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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.

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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