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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.
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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