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Weak and strong convergence theorems for maximal monotone operators in a Banach space. (English) Zbl 1078.47050
Let E be a smooth and uniformly convex Banach space with the normalized duality mapping J:EE * and TE×E * a maximal monotone operator with T -1 0. Let J r =(J+rT) -1 J, r>0, and P(x)=argmin yT -1 0 (x 2 -2x,Jy+y 2 ). The authors study the iterative process x n+1 =J -1 (α n J(x n )+(1-α n )J(J r n x n )), α n [0,1], r n (0,) (n=1,2,) and prove that the sequence {P(x n )} converges strongly to an element of T -1 0. The results are applied to the convex minimization problem and the variational inequality problem.

MSC:
47J25Iterative procedures (nonlinear operator equations)
47H05Monotone operators (with respect to duality) and generalizations
90C48Programming in abstract spaces
90C25Convex programming
47N10Applications of operator theory in optimization, convex analysis, programming, economics
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