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Approximation of a mixed functional equation in quasi-Banach spaces. (English) Zbl 1167.39017

The authors deal with the functional equation \[ f(2x+y)+f(x+2y)=6f(x+y)+f(2x)+f(2y)-5[f(x)+f(y)](*) \]
showing that its general solution between real vector spaces is a sum of a cubic function and an additive one. A cubic function is the diagonalization of a three-place mapping which is symmetric for each fixed one variable and additive for each fixed two variables. Such a function is the general solution to the functional equation
\[ f(2x+y)+f(2x-y)=2f(x+y)+2f(x-y)+12f(x) \]
which has been investigated by K.-W. Jun and H.-M. Kim [J. Math. Anal. Appl. 274, No. 2, 867–878 (2002; Zbl 1021.39014)]. Their result is a main tool used by the authors to obtain the solution of (\(\ast\)).
In the paper one may also find several results on the Hyers-Ulam-Rassias stability of equation (\(\ast\)). Namely, let \(X\) be a quasi-normed space and let \(Y\) be a \(p\)-Banach space with a modulus of concativity \(K\). Assuming, for instance, that \(f: X\to Y\) and \(\varphi: X\times X\to [0,\infty)\) satisfy
\[ \| f(2x+y)+f(x+2y)-6f(x+y)-f(2x)-f(2y)+5[f(x)+f(y)]\|\leq\varphi(x,y),(**) \]
\[ \| f(x)+f(-x)\|\leq\varphi(x,0), \]
\[ \lim_{n\to\infty}\varphi(2^nx,2^ny)=0, \]
and
\[ \sum_{i=0}^\infty{1\over 2^{ip}}\varphi(2^ix,2^iy)^p<\infty , \]
for all \(x,y\in X\), it is proved that there exist: a unique additive function \(A: X\to Y\) and a unique cubic function \(C: X\to Y\) such that
\[ \| f(x)-A(x)-C(x)-\tfrac 17 f(0)\|\leq\delta, \]
where \(\delta\) depends only on the \`\` control mapping\'\'\(\varphi\) and on the parameters \(p\), \(K\) of the space \(Y\).
The obtained results allow to establish a stability-type theorem in the case where an odd mapping \(f: X\to Y\) satisfies (\(\ast\ast\)) with \(\varphi(x,y)=\vartheta\| x\|^r\| y\|^s\), where \(\vartheta\geq 0\), \(r,s>0\) and \(1<r+s<3\).

MSC:

39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges

Citations:

Zbl 1021.39014
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