## 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
(i)
$$\|x\|\geq 0$$ for all $$x\in X$$, and $$\|x\|=0$$ if and only if $$x=0$$;
(ii)
$$\|\lambda x\|=|\lambda|\|x\|$$ for all $$\lambda \in {\mathbb R}$$ and all $$x\in X$$;
(iii)
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.

### MSC:

 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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