一种广义二阶导数和二阶次微分  

THE GENERALIZED SECOND-ORDER DERIVATIVE AND SECOND-ORDER SUBDIFFERENTIAL

作　　者：阮国桢[1] 

机构地区：[1]湘潭大学数学系

出　　处：《湘潭大学自然科学学报》1993年第1期39-47,共9页Natural Science Journal of Xiangtan University

摘　　要：给出了Banach空间X上的实值函数f的一种广义二阶方向导数f^(00)(x;υ;ω)的定义,f^(00)(X;·;·)是X×X上的双正齐次凸泛函.在f^(00)(x;·;·)取有限值时给出了二阶次微分(?)f(x)的定义,当f^(00)(x;·;·)有界时,(?)f(x)是一个非空的有界闭凸集。当X=R^n时,(?)f(x)与Urruty的广义Hessian矩阵一致。Let X be Banach space,f:X→R be locally Lipschitzan.(1)The generalized second-order derivatives of fat x in directions υ∈X,ω∈X is defined as ■The bifunction f°° (x;°;°):XX→R∪{+∞}is positively homogeneous andsubadditive with respect to υ and ω respectively.Andf°°(x;-υ;ω)=f°°(x;υ;ω),υ∈X,ω∈X.(2)The second-order subdifferential of f at x is defined as~2f (x)={A∈LL(X,X):-f°°(x;-υ;ω)≤A (ω,υ)f°°(x;υ;ω) for all υ,ω in X}.where LL(X.X) is the Banach space of the bounded bilinear functionalon XX.(3)When f°°(x;·;·)is bounded,then (?)~2 f(x) is α nonempty,covex closedoubset of LL(X,X).And for every υ,ω in X we have

关 键 词：双线性泛函 二阶次导数 二阶次微分 

分 类 号：O177.91[理学—数学]