Ulam Stabilities and Instabilities of Euler–Lagrange-Rassias Quadratic Functional Equation in Non-Archimedean IFN Spaces

Abstract

In this paper, we use direct and fixed-point techniques to examine the generalised Ulam–Hyers stability results of the general Euler–Lagrange quadratic mapping in non-Archimedean IFN spaces (briefly, non-Archimedean Intuitionistic Fuzzy Normed spaces) over a field.

1. Introduction

One of the interesting questions concerning the stability problems of functional equations is as follows: When is it true that a mapping approximately satisfying a functional equation must be close to the solution of the given functional equation? Such an idea was suggested in 1940 by Ulam [1] as follows.

Given a group $(G,*)$, a metric group $(G^{{}^{\prime }},·)$ with the metric d, and a mapping f from G and $G^{{}^{\prime }}$, does $\delta >0$ exist such that

$$d\left(f\right(x*y),f(x)·f(y\left)\right)\le \delta$$

for all $x,y\in G$? If such a mapping exists, then, does a homomorphism $h:G\to G^{{}^{\prime }}$ exist such that

$$d\left(f\right(x),h(x\left)\right)\le ϵ$$

for all $x\in G$?

In 1941, Hyers [2] studied nearly additive mappings in Banach spaces that satisfied the very weak Hyers stability defined by a positive constant. The classic Hyers stability finding (in [2]) was generalized by Aoki [3] in the stability involving a sum of powers of norms. Rassias [4] proposed an extension of the Hyers Theorem in 1978, allowing for an unbounded Cauchy difference. A number of authors have examined and generalized stability problems of various functional equations that have been discussed in different normed spaces by using a fixed-point approach over the last few decades (see [5,6,7,8,9,10,11,12]).

The quadratic functional equation is defined by $\varphi (u+v)+\varphi (u-v)=2\varphi \left(u\right)+2\varphi \left(v\right)$. In particular, every solution of the quadratic functional equation is called a quadratic function. Skof [13] demonstrated the stability of quadratic functional equations for mappings between normed space and Banach space. Cholewa [14] observed that, if the appropriate domain normed space is substituted by an Abelian group, the Skof theorem still holds.

Since the work [15], the stability of different functional equations in the setting of random normed spaces or FN- spaces (briefly, Fuzzy Normed-spaces) has been explored (for example, [16,17]). The paper [18] demonstrates the stability of the additive Cauchy equation in non-Archimedean FN-spaces under the strongest t-norm $T_M$. When the field of scalars is $\mathbb{R}$ or $\mathbb{C}$, however, the findings in [18] do not apply since the vector space consists only of a single element.

Rassias [19] introduced the Euler–Lagrange-type quadratic functional equation as

$$\varphi (\alpha v_1+\beta v_2)+\varphi (\beta v_1-\alpha v_2)=({\alpha }^2+{\beta }^2)[\varphi \left(v_1\right)+\varphi \left(v_2\right)],$$

(1)

where $\alpha$ and $\beta$ are fixed reals with $\alpha \ne 0,\beta \ne 0$. The paper [20] also demonstrated that this kind of cubic functional equation is Hyers-Ulam-Rassias stable. Numerous mathematicians have studied several Euler–Lagrange-type functional equations (for example, [21,22,23,24,25,26,27,28,29]).

In this work, we use direct and fixed-point techniques to examine the generalised Ulam–Hyers stability results of the Equation (1) in non-Archimedean IFN spaces over a field. Additionally, we examine some of the main theorems’ applications.

2. Preliminaries

We were able to refer to some needed preliminaries in the Refs. [28,30,31,32], and utilized the alternative fixed-point theorem which gave some important results in the fixed-point theory.

A map $|·|:\mathbb{K}\to [0,\infty )$ is a valuation such that zero is the only one element having the zero valuation, $|k_1{k}_2|=|k_1\left|\right|k_2|$, and the inequality of the triangle holds true, that is, $|k_1+k_2|\le |k_1|+|k_2|$, for all $k_1,k_2\in \mathbb{K}$.

We call a field $\mathbb{K}$ valued if $\mathbb{K}$ holds a valuation. Examples of valuations include the typical absolute values of $\mathbb{R}$ and $\mathbb{C}$.

Consider a valuation that satisfies a criterion that is stronger than the triangle inequality. A $|·|$ is called a non-Archimedean valuation if the triangle inequality is replaced by $|k_1+k_2|\le max\{|k_1|,|k_2\left|\right\}$, for all $k_1,k_2\in \mathbb{K}$, and a field is called a non-Archimedean field. Evidently, $|-1|=1=\left|1\right|$ and $\left|n\right|$ are greater than or equal to 1, for all n in $\mathbb{N}$. The map $|·|$ takes everything except 0 for 1, and $\left|0\right|=0$ is a basic example of a non-Archimedean valuation.

Definition 1.

[28] Let E be a linear space over $\mathbb{K}$ with $|·|$. A mapping $\parallel ·\parallel :E\to [0,\infty )$ is known as a non-Archimedean norm if it satisfies:

(i) 

$\parallel v\parallel =0$ if, and only if $v=0$;

(ii) 

$\parallel rv\parallel =\left|r\right|\parallel v\parallel$, $v\in E$ and $r\in \mathbb{K}$;

(iii) 

the strong triangle inequality

$$\parallel v_1+v_2\parallel \le max\{\parallel v_1\parallel ,\parallel v_2\parallel \},v_1,v_2\in E.$$

Then, $(E,\parallel ·\parallel )$ is called a non-Archimedean normed space. Every Cauchy sequence converges in a complete non-Archimedean normed space, which we call a complete non-Archimedean normed space.

A triangular norm (shorter t-norm) is a binary operation $T:[0,1]\times [0,1]\to [0,1]$ which satisfies the following conditions: (a) T is commutative and associative; (b) $T(a,1)=a$ for all $a\in [0,1]$; (c) $T(a,b)\le T(c,d)$ whenever $a\le c$ and $b\le d$ for all $a,b,c,d\in [0,1]$. Basic examples of continuous t-norms are the Lukasiewicz $t-$norm $T_L$, $T_L(a,b)=max\{a+b-1,0\}$, the product t-norm $T_P$, $T_P(a,b)=ab$ and the strongest triangular norm $T_M$, $T_M(a,b)=min\{a,b\}$. A t-norm is called continuous if it is continuous with respect to the product topology on the set $[0,1]\times [0,1]$.

A t-norm T can be extended (by associativity) in a unique way to an m-array operation taking for $(x_1,\cdots ,x_m)\in {[0,1]}^m$, the value $T(x_1,\cdots ,x_m)$ defined recurrently by $T_{i=1}^0{x}_i=1$ and $T_{i=1}^m{x}_i=T(T_{i=1}^{m-1}{x}_i,x_m)$ for $m\in \mathbb{N}$. T can also be extended to a countable operation, taking for any sequence ${\left\{x_i\right\}}_{i\in \mathbb{N}}$ in $[0,1]$, the value $T_{i=1}^{\infty }{x}_i$ is defined as ${lim}_{m\to \infty }{T}_{i=1}^m{x}_i$. The limit exists since the sequence ${\left\{T_{i=1}^m{x}_i\right\}}_{m\in \mathbb{N}}$ is non-increasing and bounded from below. $T_{i=m}^{\infty }{x}_i$ is defined as $T_{i=1}^{\infty }{x}_{m+i}$.

Definition 2.

[28] A t-norm T is called a Hadžić-type (H-type, denoted by $T\in H$) if a family of functions $\left\{T_{i=1}^m\left(t\right)\right\}$ for every $m\in \mathbb{N}$ is equicontinuous at $t=1$, that is, for every $ϵ\in (0,1)$ there exist $\delta \in (0,1)$ that satisfies

$$t>1-\delta \Rightarrow T_{i=1}^m\left(t\right)>1-ϵm\in \mathbb{N}.$$

The t-norm $T_M$ is a t-norm of Hadžić-type.

Proposition 1.

[28]

(1) 

If $T=T_L$ or $T=T_P$, then

$$\underset{m\to \infty }{lim}{T}_{i=1}^{\infty }{v}_{m+i}=1\iff \sum _{i=1}^{\infty }(1-v_i)<\infty .$$

(2) 

If T is of Hadžić-type, then

$$\underset{m\to \infty }{lim}{T}_{i=m}^{\infty }{v}_i=\underset{m\to \infty }{lim}{T}_{i=1}^{\infty }{v}_{m+i}=1$$

for all ${\left\{v_i\right\}}_{i\in \mathbb{N}}$ in $[0,1]$, such that ${lim}_{i\to \infty }{v}_i=1.$

Lemma 1.

