Stability results in non-Archimedean $\mathcal{L}$-fuzzy normed spaces for a cubic functional equation

Abstract

We establish some stability results concerning the functional equation

$$nf(x+ny)+f(nx-y)=\frac{n(n^2+1)}{2}[f(x+y)+f(x-y)]+(n^4-1)f(y),$$

where $n\ge 2$ is a fixed integer in the setting of non-Archimedean $\mathcal{L}$-fuzzy normed spaces.

MSC:39B52, 46S10, 46S40, 47S10, 47S40.

1 Introduction

The theory of fuzzy sets was introduced by Zadeh [34] in 1965. After the pioneering work of Zadeh, there has been a great effort to obtain fuzzy analogues of classical theories. Among other fields, a progressive development has been made in the field of fuzzy topology [1, 8, 9, 12–14, 18, 22, 30]. One of the problems in $\mathcal{L}$-fuzzy topology is to obtain an appropriate concept of $\mathcal{L}$-fuzzy metric spaces and $\mathcal{L}$-fuzzy normed spaces. In 2004, Park [23] introduced and studied the notion of intuitionistic fuzzy metric spaces. In 2006, Saadati and Park [28] introduced and studied the notion of intuitionistic fuzzy normed spaces.

On the other hand, the study of stability problems for a functional equation is related to the question of Ulam [33] concerning the stability of group homomorphisms and affirmatively answered for Banach spaces by Hyers [19]. Subsequently, the result of Hyers was generalized by Aoki [2] for additive mappings and by Rassias [24] for linear mappings by considering an unbounded Cauchy difference. We refer the interested readers for more information on such problems to the papers [4, 6, 11, 20, 21, 25, 29, 32, 33].

Let X and Y be real linear spaces and $f:X\to Y$ a mapping. If $X=Y=\mathbb{R}$, the cubic function $f(x)=c{x}^3$, where c is a real constant, clearly satisfies the functional equation

$$f(2x+y)+f(2x-y)=2f(x+y)+2f(x-y)+12f(x).$$

Hence, the above equation is called the cubic functional equation. Recently, Cho, Saadati and Wang [5] introduced the functional equation

$$3f(x+3y)+f(3x-y)=15[f(x+y)+f(x-y)]+80f(y),$$

which has $f(x)=c{x}^3$ ($x\in \mathbb{R}$) as a solution for $X=Y=\mathbb{R}$.

In this paper, we investigate the Hyers-Ulam stability of the functional equation as follows:

$$nf(x+ny)+f(nx-y)=\frac{n(n^2+1)}{2}[f(x+y)+f(x-y)]+(n^4-1)f(y),$$

(1.1)

where $n\ge 2$ is a fixed integer.

2 Preliminaries

In this section, we recall some definitions and results for our main result in this paper.

A triangular norm (shorter t-norm) is a binary operation on the unit interval $[0,1]$, i.e., a function $T:[0,1]\times [0,1]\to [0,1]$ satisfying the following four axioms: for all $a,b,c\in [0,1]$,

• (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) $T(a,b)\le T(a,c)$ whenever $b\le c$ (monotonicity).

Basic examples are the Łukasiewicz t-norm $T_L$ and the t-norms $T_P$, $T_M$ and $T_D$, where $T_L(a,b):=max\{a+b-1,0\}$, $T_P(a,b):=ab$, $T_M(a,b):=min\{a,b\}$ and

$$T_D(a,b):=\{\begin{array}{cc}min\{a,b\}\hfill & \text{if }max\{a,b\}=1,\hfill \\ 0\hfill & \text{otherwise}\hfill \end{array}$$

for all $a,b\in [0,1]$.

For all $x\in [0,1]$ and all t-norms T, let $x_T^{(0)}:=1$. For all $x\in [0,1]$ and all t-norms T, define $x_T^{(r)}$ by the recursion equation $x_T^{(r)}=T(x_T^{(r-1)},x)$ for all $r\in \mathbb{N}$. A t-norm T is said to be Hadžić type (we denote it by $T\in \mathcal{H}$) if the family ${(x_T^{(r)})}_{r\in \mathbb{N}}$ is equicontinuous at $x=1$ (see [15]).

