AN ITERATIVE METHOD FOR A FAMILY OF NONEXPANSIVE MAPPINGS

作者：Razani Abdolrahman^*; Yazdi Maryam

来源：Mathematical Reports, 2014, 16(1): 7-23.

• fixed point
• nonexpansive mapping
• weakly contractive mapping
• variational inequality

Let H be a real Hilbert space. Consider the iterative sequence
x(n+1) = alpha(n)gamma f (x(n)) + beta(n)x(n) + ((1 On)I alpha(n) A)W xn, for all n >= 0,
where gamma > 0 is some constant, f : H H is a given weakly contractive mapping, A is a strongly positive bounded linear operator on H and W-n is the W-mapping generated by an infinite countable family of nonexpansive mappings T-1, T-2, T-n,.. and lambda 1, lambda(2),, lambda n,... such that the common fixed points set F := n boolean AND(infinity)(n=1) F(Tn) not equal phi. Under very mild conditions on the parameters, it is proved, the sequence {xn} converges strongly to p is an element of F where p is the unique solution in F of the following variational inequality:
<(A -gamma f)p, p-x*)> <= 0, for all x* is an element of F,
which is the optimality condition for minimization problem
min x is an element of F 1/2 < Ax, x > - h(x),
where h is a potential function for gamma f.

• 出版日期2014