[33] Consider the $L^{*}$ and the order relation ${\le }_{L^{*}}$ defined by

$$L^{*}=\{(x_1,x_2):(x_1,x_2)\in {[0,1]}^{2}and{x}_1+x_2\le \},$$

$$(x_1,x_2){\le }_{L^{*}}(y_1,y_2)\iff x_1\le y_1,x_2\ge y_2,\forall (x_1,x_2),(y_1,y_2)\in L^{*}.$$

Then, $(L^{*},{\le }_{L^{*}})$ is a complete lattice. We denote its units by $0_{L^{*}}=(0,1)$ and $1_{L^{*}}=(1,0)$.

Definition 3.

[34] A triangular norm (t-norm) on $L^{*}$ is a mapping $\tau :{\left(L^{*}\right)}^2\to L^{*}$ satisfying the following conditions:

(a) 

Boundary condition

$\tau (x,1_{L^{*}})=x\forall x\in L^{*};$

(b) 

Commutativity

$\tau (x,y)=\tau (y,x)\forall (x,y)\in {\left(L^{*}\right)}^2;$

(c) 

Associativity

$\tau (x,\tau (y,z))=\tau (\tau (x,y),z))\forall (x,y,z)\in {\left(L^{*}\right)}^3;$

(d) 

Monotonicity

$x{\le }_{L^{*}}{x}^{{}^{\prime }}andy{\le }_{L^{*}}{y}^{{}^{\prime }}\Rightarrow \tau (x,y){\le }_{L^{*}}\tau (x^{{}^{\prime }},y^{{}^{\prime }})\forall (x,x^{{}^{\prime }},y,y^{{}^{\prime }})\in {\left(L^{*}\right)}^4.$

A t-norm τ on $L^{*}$ is said to be continuous if for any $x,y\in L^{*}$ and any sequences $\left\{x_n\right\}$ and $\left\{y_n\right\}$ converge to x and y, respectively.

$${lim}_{n\to \infty }\tau (x_n,y_n)=\tau (x,y).$$

Definition 4.

[34] A continuous t-norm τ on $L^{*}$ is said to be continuous t-representable if there exists a continuous t-norm * and a continuous t-conorm ◊ on $[0,1]$ such that, for all $x=(x_1,x_2)$, $y=(y_1,y_2)\in L^{*}$,

$$\tau (x,y)=(x_1\ast y_1,x_2◊y_2).$$

Definition 5.

[35] A negator on $L^{*}$ is any decreasing mapping $N:L^{*}\to L^{*}$ satisfying $N\left(0_{L^{*}}\right)=1_{L^{*}}$ and $N\left(1_{L^{*}}\right)=0_{L^{*}}$. If $N\left(N\right(x\left)\right)=x$ for all $x\in L^{*}$, then, N is called an involutive negator. A negator on $[0,1]$ is a decreasing mapping $N:[0,1]\to [0,1]$ satisfying $N\left(0\right)=1$ and $N\left(1\right)=0$. $N_s$ denotes the standard negator on $[0,1]$ defined by $N_s\left(x\right)=1-x$ for all $x\in [0,1]$.

We should also remark that the Definition 6 of a non-Archimedean Menger norm is broader than the definition in [18,36], which only considers fields with $|·|$.

Definition 6.

Let the membership degree μ and non-membership degree ν of an intuitionistic fuzzy set from $E\times (0,+\infty )$ to $[0,1]$ satisfies ${\mu }_v\left(t\right)+{\nu }_v\left(t\right)\le 1$ for all $v\in E$ and all $t>0$. The triple $(E,N_{\mu ,\nu },T)$ is called a non-Archimedean intuitionistic fuzzy Menger norm if a vector space E, a continuous t-representable T and $N_{\mu ,\nu }:E\times (0,+\infty )\to L^{*}$ satisfying the follows: for all $v_1,v_2\in E$ and all $s,t>0$,

(IFN1) 

$N_{\mu ,\nu }(v_1,t)=0$ for all $t\le 0$;

(IFN2) 

$v_1=0\iff N_{\mu ,\nu }(v_1,t)=1,t>0$;

(IFN3) 

$N_{\mu ,\nu }(\alpha v_1,t)=N_{\mu ,\nu }(v_1,\frac{t}{\mid \alpha \mid })\forall \alpha \ne 0$;

(IFN4) 

$N_{\mu ,\nu }(v_1+v_2,max\{s,t\})\ge T(N_{\mu ,\nu }(v_1,s),N_{\mu ,\nu }(v_2,t))$.

(IFN5) 

${lim}_{t\to \infty }{N}_{\mu ,\nu }(v_1,t)=1$.

If $N_{\mu ,\nu }$ is a non-Archimedean intuitionistic fuzzy Menger norm on E, then, $(E,N_{\mu ,\nu },T)$ is said to be a non-Archimedean IFN space. It is important to note that the condition (IFN4) implies

$$N_{\mu ,\nu }(v_1,t)\ge T(N_{\mu ,\nu }(0,t),N_{\mu ,\nu }(v_1,s))=N_{\mu ,\nu }(v_1,s)$$

for all $0<s<t$ and $v_1,v_2\in E$, that is, $(N_{\mu ,\nu },·)$ is increasing for every $v_1$, which gives

$$N_{\mu ,\nu }(v_1,s+t)\ge N_{\mu ,\nu }(v_1,max\{s,t\}).$$

If (IFN4) holds, then,

$$\left(IFN6\right)N_{\mu ,\nu }(v_1+v_2,s+t)\ge T(N_{\mu ,\nu }(v_1,s),N_{\mu ,\nu }(v_2,t)).$$

We frequently employ that

$$N(-v_1,t)=N(v_1,t),v_1\in E,t>0,$$

which is derived from (IFN3).

Definition 7.

Let a non-Archimedean IFN space $(E,N_{\mu ,\nu },T)$ and ${\left\{v_m\right\}}_{m\in \mathbb{N}}$ in E. Then, ${\left\{v_m\right\}}_{m\in \mathbb{N}}$ is called convergent if there is $v\in E$ that satisfies

$$\underset{m\to \infty }{lim}{N}_{\mu ,\nu }(v_m-v,t)=1$$

for every $t>0$.

Here, v is said to be a limit of ${\left\{v_m\right\}}_{m\in \mathbb{N}}$, and we refer to it as

$$\underset{m\to \infty }{lim}{v}_m=v.$$

The sequence ${\left\{v_m\right\}}_{m\in \mathbb{N}}$ in E is called a Cauchy sequence if

$$\underset{m\to \infty }{lim}{N}_{\mu ,\nu }(v_{m+i}-v_m,t)=1$$

for every $t>0$ and $i=1,2,3,\cdots$.

A complete non-Archimedean IFN space is defined as one in which every Cauchy sequence in E is convergent.

Example 1.

[37] Let $(E,\parallel ·\parallel )$ be a normed space. Let $T(u,v)=(uv,min(u_2+v_2,1))$ for all $u=(u_1,u_2)$, $v=(v_1,v_2)\in L^{*}$ and let $\mu ,\nu$ be membership and non-membership degrees of an intuitionistic fuzzy set defined by

$$N_{\mu ,\nu }(v,t)=\left(\frac{t}{t+\parallel v\parallel },\frac{\parallel v\parallel }{t+\parallel v\parallel }\right),\forall t\in R^{+}.$$

Then, the triple $(E,N_{\mu ,\nu },T)$ is an IFN space.

For specific later use, we note the subsequent results by Diaz and Margolis [38].

Theorem 1.

Let $(W,d)$ be a generalized complete metric space and a strictly contractive mapping $M:W\to W$ with Lipschitz constant $0<L<1$. Then, for all $v_1\in W$, either

$$\begin{array}{c}\hfill d\left(M^m{v}_1,M^{m+1}{v}_1\right)=\infty ,m\ge m_0;\end{array}$$

or there exists a positive integer $m_0$ such that

(i) 

$d\left(M^m{v}_1,M^{m+1}{v}_1\right)<\infty ,m\ge m_0$;

(ii) 

the sequence ${\left\{M^m{v}_1\right\}}_{m\in \mathbb{N}}$ converges to a fixed-point $v_1^{*}$ of M;

(iii) 

$v_1^{*}$ is the unique fixed-point of M in $W^{*}=\{v_2\in W|d(M^{m_0}{v}_1,v_2)<\infty \}$;

(iv) 

$d(v_2,v_1^{*})\le \frac{1}{1-L}d(M{v}_2,v_2)$, for every $v_2\in W^{*}$.