Other important triangular norms are as follows (see [16]):

• The Sugeno-Weber family${\{T_{\lambda }^{SW}\}}_{\lambda \in [-1,\mathrm{\infty }]}$ is defined by $T_{-1}^{SW}:=T_D$, $T_{\mathrm{\infty }}^{SW}:=T_P$ and

  $$T_{\lambda }^{SW}(x,y):=max\{0,\frac{x+y-1+\lambda xy}{1+\lambda }\}$$

if $\lambda \in (-1,\mathrm{\infty })$.

• The Domby family${\{T_{\lambda }^D\}}_{\lambda \in [0,\mathrm{\infty }]}$ is defined by $T_D$, if $\lambda =0$, $T_M$, if $\lambda =\mathrm{\infty }$ and

  $$T_{\lambda }^D(x,y):=\frac{1}{1+{[{(\frac{1-x}{x})}^{\lambda }+{(\frac{1-y}{y})}^{\lambda }]}^{1/\lambda }}$$

if $\lambda \in (0,\mathrm{\infty })$.

• The Aczel-Alsina family${\{T_{\lambda }^{AA}\}}_{\lambda \in [0,\mathrm{\infty }]}$ is defined by $T_D$, if $\lambda =0$, $T_M$, if $\lambda =\mathrm{\infty }$ and

  $$T_{\lambda }^{AA}(x,y):=e^{-{({|logx|}^{\lambda }+{|logy|}^{\lambda })}^{1/\lambda }}$$

if $\lambda \in (0,\mathrm{\infty })$.

A t-norm T can be extended (by associativity) in a unique way to an r-array operation by taking, for any $(x_1,\dots ,x_r)\in {[0,1]}^r$, the value $T(x_1,\dots ,x_r)$ defined by

$$T_{j=1}^0{x}_j:=1,T_{j=1}^r{x}_j:=T(T_{j=1}^{r-1}{x}_j,x_r)=T(x_1,\dots ,x_r).$$

A t-norm T can also be extended to a countable operation by taking, for any sequence ${(x_r)}_{r\in \mathbb{N}}$ in $[0,1]$, the value

$$T_{j=1}^{\mathrm{\infty }}{x}_j:=\underset{r\to \mathrm{\infty }}{lim}{T}_{j=1}^r{x}_j.$$

(2.1)

The limit on the right side of (2.1) exists since the sequence ${\{T_{j=1}^r{x}_j\}}_{r\in \mathbb{N}}$ is non-increasing and bounded below.

Proposition 2.1 [16]

• (1) For $T\ge T_L$, the following equivalence holds:

  $$\underset{r\to \mathrm{\infty }}{lim}{T}_{j=1}^{\mathrm{\infty }}{x}_{r+j}=1⟺\sum _{r=1}^{\mathrm{\infty }}(1-x_r)<\mathrm{\infty }.$$

• (2) If T is of Hadžić type, then

  $$\underset{r\to \mathrm{\infty }}{lim}{T}_{j=1}^{\mathrm{\infty }}{x}_{r+j}=1$$

for all sequence ${\{x_r\}}_{r\in \mathbb{N}}$ in $[0,1]$ such that ${lim}_{r\to \mathrm{\infty }}{x}_r=1$.

• (3) If $T\in {\{T_{\lambda }^{AA}\}}_{\lambda \in (0,\mathrm{\infty })}\cup {\{T_{\lambda }^D\}}_{\lambda \in (0,\mathrm{\infty })}$, then

  $$\underset{r\to \mathrm{\infty }}{lim}{T}_{j=1}^{\mathrm{\infty }}{x}_{r+j}=1⟺\sum _{r=1}^{\mathrm{\infty }}{(1-x_r)}^{\alpha }<\mathrm{\infty }.$$

• (4) If $T\in {\{T_{\lambda }^{SW}\}}_{\lambda \in [-1,\mathrm{\infty })}$, then

  $$\underset{r\to \mathrm{\infty }}{lim}{T}_{j=1}^{\mathrm{\infty }}{x}_{r+j}=1⟺\sum _{r=1}^{\mathrm{\infty }}(1-x_r)<\mathrm{\infty }.$$

3 $\mathcal{L}$-fuzzy normed spaces

In this section, we give some definitions and related lemmas for our main result.

