关于p-凸泛函族的共鸣定理

ON THE RESONANCE THEOREMS OF P-CONVEX FUNCTIONALS

共鸣定理是泛函分析中重要的基础定理之一,其形式也有多种变化。在[1]—[5]中讨论了一类共鸣定理,即所谓凸泛函族的共鸣定理。由于凸分析的广泛运用性,凸性的概念有必要进一步推广。本文首先给出p-凸性的概念,p-凸集和p-凸泛函具有完全类似于凸集和凸泛函的性质,然后导出了在拓扑向量空间及赋范空间中p-凸泛函族的几个共鸣定理,推广了文[1]—[5]中的相应结果。

In this paper,some resonance theorems of p-convex functionals in topological rector space and normed space are given. A set EV is called p-convex if for any x、y∈E, when λ_1≥0,λ_2≥0, λ_1~ρ+λ_2~ρ=1, we have λ_1x+λ_2y∈E; A functional is called p-convex if for any x、y∈E, λ_1≥0,λ_2≥0, λ_1~ρ+λ_2~ρ=1 we have f(λ_1x+λ_2y)≤λ_1f(x)+λ_2f(y), where constant p∈[0, 1].