Throughout the Section 3 and Section 4, we consider $\mathbb{K}$ as a valued field, E and F are vector spaces over $\mathbb{K}$ and $(F,N_{\mu ,\nu },T)$ is a complete non-Archimedean IFN space over $\mathbb{K}$. Additionally, consider that $\alpha ,\beta \in \mathbb{K}$ are fixed with $\theta :={\alpha }^2+{\beta }^2\ne 0,1$ (${\theta }_1:=2\alpha \ne 0,1$ if $\alpha =\beta$) and the set of all positive integers is denoted by $\mathbb{N}$, whereas the set of all reals is denoted by $\mathbb{R}$ (or $\mathbb{Q}$) (or rationals).

Theorem 2.

If a mapping $\varphi :E\to F$ satisfies the Euler–Lagrange functional Equation (1), then, the function ϕ is quadratic.

Proof. 

Letting $v_1=v_2=0$ in (1), we obtain $\varphi \left(0\right)=0$. Setting $v_2=0$ in (1), we have

$$\varphi \left(\alpha v_1\right)+\varphi \left(\beta v_1\right)=({\alpha }^2+{\beta }^2)\varphi \left(v_1\right)$$

(2)

for all $v_1\in E$. Replacing $v_1$ by $\alpha v_1$ and $v_2$ by $\beta v_1$ in (1), respectively, we obtain

$$\varphi ({\alpha }^2{v}_1+{\beta }^2{v}_1)=({\alpha }^2+{\beta }^2)\left[\varphi \left(\alpha v_1\right)+\varphi \left(\beta v_1\right)\right]$$

(3)

for all $v_1\in E$. Using Equation (2), we obtain that

$$\varphi \left(({\alpha }^2+{\beta }^2)v_1\right)={({\alpha }^2+{\beta }^2)}^2\varphi \left(v_1\right)$$

(4)

for all $v_1\in E$. From Equation (4), we get

$$\varphi \left(\theta v_1\right)={\theta }^2\varphi \left(v_1\right)$$

for all $v_1\in E$, where $\theta ={\alpha }^2+{\beta }^2$. Now, if $\alpha =\beta$ in Equation (1), we have

$$\varphi \left(\alpha (v_1+v_2)\right)+\varphi \left(\alpha (v_1-v_2)\right)=2{\alpha }^2\left[\varphi \left(v_1\right)+\varphi \left(v_2\right)\right]$$

(5)

for all $v_1,v_2\in E$. Replacing $v_1=v_2$ in Equation (5), we have

$$\varphi \left({\theta }_1{v}_1\right)={\theta }_1^2\varphi \left(v_1\right)$$

for all $v_1\in E$, where ${\theta }_1=2\alpha$. Hence the function $\varphi$ is quadratic. □

3. Hyers-Ulam Stability: Direct Technique

Theorem 3.

Suppose that ${\mathrm{\Phi }}_{\mu ,\nu }:E^2\times [0,\infty )\to [0,1]$ is a mapping such that

$$\begin{array}{c}\hfill \underset{i\to \infty }{lim}{\mathrm{\Phi }}_{\mu ,\nu }({\theta }^i{v}_1,{\theta }^i{v}_2{,\left|\theta \right|}^{2i}t)=1\end{array}$$

(6)

and

$$\begin{array}{ccc}\hfill \underset{l\to \infty }{lim}{T}_{i=l}^{\infty }& T& ({\mathrm{\Phi }}_{\mu ,\nu }({\theta }^i{v}_1{,0,\left|\theta \right|}^{2i+1}t),{\mathrm{\Phi }}_{\mu ,\nu }(\alpha {\theta }^i{v}_1,\beta {\theta }^i{v}_1{,\left|\theta \right|}^{2(i+1)}t\left)\right)\hfill \\ & =& \underset{l\to \infty }{lim}{T}_{i=1}^{\infty }T({\mathrm{\Phi }}_{\mu ,\nu }({\theta }^{l+i-1}{v}_1{,0,\left|\theta \right|}^{2i+2l-1}t),\hfill \\ & & {\mathrm{\Phi }}_{\mu ,\nu }(\alpha {\theta }^{l+i-1}{v}_1,\beta {\theta }^{l+i-1}{v}_1{,\left|\theta \right|}^{2l+2i}t\left)\right)\hfill \\ & =& 1\hfill \end{array}$$

(7)

for all $v_1,v_2\in E$ and $t>0$. If a mapping $\varphi :E\to F$ is defined by (1), which satisfying

$$\varphi \left(0\right)=0,$$

(8)

and

$$N_{\mu ,\nu }\left(\varphi (\alpha v_1+\beta v_2)+\varphi (\beta v_1-\alpha v_2)-({\alpha }^2+{\beta }^2)[\varphi \left(v_1\right)+\varphi \left(v_2\right)],t\right)\ge {\mathrm{\Phi }}_{\mu ,\nu }(v_1,v_2,t)$$

(9)

for all $v_1,v_2\in E$ and $t\in [0,\infty )$, then there exists a unique general Euler–Lagrange quadratic mapping $Q_2:E\to F$ satisfies

$$\begin{array}{c}\hfill N_{\mu ,\nu }\left(\varphi \left(v_1\right)-Q_2\left(v_1\right),t\right)\ge T_{i=1}^{\infty }T\left({\mathrm{\Phi }}_{\mu ,\nu }({\theta }^{i-1}{v}_1{,0,\left|\theta \right|}^{2i-1}t),{\mathrm{\Phi }}_{\mu ,\nu }(\alpha {\theta }^{i-1}{v}_1,\beta {\theta }^{i-1}{v}_1{,\left|\theta \right|}^{2i}t)\right)\end{array}$$

(10)

for all $v_1\in E$ and all $t>0$.

Proof. 

Fix $v_1\in E$ and $t>0$. Putting $v_2=0$ in (9), we have

$$\begin{array}{c}\hfill N_{\mu ,\nu }\left(\varphi \left(\alpha v_1\right)+\varphi \left(\beta v_1\right)-\theta \varphi \left(v_1\right),t\right)\ge {\mathrm{\Phi }}_{\mu ,\nu }(v_1,0,t).\end{array}$$

(11)

From inequality (11), we get

$$\varphi \left(\alpha v_1\right)+\varphi \left(\beta v_1\right)=\theta \varphi \left(v_1\right).$$

(12)

Replacing $v_1$ by $\alpha v_1$ and $v_2$ by $\beta v_1$ in (9), respectively, we have

$$\begin{array}{c}\hfill N_{\mu ,\nu }\left(\varphi \left(\theta v_1\right)-\theta \varphi \left(\alpha v_1\right)-\theta \varphi \left(\beta v_1\right),t\right)\ge {\mathrm{\Phi }}_{\mu ,\nu }(\alpha v_1,\beta v_1,t).\end{array}$$

(13)

Using (12) and Definition 6 in (13), we get

$$\begin{array}{c}\hfill N_{\mu ,\nu }\left(\frac{1}{{\theta }^2}\varphi \left(\theta v_1\right)-\varphi \left(v_1\right),t\right)\ge T\left({\mathrm{\Phi }}_{\mu ,\nu }(v_1,0,\left|\theta \right|t),{\mathrm{\Phi }}_{\mu ,\nu }(\alpha v_1,\beta v_1{,\left|\theta \right|}^{2}t)\right).\end{array}$$

(14)

Therefore, one can get

$$\begin{array}{ccc}& & N_{\mu ,\nu }\left(\frac{1}{{\theta }^{2(l+m)}}\varphi \left({\theta }^{l+m}{v}_1\right)-\frac{1}{{\theta }^{2l}}\varphi \left({\theta }^l{v}_1\right),t\right)\hfill \\ & & \ge T_{i=l}^{l+m-1}T\left({\mathrm{\Phi }}_{\mu ,\nu }({\theta }^i{v}_1{,0,\left|\theta \right|}^{2i+1}t),{\mathrm{\Phi }}_{\mu ,\nu }(\alpha {\theta }^i{v}_1,\beta {\theta }^i{v}_1{,\left|\theta \right|}^{2(i+1)}t)\right),\hfill \end{array}$$

and thus from (7), it follows that the sequence ${\left\{\frac{\varphi \left({\theta }^i{v}_1\right)}{{\theta }^{2i}}\right\}}_{i\in \mathbb{N}}$ is a Cauchy sequence in a complete non-Archimedean IFN space.

