Stability of the quadratic functional equation in Lipschitz spaces
The stability of the quadratic functional equation is dealt with. Invariant means are used to obtain the main results . Let \(E\) be a (real) vector space and let~\(G\) be an Abelian group. By the quadratic difference of a~function \(f:G\to E\) we mean the map \[ Qf(x,y):=2f(x)+2f(y)-f(x+y)-f(x-y),\quad x,y\in G. \] The family \(S(E)\subset 2^E\) is called linearly invariant if (1) \(x+\alpha V\in S(E)\) for any \(x\in E\), \(\alpha\in{\mathbb R}\), \(V\in S(E)\) and (2) \(V+W\in S(E)\) for any \(V,W\in S(E)\). Let \[ B\bigl(G,S(E)\bigr):= \bigl\{ f:g\to E\;| \;\text{Im}f\subset V\text{ for some }V\in S(E) \bigr\}. \] For \(f:G\to E\) and \(a\in G\) we define \(f^a(x)=f(x+a)\). We say that \(B\bigl(G,S(E)\bigr)\) admits a left invariant mean (LIM) if there exists a~linear operator \(M:B\bigl(G,S(E)\bigr)\to E\) such that (i) \(\text{Im}f\subset V\Rightarrow M[f]\in V\) and (ii) \(M[f^a]=M[f]\) for any \(f\in B\bigl(G,S(E)\bigr)\), \(a\in G\). If \({\mathbf d}:G\times G\to S(E)\) satisfies the condition \({\mathbf d}(x+a,y+a)={\mathbf d}(a+x,a+y)={\mathbf d}(x,y)\), \(x,y,a\in G\), then a~function \(f:G\to E\) is called \({\mathbf d}\)-Lipschitz, if \(f(x)-f(y)\in{\mathbf d}(x,y)\), \(x,y\in G\). The main result of the paper is the following Theorem: Let \(G\) be an Abelian group and \(E\) a vector space. Assume that \(S(E)\) is a~linearly invariant family such that \(B\bigl(G,S(E)\bigr)\) admits LIM. Let \(f:G\to E\) be an arbitrary function. If \(Qf(x,\cdot):G\to E\) is \({\mathbf d}\)-Lipschitz for every \(x\in G\), then there exists a~quadratic function \(K:G\to E\) such that \(f-K\) is \(\frac{1}{2}{\mathbf d}\)-Lipschitz. If, moreover, \(\text{Im}Qf\subset V\) for some \(V\in S(E)\), then \(\text{Im}(f-K)\subset\frac{1}{2}V\). By an appropriate specification of \({\mathbf d}\) some results concerning the stability of a~quadratic functional equation in Lipschitz norms \(\| f\| _{\text{Lip}}:=\| f\| _{\sup}+\text{lip}(f)\) and \(\| f\| _{\text{Lip}^0}:=\| f(0)\| +\text{lip}(f)\) are derived.
scientific article; zbMATH DE number 2082130