Banach空间中极大单调算子零点的迭代收敛定理及应用

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-1βnJx1+(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.

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

Let E be a real smooth and uniformly convex space with E~* its duality space.Let AE×E~* be a maximal monotone operator with A~(-1)0≠.Let {r_n}(0,+∞) be a real sequence with r_n→∞ as n→∞,let {α_n} satisfy ∑∞n=1(1-α_n)<+∞.For a given vector x_n∈E,find vectors_n and {e_n} such that α_nJx_n+(1-α_n)Je_n∈J_n+r_nA_n,where {e_n}E is the error sequence and satisfies some conditions.Then the iterative sequence {x_n}_(n1) is defined as follow: x_(n+1)=J~(-1)-n], n1,where {β_n} is a real sequence with β_n→0,as n→∞ and ∑∞n=1β_n=+∞,then {x_n} is strongly convergent to Q_(A~(-1)0)(x_1),where Q_(A~(-1)0) is the generalized projection operator from E onto A~(-1)0.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,Q_r 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.