Thus, we can define a mapping $Q_2:E\to F$ by

$$\underset{i\to \infty }{lim}{N}_{\mu ,\nu }\left(\frac{1}{{\theta }^{2i}}\varphi \left({\theta }^i{v}_1\right)-Q_2\left(v_1\right),t\right)=1.$$

Next, for each $l\in \mathbb{N}$ with $l\ge 1$, we have

$$\begin{array}{ccc}\hfill N_{\mu ,\nu }\left(\varphi \left(v_1\right)-\frac{1}{{\theta }^{2l}}\varphi \left({\theta }^l{v}_1\right),t\right)& \ge & T_{i=1}^l{N}_{\mu ,\nu }\left(\frac{1}{{\theta }^{2(i-1)}}\varphi \left({\theta }^{i-1}{v}_1\right)-\frac{1}{{\theta }^{2i}}\varphi \left({\theta }^i{v}_1\right),t\right)\hfill \\ & \ge & T_{i=1}^{l}T\left({\mathrm{\Phi }}_{\mu ,\nu }({\theta }^{i-1}{v}_1{,0,\left|\theta \right|}^{2i-1}t),{\mathrm{\Phi }}_{\mu ,\nu }(\alpha {\theta }^{i-1}{v}_1,\beta {\theta }^{i-1}{v}_1{,\left|\theta \right|}^{2i}t)\right).\hfill \end{array}$$

Therefore,

$$\begin{array}{ccc}\hfill N_{\mu ,\nu }\left(\varphi \left(v_1\right)-Q_2\left(v_1\right),t\right)& \ge & T\left(N_{\mu ,\nu }\left(\varphi \left(v_1\right)-\frac{1}{{\theta }^{2l}}\varphi \left({\theta }^l{v}_1\right),t\right),N_{\mu ,\nu }\left(\frac{1}{{\theta }^{2l}}\varphi \left({\theta }^l{v}_1\right)-Q_2\left(v_1\right),t\right)\right)\hfill \\ & \ge & T\left(T_{i=1}^{l}T\left({\mathrm{\Phi }}_{\mu ,\nu }\left({\theta }^{i-1}{v}_1,{\left|\theta \right|}^{2i-1}t\right),{\mathrm{\Phi }}_{\mu ,\nu }\left(\alpha {\theta }^{i-1}{v}_1,\beta {\theta }^{i-1}{v}_1,{\left|\theta \right|}^{2i}t\right)\right)\right),\hfill \\ & & N_{\mu ,\nu }\left({\theta }^{-2l}\varphi \left({\theta }^l{v}_1\right)-Q_2\left(v_1),t\right)\right).\hfill \end{array}$$

Letting $l\to \infty$ in this inequality, we obtain (10). Now, also fixing $v_2\in E$, from (6) and (9) it follows that

$$\begin{array}{ccc}& & N_{\mu ,\nu }\left(Q_2\left(\alpha v_1+\beta v_2\right)+Q_2\left(\beta v_1-\alpha v_2\right)-\left({\alpha }^2+{\beta }^2\right)\left[Q_2\left(v_1\right)+Q_2\left(v_2\right)\right],t\right)\hfill \\ & & \ge T(N_{\mu ,\nu }\left(Q_2\left(\alpha v_1+\beta v_2\right)-{\theta }^{-2i}\varphi \left({\theta }^i\left(\alpha v_1+\beta v_2\right)\right),t\right)\hfill \\ & & N_{\mu ,\nu }\left(Q_2\left(\beta v_1-\alpha v_2\right)-{\theta }^{-2i}\varphi \left({\theta }^i\left(\beta v_1-\alpha v_2\right)\right),t\right),\hfill \\ & & N_{\mu ,\nu }\left(-\theta Q_2\left(v_1\right)+{\theta }^{-2i+1}\varphi \left({\theta }^i{v}_1\right),t\right),N_{\mu ,\nu }\left(\theta Q_2\left(v_2\right)+{\theta }^{-2i+1}\varphi \left({\theta }^i{v}_2\right),t\right),\hfill \\ & & {\mathrm{\Phi }}_{\mu ,\nu }\left({\theta }^i{v}_1,{\theta }^i{v}_2,{\left|\theta \right|}^{2i}t\right))\to 1(asi\to \infty ).\hfill \end{array}$$

Hence, the function $Q_2$ is quadratic. Consider that $Q_2^{{}^{\prime }}:E\to F$ is an another Euler–Lagrange quadratic function which satisfying (10). Hence, by $\varphi \left(\theta v_1\right)={\theta }^2\varphi \left(v_1\right)$ (by Theorem 2) and (10), (7), it follows that

$$\begin{array}{ccc}\hfill N_{\mu ,\nu }\left(Q_2\left(v_1\right)-Q_2^{{}^{\prime }}\left(v_1\right),t\right)& =& N_{\mu ,\nu }\left(Q_2\left({\theta }^l{v}_1\right)-Q_2^{{}^{\prime }}\left({\theta }^l{v}_1\right),{\left|\theta \right|}^{2l+2i-1}t\right)\hfill \\ & \ge & T\left(T_{i=1}^{\infty }T\right({\mathrm{\Phi }}_{\mu ,\nu }\left({\theta }^{l+i-1}{v}_1,0,{\left|\theta \right|}^{2l+2i-1}t\right),\hfill \\ & & {\mathrm{\Phi }}_{\mu ,\nu }\left(\alpha {\theta }^{l+i-1}{v}_1,\beta {\theta }^{l+i-1}{v}_1,{\left|\theta \right|}^{2l+2i}t\right)),\hfill \\ & & T_{i=1}^{\infty }T({\mathrm{\Phi }}_{\mu ,\nu }\left({\theta }^{l+i-1}{v}_1,0,{\left|\theta \right|}^{2l+2i-1}t\right),\hfill \\ & & {\mathrm{\Phi }}_{\mu ,\nu }\left(\alpha {\theta }^{l+i-1}{v}_1,\beta {\theta }^{l+i-1}{v}_1,{\left|\theta \right|}^{2l+2i}t\right)\left)\right)\hfill \\ & & \to 1(asl\to \infty ),\hfill \end{array}$$

and therefore, $Q_2=Q_2^{{}^{\prime }}$. This ends the proof. □

Theorem 4.

Suppose that ${\mathrm{\Phi }}_{\mu ,\nu }:E^2\times [0,\infty )\to [0,1]$ is a mapping such that

$$\begin{array}{c}\hfill \underset{i\to \infty }{lim}{\mathrm{\Phi }}_{\mu ,\nu }({\theta }^{-i}{v}_1,{\theta }^{-i}{v}_2{,\left|\theta \right|}^{-2i}t)=1\end{array}$$

(15)

and

$$\begin{array}{ccc}\hfill \underset{l\to \infty }{lim}{T}_{i=l}^{\infty }& T& ({\mathrm{\Phi }}_{\mu ,\nu }({\theta }^{-i-1}{v}_1{,0,\left|\theta \right|}^{-2i-1}t),{\mathrm{\Phi }}_{\mu ,\nu }(\alpha {\theta }^{-i-1}{v}_1,\beta {\theta }^{-i-1}{v}_1{,\left|\theta \right|}^{-2i}t\left)\right)\hfill \\ & =& \underset{l\to \infty }{lim}{T}_{i=1}^{\infty }T({\mathrm{\Phi }}_{\mu ,\nu }({\theta }^{-l-i-1}{v}_1{,0,\left|\theta \right|}^{-2i-2l-1}t),\hfill \\ & & {\mathrm{\Phi }}_{\mu ,\nu }(\alpha {\theta }^{-l-i-1}{v}_1,\beta {\theta }^{-l-i-1}{v}_1{,\left|\theta \right|}^{-2l-2i}t\left)\right)\hfill \\ & =& 1\hfill \end{array}$$

(16)

for every $v_1,v_2\in E$ and $t>0$. If a mapping $\varphi :E\to F$ satisfying (8) and (9), then there exists a unique general Euler–Lagrange quadratic mapping $Q_2:E\to F$ satisfies

$$\begin{array}{c}\hfill N_{\mu ,\nu }\left(\varphi \left(v_1\right)-Q_2\left(v_1\right),t\right)\ge T_{i=1}^{\infty }T\left({\mathrm{\Phi }}_{\mu ,\nu }({\theta }^{-i-1}{v}_1,0){,\left|\theta \right|}^{-2i-1}t),{\mathrm{\Phi }}_{\mu ,\nu }(\alpha {\theta }^{-i-1}{v}_1,\beta {\theta }^{-i-1}{v}_1{,\left|\theta \right|}^{-2i}t)\right)\end{array}$$

for all $v_1\in E$ and all $t>0$.

The subsequent main theorems are investigated by considering $\alpha =\beta$ in (1).

Theorem 5.

