Back to Search
Start Over
An Implicit Hierarchical Fixed-Point Approach to General Variational Inequalities in Hilbert Spaces.
- Source :
- Fixed Point Theory & Applications; 2011, Special section p1-17, 17p
- Publication Year :
- 2011
-
Abstract
- Let C be a nonempty closed convex subset of a real Hilbert space H. Let F : C → H be a κ-Lipschitzian and η-strongly monotone operator with constants κ, η > 0, V, T : C → C be nonexpansive mappings with Fix(T) ≠ ∅ where Fix(T) denotes the fixed-point set of T, and f : C → H be a ρ-contraction with coefficient ρ ∈ [0, 1). Let 0 < µ < 2η/κ² and 0 < γ ≤ τ, where τ = √1 - µ(2η - µκ²). For each s, t ∈ (0, 1), let x<subscript>s,t</subscript> be a unique solution of the fixed-point equation x<subscript>s,t</subscript> PC[sγf(x<subscript>s,t</subscript>) + (I - sµF)(tV + (1 - t)T)x<subscript>s,t</subscript>].We derive the following conclusions on the behavior of the net {x<subscript>s,t</subscript>} along the curve t = t(s): (i) if t(s) = O(s), as s → 0, then x<subscript>s,t(s)</subscript> → z<subscript>∞</subscript> strongly, which is the unique solution of the variational inequality of finding z<subscript>∞</subscript> ∈ Fix(T) such that <[(µF - γf) + l(I - V)]z<subscript>∞</subscript>, x - z<subscript>∞</subscript>> ≥ 0, for all x ∈ Fix(T) and ?(ii) if t(s)/s → <subscript>∞</subscript>, as s → 0, then x<subscript>s,t(s)</subscript> → x<subscript>∞</subscript> strongly, which is the unique solution of some hierarchical variational inequality problem. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 16871820
- Database :
- Complementary Index
- Journal :
- Fixed Point Theory & Applications
- Publication Type :
- Academic Journal
- Accession number :
- 71892888
- Full Text :
- https://doi.org/10.1155/2011/748918