The Iterative Convergence Theorem of Zero Point for Maximal Monotone Operator in Banach Space and Its Application
Li Wei, Zhou Hai-yun · Shuxue de shijian yu renshi · 2006
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.