广义共同逼近问题的适定性

设C是实Banach空间X中有界闭凸子集且0是C的内点，G是X中非空闭的有界相对弱紧子集.记K(X)为X的非空紧凸子集全体并赋Hausdorff距离，KG(X)为集合｛A∈K(X)；A∩G=｝的闭包.称广义共同逼近问题minC(A,G)是适定的是指它有唯一解(x0,z0)，且它的每个极小化序列均强收敛到(x0,z0).在C是严格凸和Kadec的假定下，证明了｛A∈K(X)；minC(A，G)是适定的｝含有KG(X)中稠Gδ子集，这本质地推广和延拓了包括De Blasi,Myjak and Papini［1］、Li［2］和De Blasi and Myjak［3］等人在内的近期相应结果.

On Well Posedness of Generalized Mutually Approximation Problem

Let C be a closed bounded convex subset of a realBanachspace X with 0 being an interior point of C. Let G be a nonempty closed, boundedly relatively weakly compact subset of X. Let K(X) denote the space ofall nonempty compact convex subset of X endowed with the Hausdorff distance. Moreover, Let KG(X) denote the closure of the set ｛A∈K(X);A∩G=｝. A generalized mutually approximation problem minC(A,G) is said to be well posed if it hasa uniquesolution (x0,z0) and every minimizing sequence converges strongly to (x0,z0). Under the assumption that C is strictly convex and (sequentially) Kadec, that the set ｛A∈KG(X);minC(A,G) is well posed｝ contains a dense Gδ subset of KG(X) is proved. The results generalize and extend the recent corresponding results due to De Blasi, Myjak and Papini［1］,Li［2］,De Blasi andMyjak［3］ and other authors.