区域函数的广义导数及其应用
Generalized Derivatives of a Region Function and Its Applications
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摘要: 本文以[7]的基本概念为基础,并根据Clarke的广义导数[1],以及Lasotra和Strauss[6]的多值函数f(x)的广义微分Df(x)的定义.从而建立了区域函数F(x)的广义导数DF=∪∩{G(x)⊆B(R),∀x∈B(R);G(x)=Fx=F(x)}讨论了区域函数F(x)的广义导数的存在性;建立了区域函数的广义Fréchet导数存在的必要充分条件.Abstract: This paper is based on tome fundamental concepts in [7], Clarke's generalized derivatives[1], as well as Lasotra's and Strauss's definitions of differential D(x) of a multivalued function f(x)[6]. Thereby, the generalized derivatives of a region function f(x) is defined as DF=∪∩{G(x)⊆B(R),∀x∈B(R);G(x)=Fx=F(x)} The existence of the generalized derivatives of a region function F(x) is discussed; the necessary and sufficient conditions of existence of the Frechet generalized derivatives of such a function is established.
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[1] Clarke,F.H.,Generalized gradient and applications,Trans.Amer.Math.Soc.,205(1975),247-262. [2] Lebourg,M.G.,Valeur moyenne pour gradient généralisé,C.R.Acad Soc.,Ser.A,281(1975),795-797. [3] Thibaulf,L.,On generalized differentials and subdifferentials of Lipschitz vector-valued functions,Nonlinear Anal.TMA,6,10(1982),1037-1053. [4] Schröder,G.,Differentiation of interval functions,Proc.Amer.Math.Soc.,36,2(1972),485-480. [5] Clarke,F.H.,On the inverse functions theorem,Pacific J.Math.,64,1(1976),92-102. [6] Lasotra,A.and A.Strauss,Asymptotic behavior for differential equations which cannot be locally linearized J Differ.Equans.,10,10(1971),152-172. [7] 何冲,区域函数,应用数学和力学,7,2(1986),173-179 -
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