Definition 3.1 [10]

Let $\mathcal{L}=(L,{\le }_L)$ be a complete lattice and U a non-empty set called a universe. An $\mathcal{L}$-fuzzy set $\mathcal{A}$ on U is defined by a mapping $\mathcal{A}:U\to L$. For any $u\in U$, $\mathcal{A}(u)$ represents the degree (in L) to which u satisfies $\mathcal{A}$.

Lemma 3.2 [7]

Consider the set $L^{\ast }$ and the operation ${\le }_{L^{\ast }}$ defined by

for all $(x_1,x_2),(y_1,y_2)\in L^{\ast }$. Then $(L^{\ast },{\le }_{L^{\ast }})$ is a complete lattice.

Definition 3.3 [3]

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

In Section 2, we presented the classical definition of t-norms, which can be straight-forwardly extended to any lattice $\mathcal{L}=(L,{\le }_L)$. Define first $0_{\mathcal{L}}:=infL$ and $1_{\mathcal{L}}:=supL$.

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

• (i) $\mathcal{T}(x,1_{\mathcal{L}})=x$ for all $x\in L$ (boundary condition);

• (ii) $\mathcal{T}(x,y)=\mathcal{T}(y,x)$ for all $(x,y)\in L^2$ (commutativity);

• (iii) $\mathcal{T}(x,\mathcal{T}(y,z))=\mathcal{T}(\mathcal{T}(x,y),z)$ for all $(x,y,z)\in L^3$ (associativity);

• (iv) $x{\le }_L{x}^{\prime }$ and $y{\le }_L{y}^{\prime }\Rightarrow \mathcal{T}(x,y){\le }_L\mathcal{T}(x^{\prime },y^{\prime })$ for all $(x,x^{\prime },y,y^{\prime })\in L^4$ (monotonicity).

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

$${\mathcal{T}}^r(x_1,\dots ,x_{r+1})=\mathcal{T}({\mathcal{T}}^{r-1}(x_1,\dots ,x_r),x_{r+1})$$

for all $r\ge 2$ and $x_j\in L$.

The t-norm $\mathcal{M}$ defined by

$$\mathcal{M}(x,y)=\{\begin{array}{cc}x\hfill & \text{if }x{\le }_{L}y,\hfill \\ y\hfill & \text{if }y{\le }_{L}x\hfill \end{array}$$

is a continuous t-norm.

Definition 3.5 A t-norm $\mathcal{T}$ on $L^{\ast }$ is said to be t-representable if there exist a t-norm T and a t-conorm S on $[0,1]$ such that

$$\mathcal{T}(x,y)=(T(x_1,y_1),S(x_2,y_2))$$

for all $x=(x_1,x_2)$, $y=(y_1,y_2)\in L^{\ast }$.

Definition 3.6

• (1) A negator on $\mathcal{L}$ is any decreasing mapping $\mathcal{N}:L\to L$ satisfying $\mathcal{N}(0_{\mathcal{L}})=1_{\mathcal{L}}$ and $\mathcal{N}(1_{\mathcal{L}})=0_{\mathcal{L}}$.

• (2) If a negator $\mathcal{N}$ on $\mathcal{L}$ satisfies $\mathcal{N}(\mathcal{N}(x))=x$ for all $x\in L$, then $\mathcal{N}$ is called an involution negator.

• (3) The negator ${\mathcal{N}}_s$ on $([0,1],\le )$ defined as ${\mathcal{N}}_s(x)=1-x$ for all $x\in [0,1]$ is called the standard negator on $([0,1],\le )$.

