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An application of a fixed point theorem to a functional inequality. (English) Zbl 1184.39012
Let $A$ be a unital $C^*$-algebra with unitary group $U(A)$. Assume that $X,Y$ are left Banach modules over $A$ and that $Y$ is complete. Given $f: X\to Y$, $a\in A$ and $x,y,z\in X$ let $$D_af(x,y,z):=f\left(\frac{ax-ay}{2}+az\right)+f\left(\frac{ay-az}{2}+ax\right)+a f\left(\frac{z-x}{2}+y\right).$$ The authors prove the following theorem. Let $f: X\to Y$ be such that there is some $\varphi: X^3\to[0,\infty)$ with $$\lim_{n\to\infty} 2^n\varphi\left(\frac x{2^n},\frac y{2^n}, \frac z{2^n}\right)=0\text{ and }\Vert D_af(x,y,z)\Vert\leq\Vert f(ax+ay+az)\Vert+\varphi(x,y,z)$$ for all $x,y,z\in X$ and all $a\in U(A)$. Assume furthermore that there is some $L<1$ such that the mapping $X\ni x\mapsto \psi(x):=2\varphi(x/7,2x/7,-3x/7)+\varphi(4x/7,x/7,-5x/7)$ satisfies $2\psi(x)\leq L\psi(2x)$ for all $x\in X$. Then there is a unique $A$-linear mapping $T: X\to Y$ such that $$\Vert f(x)-T(x)\Vert\leq\frac1{1-L}\psi(x)$$ for all $x\in X$. By specializing the hypotheses, several corollaries are derived. The main tool for proving these results is a fixed point theorem in generalized metric spaces [cf. {\it J. B. Diaz} and {\it B. Margolis}, Bull. Am. Math. Soc. 74, 305--309 (1968; Zbl 0157.29904)].

39B62Functional inequalities, including subadditivity, convexity, etc. (functional equations)
47H09Mappings defined by “shrinking” properties