Banach空间中极大单调算子零点的迭代收敛定理及应用

The Iterative Convergence Theorem of Zero Point for Maximal Monotone Operator in Banach Space and Its Application

令E为实光滑、一致凸的Banach空间,E*为其对偶空间.令A E×E*为极大单调算子且A-10≠.假设{rn}(0,+∞)为实数列且满足rn→∞,n→∞,数列{αn}[0,1]满足∑∞n=1(1-αn)<+∞,对给定的向量xn∈E,寻找向量{x∧n}及{en}使之满足:αnJxn+(1-αn)Jen∈Jx∧n+rnAx∧n,其中{en}E为误差序列而且满足一定的限制条件.即而定义迭代序列{xn}n 1如下:xn+1=J-1[βnJx1+(1-βn)Jx∧n],n 1,其中数列{βn}[0,1]满足βn→0,n→∞且∑∞n=1βn=+∞,则{xn}强收敛于QA-10(x1),这里QA-10为从E到A-10上的广义投影算子.利用Lyapunov泛函,Qr算子与广义投影算子等新技巧,证明了引入的新迭代序列强收敛于极大单调算子A的零点,并讨论了此结论在求解一类凸泛函最小值上的应用.

Let E be a real smooth and uniformly convex space with E＊ its duality space. Let A （∈）E×E＊ be a maximal monotone operator with A^-10≠Ф. Let {rn}（∈）（0, ＋∞） be a real sequence with rn→∞ as n→∞, let {an}（∈）[0,1] satisfy ∞∑n=1（1 - ax） 〈＋ ∞. For a given vector xn∈E, find vectors xn and {en} such that anJxn. ＋ （1 - an）Jen∈JXn ＋ rnAXn, where {ex} （∈） E is the error sequence and satisfies some conditions. Then the iterative sequence {xn}n≥1 is defined as follow： Xn＋1=J^x-1[βnJx1＋（1-βn）JXn],n≥1, where {βn}（∈）[0,1] is a real sequence with βn→0, asn→as n →∞∑n=1=＋∞ then {xn}is strongly convergent to QA-1（X1）, where QA-lo is the generalized proonto operator form E onto A-10. A new iterative scheme is introduced which is proved to be strongly convergent to zero point of maximal monotone operator A by using the techniques of Lyapunov functional, Qr operator and generalized projection operator, etc. Moreover, the application of the new convergence theorem to solve the minimum value of one kinds of convex functional is being discussed.