Suppose that ${\mathrm{\Phi }}_{\mu ,\nu }:E^2\times [0,\infty )\to [0,1]$ is a mapping that satisfies

$$\begin{array}{c}\hfill \underset{i\to \infty }{lim}{\mathrm{\Phi }}_{\mu ,\nu }\left({\theta }_1^i{v}_1,{\theta }_1^i{v}_2,{\left|{\theta }_1\right|}^{2i}t\right)=1\end{array}$$

(17)

and

$$\begin{array}{ccc}\hfill \underset{l\to \infty }{lim}{T}_{i=l}^{\infty }{\mathrm{\Phi }}_{\mu ,\nu }\left({\theta }_1^i{v}_1,{\theta }_1^i{v}_1,{\left|{\theta }_1\right|}^{2i+2}t\right)& =& \underset{l\to \infty }{lim}{T}_{i=1}^{\infty }{\mathrm{\Phi }}_{\mu ,\nu }\left({\theta }_1^{l+i-1}{v}_1,{\theta }_1^{l+i-1}{v}_1,{\left|{\theta }_1\right|}^{2i+2l}t\right)\hfill \\ & =& 1\hfill \end{array}$$

(18)

for all $v_1,v_2\in E$ and all $t>0$. If a mapping $\varphi :E\to F$ satisfies (8) and

$$\begin{array}{c}\hfill N_{\mu ,\nu }\left(\varphi \left(\alpha \left(v_2+v_2\right)\right)+\varphi \left(\alpha \left(v_1-v_2\right)\right)-2{\alpha }^2\left[\varphi \left(v_1\right)+\varphi \left(v_2\right)\right],t\right)\ge {\mathrm{\Phi }}_{\mu ,\nu }\left(v_1,v_2,t\right)\end{array}$$

(19)

for all $v_1,v_2\in E$ and all $t\in [0,\infty )$, then there exists a unique general Euler–Lagrange quadratic mapping $Q_2:E\to F$ satisfying

$$\varphi \left(\alpha (v_1+v_2)\right)+\varphi \left(\alpha (v_1-v_2)\right)=2{\alpha }^2[\varphi \left(v_1\right)+\varphi \left(v_2\right)]$$

and such that

$$\begin{array}{c}\hfill N_{\mu ,\nu }\left(\varphi \left(v_1\right)-Q_2\left(v_1\right),t\right)\ge T_{i=1}^{\infty }{\mathrm{\Phi }}_{\mu ,\nu }\left({\theta }_1^{i-1}{v}_1,{\theta }_1^{i-1}{v}_1,{\left|{\theta }_1\right|}^{2i}t\right)\end{array}$$

(20)

for all $v_1\in E$ and all $t>0$.

Proof. 

Fix $v_1,v_2\in E$ and $t>0$. Putting $v_2=v_1$ in (19), we get

$$\begin{array}{c}\hfill N_{\mu ,\nu }\left(\frac{1}{{\theta }_1^2}\varphi \left({\theta }_1{v}_1\right)-\varphi \left(v_1\right),t\right)\ge {\mathrm{\Phi }}_{\mu ,\nu }\left(v_1,v_1,{\left|{\theta }_1\right|}^{2}t\right).\end{array}$$

(21)

Hence,

$$\begin{array}{c}\hfill N_{\mu ,\nu }\left(\frac{1}{{\theta }_1^{2(i+1)}}\varphi \left({\theta }_1^{i+1}{v}_1\right)-\frac{1}{{\theta }_1^{2i}}\varphi \left({\theta }_1^i{v}_1\right),t\right)\ge {\mathrm{\Phi }}_{\mu ,\nu }\left({\theta }_1^i{v}_1,{\theta }_1^i{v}_1,{\left|{\theta }_1\right|}^{2i+2}t\right).\end{array}$$

Therefore, one can get

$$\begin{array}{ccc}\hfill N_{\mu ,\nu }\left(\frac{1}{{\theta }_1^{2(l+m)}}\varphi \left({\theta }_1^{l+m}{v}_1\right)-\frac{1}{{\theta }_1^{2l}}\varphi \left({\theta }_1^l{v}_1\right),t\right)& \ge & T_{i=l}^{l+m-1}{N}_{\mu ,\nu }\left(\frac{1}{{\theta }_1^{2(i+1)}}\varphi \left({\theta }_1^{i+1}\right)-\frac{1}{{\theta }_1^{2i}}\varphi \left({\theta }_1^i{v}_1\right),t\right)\hfill \\ & \ge & T_{i=l}^{l+m-1}{\mathrm{\Phi }}_{\mu ,\nu }\left({\theta }_1^i{v}_1,{\theta }_1^i{v}_1,{\left|{\theta }_1\right|}^{2i+2}t\right),\hfill \end{array}$$

and thus, (18) it follows that the sequence ${\left\{{\theta }_1^{-2i}\varphi \left({\theta }_1^i{v}_1\right)\right\}}_{i\in \mathbb{N}}$ is a Cauchy sequence in a complete non-Archimedean IFN space.

Hence, we can define a mapping $Q_2:E\to F$ by

$$\underset{i\to \infty }{lim}{N}_{\mu ,\nu }\left(\frac{1}{{\theta }_1^{2i}}\varphi \left({\theta }_1^i{v}_1\right)-Q_2\left(v_1\right),t\right)=1.$$

Utilizing (21) and by induction, for any $l\in \mathbb{N}$, we obtain

$$\begin{array}{c}\hfill N_{\mu ,\nu }\left(\varphi \left(v_1\right)-\frac{1}{{\theta }_1^{2l}}\varphi \left({\theta }_1^l{v}_1\right),t\right)\ge T_{i=1}^l{\mathrm{\Phi }}_{\mu ,\nu }\left({\theta }_1^{i-1}{v}_1,{\theta }_1^{i-1}{v}_1,{\left|{\theta }_1\right|}^{2i}t\right).\end{array}$$

Therefore,

$$\begin{array}{c}\hfill N_{\mu ,\nu }\left(\varphi \left(v_1\right)-Q_2\left(v_1\right),t\right)\ge T\left(T_{i=1}^l{\mathrm{\Phi }}_{\mu ,\nu }\left({\theta }_1^{i-1}{v}_1,{\theta }_1^{i-1}{v}_1,{\left|{\theta }_1\right|}^{2i}t\right),N_{\mu ,\nu }\left({\theta }_1^{-2l}\varphi \left({\theta }_1^l{v}_1\right)-Q_2\left(v_1\right),t\right)\right).\end{array}$$

Letting $l\to \infty$ in this inequality, we obtain (20). The remaining proof of this theorem is omitted as comparable to that of Theorem 3. □

Remark 1.

Let $\alpha ,\beta \in \mathbb{N}$ and E be a commutative group, Theorems 3–5 also hold. For $\alpha =1$, consider the non-Archimedean intuitionistic fuzzy normed space $(F,N_{\mu ,\nu },T_M)$ defined as in Example 1, Theorem 5 yields Theorem 2 in [21]. If $\alpha =\beta =\pm \frac{1}{\sqrt{2}}\in \mathbb{K}$, then, ${\theta }_1=2\alpha =\pm \sqrt{2}\ne 1$ in Theorems 4 and 5 and $\theta ={\alpha }^2+{\beta }^2=1$ which is a singular case $\theta =1$ of Theorem 3.

Theorem 6.

Suppose that ${\mathrm{\Phi }}_{\mu ,\nu }:E^2\times [0,\infty )\to [0,1]$ is a mapping such that

$$\begin{array}{c}\hfill \underset{i\to \infty }{lim}{\mathrm{\Phi }}_{\mu ,\nu }\left({\theta }_1^{-i}{v}_1,{\theta }_1^{-i}{v}_2,{\left|{\theta }_1\right|}^{-2i}t\right)=1\end{array}$$

(22)

and

$$\begin{array}{ccc}\hfill \underset{l\to \infty }{lim}{T}_{i=l}^{\infty }{\mathrm{\Phi }}_{\mu ,\nu }\left({\theta }_1^{-i-1}{v}_1,{\theta }_1^{-i-1}{v}_1,{\left|{\theta }_1\right|}^{-2i}t\right)& =& \underset{l\to \infty }{lim}{T}_{i=1}^{\infty }{\mathrm{\Phi }}_{\mu ,\nu }\left({\theta }_1^{-l-i-1}{v}_1,{\theta }_1^{-l-i-1}{v}_1,{\left|{\theta }_1\right|}^{-2i-2l}t\right)\hfill \\ & =& 1\hfill \end{array}$$

(23)

for all $v_1,v_2\in E$ and $t>0$. If a mapping $\varphi :E\to F$ satisfying (8) and

$$\begin{array}{c}\hfill N_{\mu ,\nu }\left(\varphi \left(\alpha \left(v_2+v_2\right)\right)+\varphi \left(\alpha \left(v_1-v_2\right)\right)-2{\alpha }^2\left[\varphi \left(v_1\right)+\varphi \left(v_2\right)\right],t\right)\ge {\mathrm{\Phi }}_{\mu ,\nu }\left(v_1,v_2,t\right)\end{array}$$

(24)

for all $v_1,v_2\in E$ and $t\in [0,\infty )$, then there exists a unique general Euler–Lagrange quadratic mapping $Q_2:E\to F$ satisfying

$$\varphi \left(\alpha (v_1+v_2)\right)+\varphi \left(\alpha (v_1-v_2)\right)=2{\alpha }^2[\varphi \left(v_1\right)+\varphi \left(v_2\right)]$$

and such that

$$\begin{array}{c}\hfill N_{\mu ,\nu }\left(\varphi \left(v_1\right)-Q_2\left(v_1\right),t\right)\ge T_{i=1}^{\infty }{\mathrm{\Phi }}_{\mu ,\nu }\left({\theta }_1^{-i-1}{v}_1,{\theta }_1^{-i-1}{v}_1,{\left|{\theta }_1\right|}^{-2i}t\right)\end{array}$$

(25)

for all $v_1\in E$ and all $t>0$.

