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Two-step algorithms and their applications to variational problems. (English) Zbl 1050.49006

Summary: The two-step algorithm is introduced and applied to the approximation solvability of a system of nonlinear variational inequalities in a Hilbert space setting. Let H be a real Hilbert space and K be a nonempty closed convex subset of H. For arbitrarily chosen initial points x 0 , y 0 K, compute sequences {x k } and {y k } such that

x k+1 =P K a k x k + (1-a k ) P K y k - ρ T (y k )forρ>0
y k =P K b k x k + (1-b k ) P K x k - η T (x k )forη>0,

where T:KH is a nonlinear pseudococoercive mapping on K, P K is the projection of H onto K, and 0a k ,b k 1 for k0. Based on the two-step algorithm, we explore the approximation solvability of a system of nonlinear variational inequality problems: determine elements x * ,y * K such that

ρ T (y * ) + x * - y * , x - x * 0xK
η T (x * ) + y * - x * , x - y * 0xK,

where ρ, η>0.

MSC:
49J40Variational methods including variational inequalities
47J25Iterative procedures (nonlinear operator equations)