Definition 3.7 The 3-tuple $(V,\mathcal{P},\mathcal{T})$ is said to be an $\mathcal{L}$-fuzzy normed space if V is a vector space, $\mathcal{T}$ is a continuous t-norm on $\mathcal{L}$ and $\mathcal{P}$ is an $\mathcal{L}$-fuzzy set on $V\times (0,\mathrm{\infty })$ satisfying the following conditions: for all $x,y\in V$ and $t,s\in (0,\mathrm{\infty })$,

• (a) $0_{\mathcal{L}}{<}_L\mathcal{P}(x,t)$;

• (b) $\mathcal{P}(x,t)=1_{\mathcal{L}}\iff x=0$;

• (c) $\mathcal{P}(\alpha x,t)=\mathcal{P}(x,\frac{t}{|\alpha |})$ for all $\alpha \ne 0$;

• (d) $\mathcal{T}(\mathcal{P}(x,t),\mathcal{P}(y,s)){\le }_L\mathcal{P}(x+y,t+s)$;

• (e) $\mathcal{P}(x,\cdot ):(0,\mathrm{\infty })\to L$ is continuous;

• (f) ${lim}_{t\to 0}\mathcal{P}(x,t)=0_{\mathcal{L}}$ and ${lim}_{t\to \mathrm{\infty }}\mathcal{P}(x,t)=1_{\mathcal{L}}$.

In this case, $\mathcal{P}$ is called an $\mathcal{L}$-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 $(V,{\mathcal{P}}_{\mu ,\nu },\mathcal{T})$ is said to be an intuitionistic fuzzy normed space.

Definition 3.8 (see [27])

Let $(V,\mathcal{P},\mathcal{T})$ be an $\mathcal{L}$-fuzzy normed space.

• (1) A sequence ${\{x_r\}}_{r\in \mathbb{N}}$ in $(V,\mathcal{P},\mathcal{T})$ is called a Cauchy sequence if, for any $\epsilon \in L\setminus \{0_{\mathcal{L}}\}$ and for any $t>0$, there exists a positive integer $r_0$ such that

  $$\mathcal{N}(\epsilon ){<}_L\mathcal{P}(x_{r+p}-x_r,t)$$

for all $r\ge r_0$ and $p>0$, where $\mathcal{N}$ is a negator on $\mathcal{L}$.

• (2) A sequence ${\{x_r\}}_{r\in \mathbb{N}}$ in $(V,\mathcal{P},\mathcal{T})$ is said to be convergent to a point $x\in V$ in the $\mathcal{L}$-fuzzy normed space $(V,\mathcal{P},\mathcal{T})$ (denoted by\hspace*{-6pt} \hspace*{-6pt}) if $\mathcal{P}(x_r-x,t)\to 1_{\mathcal{L}}$ wherever $r\to \mathrm{\infty }$ for all $t>0$.

• (3) If every Cauchy sequence in $(V,\mathcal{P},\mathcal{T})$ is convergent in V, then the $\mathcal{L}$-fuzzy normed space $(V,\mathcal{P},\mathcal{T})$ is said to be complete and the $\mathcal{L}$-fuzzy normed space is called an $\mathcal{L}$-fuzzy Banach space.

Lemma 3.9 [26]

Let $\mathcal{P}$ be an $\mathcal{L}$-fuzzy norm on V. Then we have the following:

• (1) $\mathcal{P}(x,t)$ is non-decreasing with respect to $t\in (0,\mathrm{\infty })$ for all x in V.

• (2) $\mathcal{P}(x-y,t)=\mathcal{P}(y-x,t)$ for all x, y in V and all $t\in (0,\mathrm{\infty })$.

Definition 3.10 Let $(V,\mathcal{P},\mathcal{T})$ be an $\mathcal{L}$-fuzzy normed space. For any $t\in (0,\mathrm{\infty })$, we define the open ball $B(x,r,t)$ with center $x\in V$ and radius $r\in L\setminus \{0_{\mathcal{L}},1_{\mathcal{L}}\}$ as

$$B(x,r,t)=\{y\in V:\mathcal{N}(r){<}_L\mathcal{P}(x-y,t)\}.$$

A subset $A\subseteq V$ is called open if, for all $x\in A$, there exist $t>0$ and $r\in L\setminus \{0_{\mathcal{L}},1_{\mathcal{L}}\}$ such that $B(x,r,t)\subseteq A$.