4. Hyers–Ulam Stability: Fixed-Point Technique

Theorem 7.

Suppose that ${\mathrm{\Phi }}_{\mu ,\nu }:E^2\times [0,\infty )\to [0,1]$ is a mapping such that (6) holds and

$${\mathrm{\Phi }}_{\mu ,\nu }\left(\theta v_1,\theta v_1,{\left|\theta \right|}^{2}Lt\right)\ge {\mathrm{\Phi }}_{\mu ,\nu }\left(v_1,v_1,t\right),v_1\in E$$

(26)

for $L\in (0,1)$. If $\varphi :E\to F$ is a mapping satisfying (8) and (9), then there exists a unique general Euler–Lagrange quadratic mapping $Q_2:E\to F$ satisfies

$$\begin{array}{c}\hfill N_{\mu ,\nu }\left(\varphi \left(v_1\right)-Q_2\left(v_1\right),t\right)\ge T\left({\mathrm{\Phi }}_{\mu ,\nu }\left(v_1,0,\left|\theta \right|(1-L)t\right),{\mathrm{\Phi }}_{\mu ,\nu }\left(\alpha v_1,\beta v_1,{\left|\theta \right|}^2(1-L)t\right)\right)\end{array}$$

(27)

for all $v_1\in E$ and all $t>0$.

Proof. 

Defining the set $W:=\{p:E\to F\}$ and introducing the generalized metric on W:

$$\begin{array}{ccc}\hfill d(p,q)& =& inf\{c\in [0,\infty ]:N_{\mu ,\nu }\left(p\left(v_1\right)-q\left(v_1\right),ct\right)\hfill \\ & \ge & T\left({\mathrm{\Phi }}_{\mu ,\nu }\left(v_1,0,\left|\theta \right|t\right),{\mathrm{\Phi }}_{\mu ,\nu }\left(\alpha v_1,\beta v_1,{\left|\theta \right|}^{2}t\right)\right),v_1\in E,t>0\}\hfill \end{array}$$

for all $p,q\in W$. A standard verification (see for instance [39]) proves that $(W,d)$ is a complete generalized metric space. Now, we can define a function $S:W\to W$ by

$$Sp\left(v_1\right)=\frac{1}{{\theta }^2}p\left(\theta v_1\right),p\in W,v_1\in E.$$

Let $p,q\in W$ and $c_{p,q}\in [0,\infty ]$ with $d(p,q)\le c_{p,q}$. Then,

$$\begin{array}{ccc}\hfill N_{\mu ,\nu }\left(p\left(v_1\right)-q\left(v_1\right),c_{p,q}t\right)& \ge & T\left({\mathrm{\Phi }}_{\mu ,\nu }\left(v_1,0,\left|\theta \right|t\right),{\mathrm{\Phi }}_{\mu ,\nu }\left(\alpha v_1,\beta v_1,{\left|\theta \right|}^{2}t\right)\right),\hfill \end{array}$$

which together with (26) gives

$$\begin{array}{cc}\hfill N_{\mu ,\nu }\left(Sp\left(v_1\right)-Sq\left(v_1\right),t\right)& \ge T\left({\mathrm{\Phi }}_{\mu ,\nu }\left(v_1,0,\frac{\left|\theta \right|t}{L{c}_{p,q}}\right),{\mathrm{\Phi }}_{\mu ,\nu }\left(\alpha v_1,\beta v_1,\frac{{\left|\theta \right|}^{2}t}{L{c}_{p,q}}\right)\right),\end{array}$$

and consequently, $d(Sp,Sq)\le L{c}_{p,q}$, this indicates that S is strictly contractive. In addition, it follows from (14) that

$$\begin{array}{c}\hfill N_{\mu ,\nu }\left(S\varphi \left(v_1\right)-\varphi \left(v_1\right),t\right)\ge T\left({\mathrm{\Phi }}_{\mu ,\nu }\left(v_1,0,\left|\theta \right|t\right),{\mathrm{\Phi }}_{\mu ,\nu }\left(\alpha v_1,\beta v_1,{\left|\theta \right|}^{2}t\right)\right)\end{array}$$

and thus, $d(S\varphi ,\varphi )\le 1<\infty$. Thus, by Theorem 1, S has a unique fixed-point $Q_2:E\to F$ in the set $W^{*}=\{p\in W:d(\varphi ,p)<\infty \}$ such that

$$\begin{array}{c}\hfill \frac{1}{{\theta }^2}{Q}_2\left(\theta v_1\right)=Q_2\left(v_1\right)\end{array}$$

(28)

and

$$Q_2\left(v_1\right)=\underset{i\to \infty }{lim}\frac{1}{{\theta }^{2i}}\varphi \left({\theta }^i{v}_1\right),v_1\in E.$$

In addition, the fact that $\varphi \in W^{*}$, Theorem 1, and $d(S\varphi ,\varphi )\le 1$, we get

$$d(\varphi ,Q_2)\le \frac{1}{1-L}d\left(S\varphi ,\varphi \right)\le \frac{1}{1-L}$$

(29)

and (27) follows. The proof of Theorem 3 may also be used to show that the function $Q_2$ is quadratic.

At the end, consider that $Q_2^{{}^{\prime }}:E\to F$ is another general Euler–Lagrange quadratic mapping which satisfying (27). Then, $Q_2^{{}^{\prime }}$ satisfies (28). Therefore, it is a fixed point of S.

Thus, by (27), we obtain

$$d\left(\varphi ,Q_2^{{}^{\prime }}\right)\le \frac{1}{1-L}<\infty ,$$

and hence $Q_2^{{}^{\prime }}\in W^{*}$. Theorem 1 proves that $Q_2^{{}^{\prime }}=Q_2$, that is, the function $Q_2$ is unique, which ends the proof of the Theorem. □

Theorem 8.

Suppose that ${\mathrm{\Phi }}_{\mu ,\nu }:E^2\times [0,\infty )\to [0,1]$ is a mapping such that (15) holds and

$${\mathrm{\Phi }}_{\mu ,\nu }\left({\theta }^{-1}{v}_1,{\theta }^{-1}{v}_1,{\left|\theta \right|}^{-2}Lt\right)\ge {\mathrm{\Phi }}_{\mu ,\nu }\left(v_1,v_1,t\right),v_1\in E$$

for $L\in (0,1)$. If $\varphi :E\to F$ is a mapping satisfying (8) and (9), then there exists a unique general Euler–Lagrange quadratic mapping $Q_2:E\to F$ satisfies

$$\begin{array}{c}\hfill N_{\mu ,\nu }\left(\varphi \left(v_1\right)-Q_2\left(v_1\right),t\right)\ge T\left({\mathrm{\Phi }}_{\mu ,\nu }\left(v_1,0,\left|\theta \right|(L^{-1}-1)t\right),{\mathrm{\Phi }}_{\mu ,\nu }\left(\alpha v_1,\beta v_1,{\left|\theta \right|}^2(L^{-1}-1)t\right)\right)\end{array}$$

for all $v_1\in E$ and all $t>0$.

Corollary 1.

