{\left({3}^{k}\right)}^{i}}\right)\phantom{\rule{0.3em}{0ex}}-{\left({3}^{3k}\right)}^{i+1}f\phantom{\rule{0.3em}{0ex}}\left(\frac{x}{{\left({3}^{k}\right)}^{i+1}}\right)\phantom{\rule{0.3em}{0ex}},\phantom{\rule{0.3em}{0ex}}t\right)\right)\\ \phantom{\rule{1em}{0ex}}{\ge }_{L}\phantom{\rule{0.3em}{0ex}}{\mathcal{T}}_{i=0}^{n-1}\phantom{\rule{0.3em}{0ex}}\mathcal{M}\left(x,\phantom{\rule{0.3em}{0ex}}\frac{{\alpha }^{i+1}\phantom{\rule{0.3em}{0ex}}t}{|{3}^{3k}{|}^{i}}\right).\end{array}$
(6.21)

Therefore, we have

$\begin{array}{c}\mathcal{P}\left(f\left(x\right)\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}C\left(x\right),\phantom{\rule{0.3em}{0ex}}t\right)\\ \phantom{\rule{1em}{0ex}}\ge {L}_{}\phantom{\rule{0.3em}{0ex}}\mathcal{T}\phantom{\rule{0.3em}{0ex}}\left(\mathcal{P}\phantom{\rule{0.3em}{0ex}}\left(f\left(x\right)\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}{\left({3}^{3k}\right)}^{n}\phantom{\rule{0.3em}{0ex}}f\phantom{\rule{0.3em}{0ex}}\left(\frac{x}{{\left({3}^{k}\right)}^{n}}\right)\phantom{\rule{0.3em}{0ex}},\phantom{\rule{0.3em}{0ex}}t\right)\phantom{\rule{0.3em}{0ex}},\phantom{\rule{0.3em}{0ex}}\mathcal{P}\phantom{\rule{0.3em}{0ex}}\left({\left({3}^{3k}\right)}^{n}\phantom{\rule{0.3em}{0ex}}f\phantom{\rule{0.3em}{0ex}}\left(\frac{x}{{\left({3}^{k}\right)}^{n}}\right)\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}C\left(x\right),\phantom{\rule{0.3em}{0ex}}t\right)\right)\\ \phantom{\rule{1em}{0ex}}{\ge }_{L}\phantom{\rule{0.3em}{0ex}}\mathcal{P}\phantom{\rule{0.3em}{0ex}}\left({\mathcal{T}}_{i=0}^{n-1}\phantom{\rule{0.3em}{0ex}}\mathcal{M}\phantom{\rule{0.3em}{0ex}}\left(x,\phantom{\rule{0.3em}{0ex}}\frac{{\alpha }^{i+1}\phantom{\rule{0.3em}{0ex}}t}{{\left|{3}^{3k}\right|}^{i}}\right)\phantom{\rule{0.3em}{0ex}},\phantom{\rule{0.3em}{0ex}}\mathcal{P}\phantom{\rule{0.3em}{0ex}}\left({\left({3}^{3k}\right)}^{n}\phantom{\rule{0.3em}{0ex}}f\phantom{\rule{0.3em}{0ex}}\left(\frac{x}{{\left({3}^{k}\right)}^{n}}\right)\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}C\phantom{\rule{0.3em}{0ex}}\left(x\right),\phantom{\rule{0.3em}{0ex}}t\right)\right).\end{array}$

By letting n → ∞ in the above inequality, we obtain

$\mathcal{P}\left(f\left(x\right)\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}C\left(x\right),\phantom{\rule{0.3em}{0ex}}t\right)\phantom{\rule{0.3em}{0ex}}{\ge }_{L}\phantom{\rule{0.3em}{0ex}}{\mathcal{T}}_{i=1}^{\infty }\phantom{\rule{0.3em}{0ex}}\mathcal{M}\phantom{\rule{0.3em}{0ex}}\left(x,\phantom{\rule{0.3em}{0ex}}\frac{{\alpha }^{i+1}t}{{\left|{}^{}\right|}^{}}\right)$