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Projection methods, algorithms, and a new system of nonlinear variational inequalities. (English) Zbl 0995.47042
Let H be a real Hilbert space and T:KH a strongly monotone and Lipschitz continuous mapping from a closed convex subset KH into H. The paper is concerned with the problem of finding elements x * ,y * K such that x * =P K [y * -ρT(y * )],y * =P K [x * -γT(x * )], where ρ,γ>0 and P K is the projection of H onto K. To solve the problem the author proposes and studies the following iterative algorithm: x k+1 =(1-a k )x k +a k P K [y k -ρT(y k )],y k =P K [x k -γT(x k )], where 0a k <1 and k=0 a k =. The strong convergence of {x k } to x * is established provided that ρ and γ are sufficiently small.

47J20Inequalities involving nonlinear operators
47J25Iterative procedures (nonlinear operator equations)
49J40Variational methods including variational inequalities