Lipschitz stability of the Cauchy and Jensen equations
Let \(G\) be a semigroup and let \(E\) be a normed space. Let \(\mathcal F\) be a given set of functions from \(G\) into \(E\), and let \(\widetilde{\mathcal F}\) be a given set of functions from \(G\times G\) into \(E\). The pair \(({\mathcal F},\widetilde{\mathcal F})\) has the double difference property if for every \(f:G\to E\) such that \({\mathcal C}f\in\widetilde{\mathcal F}\) (\({\mathcal C}f(x,y)=f(x+y)-f(x)-f(y)\)) there exists an additive function \(A:G\to E\) such that \(f-A\in{\mathcal F}\). If \(E\) is a vector space, a family \(S(E)\) of subsets of \(E\) is linearly invariant if (i) \(x\in E\), \(\alpha\in\mathbb{R}\), \(V\in S(E)\) imply \(x+\alpha V\in S(E)\); (ii) \(V,W\in S(E)\) implies \(V+W\in S(E)\). Let \(G\) be a semigroup, let \(E\) be a vector space and let \(S(E)\) be a linearly invariant family of subsets of \(E\). The set \({\mathcal L}(G,S(E))\) of functions \(f:G\to E\) with \(\text{im }f\subset V\) for some \(V\in S(E)\), admits a left invariant mean (LIM) if there exists a linear operator \(M:{\mathcal L}(G,S(E))\to E\) such that (i) \(\text{im }f\subset V\in S(E)\) implies \(M[f]\in V\); (ii) \(f\in{\mathcal L}(G,S(E))\), \(a\in G\) imply \(M[_af]=M[f](_af(x)=f(ax))\). Let \(G\) be a semigroup with a metric \(d\) and let \(E\) be a normed space. A function \(f:G\to E\) is Lipschitz if \[ |f(x)-f(y)|\leq Ld(x,y) \] for some \(L\in\mathbb{R}\); the smallest constant with this property is denoted by \(\text{lip }f\). By \(\text{Lip}(G,E)\) we denote the space of all bounded Lipschitz functions with the norm \[ |f|_{\text{Lip}}=|f|_{\text{Sup}}+\text{lip }f. \] Among the results proved by the author are the following. Theorem. Let \(G\) be a semigroup with an invariant metric \(d\) \((d(x+a,y+a)=d(a+x,a+y)=d(x,y))\) and with nonempty centre, and let \(E\) be a normed space such that \({\mathcal L}(G,B(E))\) admits LIM (\(B(E)\) is the family of balls in \(E)\). (i) Denote by \(\mathcal F\) the space of uniformly continuous functions from \(G\) into \(E\), and by \(\widetilde{\mathcal F}\) the space of uniformly continuous functions from \(G\times G\) into \(E\) with the product metric on \(G\times G\). Then the pair \(({\mathcal F},\widetilde{\mathcal F})\) has the double difference property. (ii) If \(f:G\to E\) is such that \({\mathcal C}f\in\text{Lip}(G\times G,E)\), then there exists an additive function \(A:G\to E\) such that \[ |f-A|_{\text{Lip}}\leq|{\mathcal C}f|_{\text{Lip}}. \] Similar results are proved for the Jensen equation.
scientific article; zbMATH DE number 1112245