an:06485793
Zbl 1330.39029
Khodaei, Hamid
On the stability of additive, quadratic, cubic and quartic set-valued functional equations
EN
Result. Math. 68, No. 1-2, 1-10 (2015).
00348278
2015
j
39B82 39B52 54C60 39B55
set-valued mapping; subhomogeneous mapping; expansively superhomogeneous mapping; stability of functional equations; fixed point method; additive, quadratic, cubic and quartic set-valued functional equations; Banach space
Let \(X\) be a real vector space and \(Y\) a Banach space. By \(C_{cb}(Y)\) we denote the set of all nonempty, closed, bounded and convex subsets of \(Y\). By \(A\oplus B\) we mean the closure of \(A+B\). For \(f: X\to C_{cb}(Y)\), a fixed integer \(a>1\) and \(m=1,2,3,4\), the following equation is considered and its stability is proved.
\[
\begin{multlined} f(ax+y)\oplus f(ax-y)= \\ a^{m-2}[f(x+y)\oplus f(x-y)]\oplus 2(a^2-1)\left[ a^{m-2}f(x)\oplus\frac{(m-2)(1-(m-2)^2)}{6}f(y)\right] \end{multlined}
\]
for all \(x,y\in X\). Namely, if the Hausdorff distance between the left and right hand sides of the above equation is bounded by a suitably contractively subhomogeneous function, then there exists a unique additive, quadratic, cubic or quartic (for \(m=1,2,3,4\), respectively) mapping which is sufficiently close to \(f\).
Jacek Chmieli??ski (Krak??w)