Let ${\tau }_{\mathcal{P}}$ denote the family of all open subsets of V. Then ${\tau }_{\mathcal{P}}$ is called the topology induced by the $\mathcal{L}$-fuzzy norm $\mathcal{P}$.

4 $\mathcal{L}$-fuzzy Hyers-Ulam stability for cubic functional equations in non-Archimedean $\mathcal{L}$-fuzzy normed spaces

In 1897, Hensel [17] introduced the field with valuation in which does not have the Archimedean property.

Definition 4.1 Let $\mathcal{K}$ be a field. A non-Archimedean absolute value on $\mathcal{K}$ is a function $|\cdot |:\mathcal{K}\to [0,\mathrm{\infty })$ such that, for any $a,b\in \mathcal{K}$,

• (1) $|a|\ge 0$ and equality holds if and only if $a=0$,

• (2) $|ab|=|a||b|$,

• (3) $|a+b|\le max\{|a|,|b|\}$ (the strict triangle inequality).

Note that $|n|\le 1$ for each integer n. We always assume, in addition, that $|\cdot |$ is non-trivial, i.e., there is $a_0\in \mathcal{K}$ such that $|a_0|\notin \{0,1\}$.

Definition 4.2 A non-Archimedean $\mathcal{L}$ -fuzzy normed space is a triple $(V,\mathcal{P},\mathcal{T})$, where V is a vector space, $\mathcal{T}$ is a continuous t-norm on $\mathcal{L}$ and $\mathcal{P}$ is an $\mathcal{L}$-fuzzy set on $V\times (0,\mathrm{\infty })$ satisfying the following conditions: for all $x,y\in V$ and $t,s\in (0,\mathrm{\infty })$,

• (a) $0_{\mathcal{L}}{<}_L\mathcal{P}(x,t)$;

• (b) $\mathcal{P}(x,t)=1_{\mathcal{L}}\iff x=0$;

• (c) $\mathcal{P}(\alpha x,t)=\mathcal{P}(x,\frac{t}{|\alpha |})$ for all $\alpha \ne 0$;

• (d) $\mathcal{T}(\mathcal{P}(x,t),\mathcal{P}(y,s)){\le }_L\mathcal{P}(x+y,max\{t,s\})$;

• (e) $\mathcal{P}(x,\cdot ):(0,\mathrm{\infty })\to L$ is continuous;

• (f) ${lim}_{t\to 0}\mathcal{P}(x,t)=0_{\mathcal{L}}$ and ${lim}_{t\to \mathrm{\infty }}\mathcal{P}(x,t)=1_{\mathcal{L}}$.

From now on, let $\mathcal{K}$ be a non-Archimedean field, X a vector space over $\mathcal{K}$ and $(Y,\mathcal{P},\mathcal{T})$ a non-Archimedean $\mathcal{L}$-fuzzy Banach space over $\mathcal{K}$. We investigate the Hyers-Ulam stability of the cubic functional equation (1.1).

Next, we define an $\mathcal{L}$-fuzzy approximately cubic mapping. Let Ψ be an $\mathcal{L}$-fuzzy set on $X\times X\times (0,\mathrm{\infty })$ such that $\mathrm{\Psi }(x,y,\cdot )$ is non-decreasing,

$$\mathrm{\Psi }(cx,cx,t){\ge }_L\mathrm{\Psi }(x,x,\frac{t}{|c|})$$

and

$$\underset{t\to \mathrm{\infty }}{lim}\mathrm{\Psi }(x,y,t)=1_{\mathcal{L}}$$

for all $x,y\in X$, all $t>0$ and all $c\in \mathcal{K}\setminus \{0\}$.

Definition 4.3 A mapping $f:X\to Y$ is said to be Ψ-approximately cubic if

(4.1)

for all $x,y\in X$ and all $t>0$.

