Solution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasi-Banach spaces. (English) Zbl 1179.39034

Let \(X\) be a real linear space. A quasi-norm is a real-valued function \(\|\cdot\|\) on \(X\) satisfying the following: 7mm
\(\|x\|\geq 0\) for all \(x\in X\), and \(\|x\|=0\) if and only if \(x=0\);
\(\|\lambda x\|=|\lambda|\|x\|\) for all \(\lambda \in {\mathbb R} \) and all \(x\in X\);
There is a constant \(K\geq 1\) such that \(\|x+y\|\leq K(\|x\|+\|y\|)\) for all \(x, y\in X\).
Then \((X,\|.\|)\) is called a quasi-normed space. A quasi-Banach space is a complete quasi-normed space. In this paper the authors investigate the generalized Hyers-Ulam-Rassias stability of the following equation \[ f(x+ky)+f(x-ky)=k^2f(x+y)+k^2f(x-y)+2(1-k^2)f(x) \] where \(k\neq 0,+1,-1\), and \(f\) is a mapping between vector spaces, and establish the generalized Hyers-Ulam-Rassias stability of the functional equation above whenever \(f\) is a function between two quasi-Banach spaces.


39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges
46B99 Normed linear spaces and Banach spaces; Banach lattices
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