Intuitionistic Fuzzy Stability of an Euler–Lagrange Symmetry Additive Functional Equation via Direct and Fixed Point Technique (FPT)

Let $\mathbb{A},\mathbb{B}$ be non-empty sets and the mapping $\mathfrak{Z}:\mathbb{A}\to \mathbb{B}$. From (3), the following conditions are derived:

From (3) Replacing $({\theta }_1,{\theta }_2,{\theta }_3)$ by $\left(\varsigma \mathfrak{p},\wp \mathfrak{q},ħ\mathfrak{r}\right)$ in (3) and using condition $\left(4\right)$, Multiplying (6) by $\left(\varsigma +\wp +ħ\right)$, Replacing $({\theta }_1,{\theta }_2)$ by $\left(\wp ħ\mathfrak{p},-\wp ħ\mathfrak{q}\right)$ in (3) and by (4) Multiplying by $\varsigma$ in (8), and by (2), the following result is obtained. Replacing $({\theta }_1,{\theta }_2)$ by $(\varsigma ħ\mathfrak{q},-\varsigma ħ\mathfrak{r})$ in (3), and by (4), Multiplying by ℘ in (10) and applying condition (2), Replacing $({\theta }_1,{\theta }_2)$ by $(\varsigma \wp \mathfrak{r}-\varsigma \wp \mathfrak{p})$ in (3), and using condition (4), Multiplying by ħ on both sides of (10) and by condition (2), the following result is obtained. Finally, adding these derived Equations (7), (9), (11), and (13), Equation (4) can be obtained. □

Let $\left(\mathbb{A},∥·∥\right)$ be a normed space. Let $a\ast b=ab$ and $a\diamond b=min\left\{a+b,1\right\}$ for all $a,b\in [0,1]$. For all $\mathfrak{p}\in \mathbb{A}$ and every $\kappa >0$, considerThen, $\left(\mathbb{A},{\mu }_a,{\nu }_a,\ast ,\diamond \right)$ is an IFNS.

Replacing the variables $(\mathfrak{p},\mathfrak{q},\mathfrak{r})$ by $(\mathfrak{p},\mathfrak{p},\mathfrak{p})$ in (16), Using intuitionistic fuzzy conditions (A4) and (A10), the following result is arrived. Replacing $\mathfrak{p}$ by ${\left(\varsigma +\wp +ħ\right)}^n\mathfrak{p}$ in (19), It is understood from (20) and the intuitionistic fuzzy conditions (A4) and (A10) that

Taking $\kappa$ into ${\mathfrak{p}}^n\kappa$ in (21), the following result is obtained, From the above two equations, From the above equations, where Finally, Replacing $\mathfrak{p}$ by ${\left(\varsigma +\wp +ħ\right)}^m\mathfrak{p}$ in (26) and using (15), (A4), (A10), Replacing $\kappa$ by ${\mathfrak{p}}^m\kappa$ in (27), Since $0<\mathfrak{p}<1$ and $\sum _{i=0}^n{\left(\frac{\mathfrak{p}}{1}\right)}^i<\infty$. The sequence ${\displaystyle \left\{\frac{\mathfrak{Z}\left({\left(\varsigma +\wp +ħ\right)}^n\mathfrak{p}\right)}{{\left(\varsigma +\wp +ħ\right)}^n}\right\}}$ is Cauchy in $\left(\mathbb{B},{\mu }_a,{\nu }_a\right)$. Since $\left(\mathbb{B},{\mu }_a,{\nu }_a\right)$ is a complete IFNS, this sequence converges to some point $\mathfrak{L}\left(\mathfrak{p}\right)\in \mathbb{B}$. Defining $\mathfrak{L}:\mathbb{A}⟶\mathbb{B}$ by Then, Substituting $m=0$ in (29), As $n⟶\infty$ in (30), Finally, $\mathfrak{L}$ satisfies (2). Replacing $(\mathfrak{p},\mathfrak{q},\mathfrak{r})$ by in (16), Here, and Also, As $n\to \infty$ in (33), (34), and applying (35), $\mathfrak{L}$ fulfills (2). Hence, $\mathfrak{L}$ is an additive function. Here, it is proven that $\mathfrak{L}\left(\mathfrak{p}\right)$ is unique. Let ${\mathfrak{L}}^{\prime }\left(\mathfrak{p}\right)$ be another additive map (2) and (17). Therefore, Since ${\displaystyle \underset{n\to \infty }{lim}\frac{\kappa {\left(\varsigma +\wp +ħ\right)}^{n+1}|\left(\varsigma +\wp +ħ\right)-\mathfrak{p}|}{2{\mathfrak{p}}^n}=\infty ,}$ Hence, Thus, $\mathfrak{L}\left(\mathfrak{p}\right)={\mathfrak{L}}^{\prime }\left(\mathfrak{p}\right)$. Hence, $\mathfrak{L}\left(\mathfrak{p}\right)$ is unique.