The following is the main result in this paper.

Theorem 4.4 Let $f:X\to Y$ be a Ψ-approximately cubic mapping. If $n\in \mathcal{K}\setminus \{0\}$ and there are $\alpha \in (0,\mathrm{\infty })$ and an integer k ($k\ge 2$) with ${|n|}^k<\alpha$ such that

$$\mathrm{\Psi }(n^{-k}x,n^{-k}y,t){\ge }_L\mathrm{\Psi }(x,y,\alpha t)$$

(4.2)

and

$$\underset{r\to \mathrm{\infty }}{lim}{\mathcal{T}}_{j=r}^{\mathrm{\infty }}\mathcal{M}(x,\frac{{\alpha }^{j}t}{{|n|}^{kj}})=1_{\mathcal{L}}$$

for all $x,y\in X$ and all $t>0$, then there exists a unique cubic mapping $C:X\to Y$ such that

$$\mathcal{P}(f(x)-C(x),t){\ge }_L{\mathcal{T}}_{j=1}^{\mathrm{\infty }}\mathcal{M}(x,\frac{{\alpha }^{j+1}}{{|n|}^{kj}}t)$$

(4.3)

for all $x\in X$ and all $t>0$, where

$$\mathcal{M}(x,t):=\mathcal{T}(\mathrm{\Psi }(x,0,t),\mathrm{\Psi }(nx,0,t),\dots ,\mathrm{\Psi }(n^{k-1}x,0,t))$$

for all $x\in X$ and all $t>0$.

Proof Let ${\mathcal{T}}^0:L\to L$ be the identity mapping $I_L:L\to L$. First, we show, by induction on j, that

$$\mathcal{P}(f\left(n^{j}x\right)-n^{3j}f(x),t){\ge }_L{\mathcal{M}}_j(x,t):={\mathcal{T}}^{j-1}(\mathrm{\Psi }(x,0,t),\dots ,\mathrm{\Psi }(n^{j-1}x,0,t))$$

(4.4)

for all $x\in X$, all $t>0$ and all $j\ge 1$. Putting $y=0$ in (4.1), we obtain

$$\mathcal{P}(f(nx)-n^{3}f(x),t){\ge }_L\mathrm{\Psi }(x,0,t)$$

for all $x\in X$ and all $t>0$. This proves (4.4) for $j=1$. Let (4.4) hold for some $j\ge 1$. Replacing y by 0 and x by $3^{j}x$ in (4.1), we get

$$\mathcal{P}(f\left(n^{j+1}x\right)-n^{3}f\left(n^{j}x\right),t){\ge }_L\mathrm{\Psi }(n^{j}x,0,t)$$

for all $x\in X$ and all $t>0$. Since ${|n|}^3\le 1$, it follows that

for all $x\in X$ and all $t>0$. Thus (4.4) holds for all $j\ge 1$. In particular, we have

$$\mathcal{P}(f\left(n^{k}x\right)-n^{3k}f(x),t){\ge }_L\mathcal{M}(x,t)$$

for all $x\in X$ and all $t>0$. Replacing x by $\frac{x}{n^{k(r+1)}}$ in the above inequality and using the inequality (4.2), we obtain

$$\mathcal{P}(f\left(\frac{x}{n^{kr}}\right)-n^{3k}f\left(\frac{x}{n^{k(r+1)}}\right),t){\ge }_L\mathcal{M}(\frac{x}{n^{k(r+1)}},t){\ge }_L\mathcal{M}(x,{\alpha }^{r+1}t)$$

for all $x\in X$, all $t>0$ and all $r\ge 0$, and so

for all $x\in X$, all $t>0$ and all $r\ge 0$. Hence it follows that

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

$$\underset{r\to \mathrm{\infty }}{lim}\mathcal{P}(n^{3kr}f\left(\frac{x}{n^{kr}}\right)-C(x),t)=1_{\mathcal{L}}$$

