Abstract
AbstractWe use strongly pseudocontraction to regularize the following ill-posed monotone variational inequality: finding a point "Equation missing" with the property "Equation missing" such that "Equation missing", "Equation missing" where "Equation missing", "Equation missing" are two pseudocontractive self-mappings of a closed convex subset "Equation missing" of a Hilbert space with the set of fixed points "Equation missing". Assume the solution set "Equation missing" of (VI) is nonempty. In this paper, we introduce one implicit scheme which can be used to find an element "Equation missing". 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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