Abstract
We introduce a new implicit iteration method for finding a solution for a variational inequality involving Lipschitz continuous and strongly monotone mapping over the set of common fixed points for a finite family of nonexpansive mappings on Hilbert spaces.
Highlights
Let C be a nonempty closed and convex subset of a real Hilbert space H with inner product ·, · and norm ·, and let F : H → H be a nonlinear mapping
The variational inequality problem is formulated as finding a point p∗ ∈ C such that
Variational inequalities were initially studied by Kinderlehrer and Stampacchia in 1 and ever since have been widely investigated, since they cover as diverse disciplines as partial differential equations, optimal control, optimization, mathematical programming, mechanics, and finance see 1–3
Summary
Let C be a nonempty closed and convex subset of a real Hilbert space H with inner product ·, · and norm · , and let F : H → H be a nonlinear mapping. For an arbitrary initial point x0 ∈ H, the sequence {xk}∞k 1 is generated as follows: x1 β1x0 1 − β1 T1x1 − λ1μF T1x1 , x2 β2x1 1 − β2 T2x2 − λ2μF T2x2 , xN βN xN−1 1 − βN TN xN − λN μF TN xN , xN 1 βN 1xN 1 − βN 1 T1xN 1 − λN 1μF T1xN 1 , The scheme is written in a compact form as xk βkxk−1 1 − βk T k xk − λkμF T k xk , k ≥ 1 I denotes the identity mapping of H, and the parameters {λt}, {βti} ⊂ 0, 1 for all t ∈ 0, 1 satisfy the following conditions: λt → 0 as t → 0 and 0 < lim inft → 0βti ≤ lim supt → 0βti < 1, i 1, . . . , N
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