(4.5)

for all $x\in X$ and all $t>0$. Next, for all $r\ge 1$, all $x\in X$ and all $t>0$, we have

$$\begin{array}{rl}\mathcal{P}(f(x)-n^{3kr}f\left(\frac{x}{n^{kr}}\right),t)& =\mathcal{P}(\sum _{j=0}^{r-1}[n^{3kj}f\left(\frac{x}{n^{kj}}\right)-n^{3k(j+1)}f\left(\frac{x}{n^{k(j+1)}}\right)],t)\\ {\ge }_L{\mathcal{T}}_{j=0}^{r-1}\mathcal{P}(n^{3kj}f\left(\frac{x}{n^{kj}}\right)-n^{3k(j+1)}f\left(\frac{x}{n^{k(j+1)}}\right),t)\\ {\ge }_L{\mathcal{T}}_{j=0}^{r-1}\mathcal{M}(x,\frac{{\alpha }^{j+1}}{{|n|}^{kj}}t)\end{array}$$

for all $x\in X$ and all $t>0$, and so

$$\begin{array}{rcl}\mathcal{P}(f(x)-C(x),t)& {\ge }_L& \mathcal{T}(\mathcal{P}(f(x)-n^{3kr}f\left(\frac{x}{n^{kr}}\right),t),\\ \mathcal{P}(n^{3kr}f\left(\frac{x}{n^{kr}}\right)-C(x),t))\\ {\ge }_L& \mathcal{T}({\mathcal{T}}_{j=0}^{r-1}\mathcal{M}(x,\frac{{\alpha }^{j+1}}{{|n|}^{kj}}t),\\ \mathcal{P}(n^{3kr}f\left(\frac{x}{n^{kr}}\right)-C(x),t))\end{array}$$

for all $x\in X$ and all $t>0$. Taking the limit as $r\to \mathrm{\infty }$ in the above inequality, we obtain

$$\mathcal{P}(f(x)-C(x),t){\ge }_L{\mathcal{T}}_{j=0}^{\mathrm{\infty }}\mathcal{M}(x,\frac{{\alpha }^{j+1}}{{|n|}^{kj}}t)$$

for all $x\in X$ and all $t>0$, which proves (4.3).

Since $\mathcal{T}$ is continuous, from the well-known result in an $\mathcal{L}$-fuzzy (probabilistic) normed space (see [31], Chapter 12), it follows that

for all $x,y\in X$ and all $t>0$. On the other hand, replacing x, y by $n^{-kr}x$, $n^{-kr}y$ in (4.1) and (4.2), we get

for all $x,y\in X$ and all $t>0$. Since ${lim}_{r\to \mathrm{\infty }}\mathrm{\Psi }(x,y,\frac{{\alpha }^{r}t}{{|n|}^{3kr}})=1_{\mathcal{L}}$, we infer that C is a cubic mapping.

For the uniqueness of C, let $C^{\prime }:X\to Y$ be another cubic mapping such that

$$\mathcal{P}(C^{\prime }(x)-f(x),t){\ge }_L\mathcal{M}(x,t)$$

for all $x\in X$ and all $t>0$. Then we have, for all $x,y\in X$ and all $t>0$,

Therefore, from (4.5), we conclude that $C=C^{\prime }$. This completes the proof. □

Corollary 4.5 Let $\mathcal{T}\in \mathcal{H}$ and let $f:X\to Y$ be a Ψ-approximately cubic mapping. If $n\in \mathcal{K}\setminus \{0\}$ and there are $\alpha \in (0,\mathrm{\infty })$ and an integer k ($k\ge 2$) with ${|n|}^k<\alpha$ such that

$$\mathrm{\Psi }(n^{-k}x,n^{-k}y,t){\ge }_L\mathrm{\Psi }(x,y,\alpha t)$$

and

$$\underset{r\to \mathrm{\infty }}{lim}\mathcal{M}(x,\frac{{\alpha }^{r}t}{{|n|}^{kr}})=1_{\mathcal{L}}$$

for all $x,y\in X$ and all $t>0$, then there exists a unique cubic mapping $C:X\to Y$ such that