Let E be a real normed space, F be a real Banach space (or E be a non-Archimedean normed space and F be a complete non-Archimedean normed space over a non-Archimedean filed $\mathbb{K}$, respectively). Let $ϵ,\lambda >0$ and $s\in (0,2)\cup (2,\infty )$. If $\varphi :E\to F$ is a mapping satisfying (8) and

$$\begin{array}{c}\hfill \parallel \varphi (\alpha v_1+\beta v_2)+\varphi (\beta v_1+\alpha v_2)-({\alpha }^2+{\beta }^2)\left[\varphi \left(v_1\right)+\varphi \left(v_2\right)\right]\parallel \le \lambda \left[\parallel v_1{\parallel }^s+{\parallel v_2\parallel }^s\right]\end{array}$$

Then, there exists a unique general Euler–Lagrange quadratic mapping $Q_2:E\to F$ such that

$$\begin{array}{c}\hfill \parallel \varphi \left(v_1\right)-Q_2\left(v_1\right)\parallel \le \frac{{max\left\{\right|\theta |,|\alpha |}^s+{\left|\beta \right|}^s\}\lambda \left(\parallel v_1{\parallel }^s\right)}{{\left|\right|\theta |}^s-{\left|\theta \right|}^2|}\end{array}$$

for all $v_1\in E$.

Proof. 

Consider the non-Archimedean intuitionistic fuzzy normed space $(F,N_{\mu ,\nu }^1,T_M)$ defined in the first example in the preliminaries, ${\mathrm{\Phi }}_{\mu ,\nu }$ be defined by

$$\begin{array}{c}\hfill {\mathrm{\Phi }}_{\mu ,\nu }(v_1,v_2,t):=\frac{t}{t+\left[\parallel v_1{\parallel }^s+{\parallel v_2\parallel }^s\right]},\end{array}$$

and apply Theorems 7 and 8. □

Remark 2.

Theorems 7 and 8 can be regarded as a generalization of the classical stability result in the framework of normed spaces (see [40]). The generalized Hyers–Ulam stability problem for the case of $s=2$ was excluded in Corollary 1 (see [41]).

5. Counterexample

Here, we examine a suitable example to justify that the stability of the Equation (1) fails for a singular case. Instigated by the excellent example provided by Z. Gajda in [42], we present the upcoming counter-example which shows the instability in a particular condition $s=2$ in Corollary 1 of the Equation (1).

Remark 3.

If a mapping $\varphi :\mathbb{R}\to \mathbb{R}$ satisfies (1), then, the following conditions hold:

(1) 

$\varphi \left(m^{c/2}v\right)=m^c\varphi \left(v\right),v\in \mathbb{R},m\in \mathbb{Q}$ and $c\in \mathbb{Z}$.

(2) 

$\varphi \left(v\right)=v^2\varphi \left(1\right),v\in \mathbb{R}$ if the function ϕ is continuous.

Example 2.

Let a mapping $\varphi :\mathbb{R}\to \mathbb{R}$ defined by:

$$\begin{array}{c}\hfill \varphi \left(v\right)=\sum _{p=0}^{\infty }\frac{\psi \left({\theta }^{p}v\right)}{{\theta }^{2p}}\end{array}$$

where

$$\begin{array}{c}\hfill \psi \left(v\right)=\left\{\begin{array}{c}\lambda v^2,-1<v<1\hfill \\ \lambda ,else,\hfill \end{array}\right.\end{array}$$

(30)

and $\theta >max\left\{\right|\alpha |,|\beta |,1\}$, then, the mapping $\varphi :\mathbb{R}\to \mathbb{R}$ satisfies

$$\begin{array}{c}\hfill |\mathrm{\Theta }\varphi (v_1,v_2,\cdots ,v_m)|\le \frac{2{\theta }^4\left({\left|\alpha \right|}^2+{\left|\beta \right|}^2+1\right)}{{\theta }^2-1}\lambda \left(|v_1{|}^2+{\left|v_2\right|}^2\right)\end{array}$$

(31)

for all $v_1,v_2,\cdots ,v_m\in \mathbb{R}$, but a general Euler–Lagrange quadratic mapping $Q_{2}doesnotexist:\mathbb{R}\to \mathbb{R}$ satisfies

$$\begin{array}{c}\hfill |\varphi \left(v\right)-Q_2{\left(v\right)|\le \delta |v|}^2,v\in \mathbb{R},\end{array}$$

(32)

where λ and δ are constants.

Proof. 

We can easily find that $\varphi$ is bounded by $\frac{{\theta }^2}{{\theta }^2-1}\lambda$ on $\mathbb{R}$. If ${\sum }_{j=1}^2{\left|v_j\right|}^2\ge \frac{1}{{\theta }^2}$ or 0, then,

$$\begin{array}{c}\hfill |\mathrm{\Theta }\varphi (v_1,v_2,\cdots ,v_m)|\le \frac{2{\theta }^4\left({\left|\alpha \right|}^2+{\left|\beta \right|}^2+1\right)}{{\theta }^2-1}\lambda \left(|v_1{|}^2+{\left|v_2\right|}^2\right).\end{array}$$

Thus, (31) is valid. Next, suppose that

$$0<\sum _{j=1}^2{\left|v_j\right|}^2<\frac{1}{{\theta }^2},$$

Then, there exists an integer $l>0$ that satisfies

$$\begin{array}{c}\hfill \frac{1}{{\theta }^{2(l+2)}}\le \sum _{j=1}^2{\left|v_j\right|}^2<\frac{1}{{\theta }^{2(l+1)}}.\end{array}$$

(33)

Hence,

$$\left.\begin{array}{ccc}\hfill {\theta }^t|\alpha v_1+\beta v_2|& <\hfill & \hfill 1,\\ \hfill {\theta }^t|\beta v_1-\alpha v_2|& <\hfill & \hfill 1,\\ \hfill {\theta }^t\left|v_1\right|& <\hfill & \hfill 1,\\ \hfill {\theta }^t\left|v_2\right|& <\hfill & \hfill 1,\end{array}\right\}\in ]-1,1[,t=0,1,\cdots ,l-1.$$

By the definition of $\varphi$ and the inequality (33), we conclude that the function $\varphi$ satisfies the (31). Now, we prove that the Equation (1) is not stable for $s=2$ in Corollary 1.

Assume that a contrary that there exists a general Euler–Lagrange quadratic mapping $Q_2:\mathbb{R}\to \mathbb{R}$ which satisfies (32).

For all $v\in \mathbb{R}$, we know $\varphi$ is bounded and continuous, and $Q_2$ is limited to an open interval of origin and continuous origin.

In the perspective of Remark 3, $Q_2$ must be $Q_2\left(v\right)=c{v}^2,v\in \mathbb{R}$. Thus, we got

$$\begin{array}{c}\hfill \left|\varphi \left(v\right)\right|\le \left(\delta +\left|c\right|\right){\left|v\right|}^2,v\in \mathbb{R}.\end{array}$$

(34)

However, we can choose $l>0$ with $l\lambda >\delta +\left|c\right|$. If $v\in \left(0,\frac{1}{{\theta }^{l-1}}\right)$, then, ${\theta }^{t}v\in (0,1)$ for every $t=0,1,\cdots ,l-1$, we obtain

$$\begin{array}{c}\hfill \varphi \left(v\right)=\sum _{t=0}^{\infty }\frac{\psi \left({\theta }^{t}v\right)}{{\theta }^{2t}}\ge \sum _{t=0}^{l-1}\frac{\lambda {\left({\theta }^{t}v\right)}^2}{{\theta }^{2t}}=l\lambda v^2>\left(\delta +\left|c\right|\right){\left|v\right|}^2,\end{array}$$

which contradicts (34). □

6. Applications

Remark 4.

Let $\mathbb{K}$ be a non-Archimedean field with $0<\left|\theta \right|<1$ and $0<|{\theta }_1|<1$, E be a normed space over $\mathbb{K}$ and $(F,N_{\mu ,\nu },T)$ be a complete non-Archimedean IFN space over $\mathbb{K}$ under a t-norm $T\in H$. Consider that $\psi :[0,\infty )\to [0,\infty )$ is a function such that

$$\psi \left({\left|\theta \right|}^{-1}\right){<\left|\theta \right|}^{-1};\psi \left(|{\theta }_1{|}^{-1}\right)<{\left|{\theta }_1\right|}^{-1}$$

and

$$\psi \left({\left|\theta \right|}^{-1}t\right)\le \psi \left({\left|\theta \right|}^{-1}\right)\psi \left(t\right);\psi \left(|{\theta }_1{|}^{-1}t\right)\le \psi \left(|{\theta }_1{|}^{-1}\right)\psi \left(t\right)$$

for all $t\in [0,\infty )$.

