A variational result for a class of vector-valued functions
Suppose X is a Banach space and J is the interval \(-\infty <t<\infty\). Let B(X) be the set of all bounded linear operators on X. An operator- valued function \(G: J\to B(X)\) is called a continuous one-parameter family if G(t)x, \(t\in J\to X\) is continuous for each \(x\in X\). We assume \(G(0)=I=the\) identity operator on X. We denote by U the set of all (continuous) X-valued functions u(t) defined by the relation \(u(t)=G(t)x+h(t)\) on J, where x varies over X. Obviously, \(u(0)=x\), and we then have \(u(t)=G(t)u(0)+h(t)\). We designate by \(U_ b\) the subset of U consisting of all those functions u(t) which are bounded on J. We assume throughout this paper that \(U_ b\) is non-empty. Now we consider the non-negative functional \(\mu\) (u) on \(U_ b\) defined by the relation \(\mu (u)=\sup_{t\in J}\| u(t)\|\), and we set (*) \({\tilde \mu}=\inf_{u\in U_ b}\mu (u).\) The following result is established. Theorem: Suppose that X is a closed subspace of \(L^ p(S,\Sigma,m)\) for some measure space (S,\(\Sigma\),m) and some p with \(2\leq p<\infty\), and that, for all \(x\in X\), \(x\neq \vartheta\), there holds \(\inf_{t\in J}\| G(t)x\| >0\). Then there exists one and only one \(\tilde u\in U_ b\) such that \(\mu(\tilde u)={\tilde\mu}\). Condition (*) is used only to prove uniqueness, while the proof of existence remains valid in any reflexive space. (There are two misprints in the paper: Line 2 from the top on page 1042 should read: \({\tilde \mu}\leq ({\tilde \mu}^ p-d^ p)^{1/p}\). Line 6 from the bottom on page 1043 should read: Since \({\tilde \mu}\leq \mu (\tilde u),\ldots)\).
scientific article; zbMATH DE number 3933871