$$\mathcal{P}(f(x)-C(x),t){\ge }_L{\mathcal{T}}_{j=1}^{\mathrm{\infty }}\mathcal{M}(x,\frac{{\alpha }^{j+1}}{{|n|}^{kj}}t)$$

for all $x\in X$ and all $t>0$, where

$$\mathcal{M}(x,t):=\mathcal{T}(\mathrm{\Psi }(x,0,t),\mathrm{\Psi }(nx,0,t),\dots ,\mathrm{\Psi }(n^{k-1}x,0,t))$$

for all $x\in X$ and all $t>0$.

Proof Since

$$\underset{r\to \mathrm{\infty }}{lim}\mathcal{M}(x,\frac{{\alpha }^r}{{|n|}^{kr}}t)=1_{\mathcal{L}}$$

for all $x\in X$ and all $t>0$ and $\mathcal{T}$ is of $Had\check{z}i\acute{c}$ type, it follows from Proposition 2.1 that

$$\underset{r\to \mathrm{\infty }}{lim}{\mathcal{T}}_{j=r+1}^{\mathrm{\infty }}\mathcal{M}(x,\frac{{\alpha }^j}{{|n|}^{kj}}t)=1_{\mathcal{L}}$$

for all $x\in X$ and all $t>0$. Therefore, if we apply Theorem 4.4, then we get the conclusion. This completes the proof. □

Example 4.6 Let $(X,\parallel \cdot \parallel )$ be a non-Archimedean Banach space, $(X,{\mathcal{P}}_{\mu ,\nu },{\mathcal{T}}_M)$ a non-Archimedean $\mathcal{L}$-fuzzy normed space (intuitionistic fuzzy normed space) in which

$${\mathcal{P}}_{\mu ,\nu }(x,t)=(\frac{t}{t+\parallel x\parallel },\frac{\parallel x\parallel }{t+\parallel x\parallel })$$

for all $x\in X$ and all $t>0$ and $(Y,{\mathcal{P}}_{\mu ,\nu },{\mathcal{T}}_M)$ a complete non-Archimedean $\mathcal{L}$-fuzzy normed space (intuitionistic fuzzy normed space). Define

$$\mathrm{\Psi }(x,y,t)=(\frac{t}{1+t},\frac{1}{1+t})$$

for all $x,y\in X$ and all $t>0$. It is easy to see that (4.2) holds for $\alpha =1$. Also, since

$$\mathcal{M}(x,t)=(\frac{t}{1+t},\frac{1}{1+t})$$

for all $x\in X$ and all $t>0$, we have

$$\begin{array}{rl}\underset{r\to \mathrm{\infty }}{lim}{({\mathcal{T}}_M)}_{j=r}^{\mathrm{\infty }}\mathcal{M}(x,\frac{{\alpha }^j}{{|n|}^{kj}}t)& =\underset{r\to \mathrm{\infty }}{lim}\left[\underset{s\to \mathrm{\infty }}{lim}{({\mathcal{T}}_M)}_{j=r}^s\mathcal{M}(x,\frac{{\alpha }^j}{{|n|}^{kj}}t)\right]\\ =\underset{r\to \mathrm{\infty }}{lim}\underset{s\to \mathrm{\infty }}{lim}(\frac{t}{t+{|n|}^{kj}},\frac{{|n|}^{kr}}{t+{|n|}^{kr}})=(1,0)=1_{L\ast }\end{array}$$

for all $x\in X$ and all $t>0$. Let $f:X\to Y$ be a Ψ-approximately cubic mapping. A straightforward computation shows that

for all $x,y\in X$ and all $t>0$. Therefore, all the conditions of Theorem 4.4 hold, and so there exists a unique cubic mapping $C:X\to Y$ such that

$${\mathcal{P}}_{\mu ,\nu }(f(x)-C(x),t){\ge }_{L^{\ast }}(\frac{t}{t+{|n|}^k},\frac{{|n|}^k}{t+{|n|}^k})$$

for all $x\in X$ and all $t>0$.