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A fixed point approach to the stability of a septic functional equation in fuzzy quasi-$$\beta$$-normed spaces. (English) Zbl 1358.39014
Suppose that $$X$$ is a real real linear space, $$T$$ is a continuous $$t$$-norm, $$\beta \in (0, 1]$$ is a fixed real number and $$N$$ is a fuzzy set on $$X\times \mathbb{R}$$ satisfying (i) $$N(x, t)=0$$ for each $$t\leq 0$$, (ii) $$x=0$$ if and only if $$N(x, t)=1$$ for each $$t >0$$, (iii) $$N(cx, t)=N(x,\frac{t}{|c|^{\beta}})$$ if $$c\neq 0$$, (iv) $$N(x+y, K(s+t))\leq N(x,s)TN(y, t)$$ for some constant $$K\leq 1$$, (v) $$\lim_{t\to \infty}N(x,t)=1$$ for each $$x, y \in X$$ and $$s, t\in \mathbb{R}$$. Then $$(X, N, T)$$ is said to be a fuzzy quasi-$$\beta$$-normed space. In this paper, the authors investigate the stability of the functional equation $$f(x + 4y) - 7f(x+3y)+21f(x+2y)-35f(x+y)-21f(x-y)+7f(x-2y)-f(x-37)+35f(x)=5040f(y)$$ in a fuzzy quasi-$$\beta$$-normed space.
##### MSC:
 39B82 Stability, separation, extension, and related topics for functional equations 39B52 Functional equations for functions with more general domains and/or ranges 46S40 Fuzzy functional analysis
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