As applications of Theorems 3–6, we get the subsequent corollaries by the support of Example 1 with the assumption that ${\mathrm{\Phi }}_{\mu ,\nu }:E^2\times [0,\infty )\to [0,1]$ is defined by

$${\mathrm{\Phi }}_{\mu ,\nu }(v_1,v_2,t):=\left(\frac{t}{t+\psi \left(\parallel v_1\parallel \right)\psi \left(\parallel v_2\parallel \right)},\frac{\psi \left(\parallel v_1\parallel \right)\psi \left(\parallel v_2\parallel \right)}{t+\psi \left(\parallel v_1\parallel \right)\psi \left(\parallel v_2\parallel \right)}\right).$$

(35)

Corollary 2.

If a mapping $\varphi :E\to F$ satisfies (8) and

$$\begin{array}{c}\hfill N_{\mu ,\nu }\left(\varphi \left(\alpha v_1+\beta v_2\right)+\varphi \left(\beta v_1-\alpha v_2\right)-\left({\alpha }^2+{\beta }^2\right)\left[\varphi \left(v_1\right)+\varphi \left(v_2\right)\right],t\right)\ge {\mathrm{\Phi }}_{\mu ,\nu }(v_1,v_2,t)\end{array}$$

(36)

for all $v_1,v_2\in E$ and all $t\in [0,\infty )$, then there exists a unique general Euler–Lagrange quadratic mapping $Q_2:E\to F$ satisfies

$$\begin{array}{ccc}& & N_{\mu ,\nu }\left(\varphi \left(v_1\right)-Q_2\left(v_1\right),t\right)\hfill \\ & \ge & T_{i=1}^{\infty }T\{\left(\frac{t}{t+{\left|\theta \right|}^{1-2i}\psi {\left({\left|\theta \right|}^{-1}\right)}^{1-i}\psi \left(\parallel v_1\parallel \right)\psi \left(0\right)},\frac{\psi {\left({\left|\theta \right|}^{-1}\right)}^{1-i}\psi \left(\parallel v_1\parallel \right)\psi \left(0\right)}{{\left|\theta \right|}^{2i-1}t+\psi {\left({\left|\theta \right|}^{-1}\right)}^{1-i}\psi \left(\parallel v_1\parallel \right)\psi \left(0\right)}\right),\hfill \\ & & \left(\frac{t}{t+{\left|\theta \right|}^{-2i}\psi {\left({\left|\theta \right|}^{-1}\right)}^{2-2i}\psi \left(\parallel \alpha v_1\parallel \right)\psi \left(\parallel \beta v_1\parallel \right)},\frac{\psi {\left({\left|\theta \right|}^{-1}\right)}^{2-2i}\psi \left(\parallel \alpha v_1\parallel \right)\psi \left(\parallel \beta v_1\parallel \right)}{{\left|\theta \right|}^{2i}t+\psi {\left({\left|\theta \right|}^{-1}\right)}^{2-2i}\psi \left(\parallel \alpha v_1\parallel \right)\psi \left(\parallel \beta v_1\parallel \right)}\right)\}\hfill \end{array}$$

for all $v_1\in E$ and all $t>0$.

Proof. 

By using (35) in Theorem 3, we obtain the result. □

Corollary 3.

If a mapping $\varphi :E\to F$ satisfying (8) and

$$\begin{array}{c}\hfill N_{\mu ,\nu }\left(\varphi \left(\alpha v_1+\beta v_2\right)+\varphi \left(\beta v_1-\alpha v_2\right)-\left({\alpha }^2+{\beta }^2\right)\left[\varphi \left(v_1\right)+\varphi \left(v_2\right)\right],t\right)\ge {\mathrm{\Phi }}_{\mu ,\nu }(v_1,v_2,t)\end{array}$$

(37)

for all $v_1,v_2\in E$ and all $t\in [0,\infty )$, then there exists a unique general Euler–Lagrange quadratic mapping where $Q_2:E\to F$ satisfies

$$\begin{array}{ccc}& & N_{\mu ,\nu }\left(\varphi \left(v_1\right)-Q_2\left(v_1\right),t\right)\hfill \\ & \ge & T_{i=1}^{\infty }T\{\left(\frac{t}{t+{\left|\theta \right|}^{2i+1}\psi {\left({\left|\theta \right|}^{-1}\right)}^{i+1}\psi \left(\parallel v_1\parallel \right)\psi \left(0\right)},\frac{\psi {\left({\left|\theta \right|}^{-1}\right)}^{i+1}\psi \left(\parallel v_1\parallel \right)\psi \left(0\right)}{{\left|\theta \right|}^{-2i-1}t+\psi {\left({\left|\theta \right|}^{-1}\right)}^{i+1}\psi \left(\parallel v_1\parallel \right)\psi \left(0\right)}\right),\hfill \\ & & \left(\frac{t}{t+{\left|\theta \right|}^{2i}\psi {\left({\left|\theta \right|}^{-1}\right)}^{2i+2}\psi \left(\parallel \alpha v_1\parallel \right)\psi \left(\parallel \beta v_1\parallel \right)},\frac{\psi {\left({\left|\theta \right|}^{-1}\right)}^{2i+2}\psi \left(\parallel \alpha v_1\parallel \right)\psi \left(\parallel \beta v_1\parallel \right)}{{\left|\theta \right|}^{-2i}t+\psi {\left({\left|\theta \right|}^{-1}\right)}^{2i+2}\psi \left(\parallel \alpha v_1\parallel \right)\psi \left(\parallel \beta v_1\parallel \right)}\right)\}\hfill \end{array}$$

for all $v_1\in E$ and all $t>0$.

Proof. 

By using (35) in Theorem 4, we obtain the result. □

Corollary 4.

If a mapping $\varphi :E\to F$ satisfies (8) and

$$\begin{array}{c}\hfill N_{\mu ,\nu }\left(\varphi \left(\alpha v_1+\beta v_2\right)+\varphi \left(\beta v_1-\alpha v_2\right)-\left({\alpha }^2+{\beta }^2\right)\left[\varphi \left(v_1\right)+\varphi \left(v_2\right)\right],t\right)\ge {\mathrm{\Phi }}_{\mu ,\nu }(v_1,v_2,t)\end{array}$$

(38)

for all $v_1,v_2\in E$ and all $t\in [0,\infty )$, then there exists a unique general Euler–Lagrange quadratic mapping $Q_2:E\to F$ that satisfies

$$\begin{array}{c}\hfill N_{\mu ,\nu }\left(\varphi \left(v_1\right)-Q_2\left(v_1\right),t\right)\ge T_{i=1}^{\infty }T\left(\frac{t}{t+|{\theta }_1{|}^{-2i}\psi {\left(|{\theta }_1{|}^{-1}\right)}^{2-2i}\psi {\left(\parallel v_1\parallel \right)}^2},\frac{\psi {\left(|{\theta }_1{|}^{-1}\right)}^{2-2i}\psi {\left(\parallel v_1\parallel \right)}^2}{|{\theta }_1{|}^{2i}t+\psi {\left(|{\theta }_1{|}^{-1}\right)}^{2-2i}\psi {\left(\parallel v_1\parallel \right)}^2}\right)\end{array}$$

for all $v_1\in E$ and all $t>0$.

Proof. 

By using (35) in Theorem 5, we obtain the result. □

Corollary 5.

If a mapping $\varphi :E\to F$ satisfying (8) and

$$\begin{array}{c}\hfill N_{\mu ,\nu }\left(\varphi \left(\alpha v_1+\beta v_2\right)+\varphi \left(\beta v_1-\alpha v_2\right)-\left({\alpha }^2+{\beta }^2\right)\left[\varphi \left(v_1\right)+\varphi \left(v_2\right)\right],t\right)\ge {\mathrm{\Phi }}_{\mu ,\nu }(v_1,v_2,t)\end{array}$$

(39)

for all $v_1,v_2\in E$ and all $t\in [0,\infty )$, then there exists a unique general Euler–Lagrange quadratic mapping $Q_2:E\to F$ satisfies

$$\begin{array}{c}\hfill N_{\mu ,\nu }\left(\varphi \left(v_1\right)-Q_2\left(v_1\right),t\right)\ge T_{i=1}^{\infty }T\left(\frac{t}{t+|{\theta }_1{|}^{2i}\psi {\left(|{\theta }_1{|}^{-1}\right)}^{2i+2}\psi {\left(\parallel v_1\parallel \right)}^2},\frac{\psi {\left(|{\theta }_1{|}^{-1}\right)}^{2i+2}\psi {\left(\parallel v_1\parallel \right)}^2}{|{\theta }_1{|}^{-2i}t+\psi {\left(|{\theta }_1{|}^{-1}\right)}^{2i+2}\psi {\left(\parallel v_1\parallel \right)}^2}\right)\end{array}$$

for all $v_1\in E$ and all $t>0$.

Proof. 

By using (35) in Theorem 6, we obtain the result. □