Abstract
We use strongly pseudocontraction to regularize the following ill-posed monotone variational inequality: finding a point with the property such that , where , are two pseudocontractive self-mappings of a closed convex subset of a Hilbert space with the set of fixed points . Assume the solution set of (VI) is nonempty. In this paper, we introduce one implicit scheme which can be used to find an element . Our results improve and extend a recent result of (Lu et al. 2009).
Highlights
Let H be a real Hilbert space with inner product ·, · and norm ·, respectively, and let C be a nonempty closed convex subset of H
If the mapping F is a monotone operator, we say that VI F, C is monotone
It is well known that if F is Lipschitzian and strongly monotone, for small enough γ > 0, the mapping PC I − γ F is a contraction on C and so the sequence {xn} of Picard iterates, given by xn PC I − γ F xn−1 n ≥ 1 converges strongly to the unique solution of the VI F, C
Summary
Let H be a real Hilbert space with inner product ·, · and norm · , respectively, and let C be a nonempty closed convex subset of H. A variational inequality problem, denoted VI F, C , is to find a point x∗ with the property x∗ ∈ C such that Fx∗, x − x∗ ≥ 0 ∀x ∈ C. It is well known that if F is Lipschitzian and strongly monotone, for small enough γ > 0, the mapping PC I − γ F is a contraction on C and so the sequence {xn} of Picard iterates, given by xn PC I − γ F xn−1 n ≥ 1 converges strongly to the unique solution of the VI F, C. Hybrid methods for solving the variational inequality VI F, C were studied by Yamada 1 , where he assumed that F is Lipschitzian and strongly monotone
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