A general iterative algorithm for monotone operators with λ-hybrid mappings in Hilbert spaces

• Published in: Journal of inequalities and applications Vol. 2014; no. 1; pp. 1 - 17
• Main Author: Hong, Chung-Chien
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 22.07.2014
• Subjects: Analysis; Applications of Mathematics; Approximation; Convergence; Hilbert space; Inequalities; Mapping; Mathematics; Mathematics and Statistics; Operators; Operators (mathematics); Theorems; maximal monotone operator; variational inequality; inverse-strongly monotone mapping; fixed point; resolvent; equilibrium problem
• ISSN: 1029-242X; 1025-5834; 1029-242X
• DOI: 10.1186/1029-242X-2014-264

Let C be a nonempty closed convex subset of a Hilbert space ℋ, let B , G be two set-valued maximal monotone operators on C into ℋ, and let g : H → H be a k -contraction with 0 < k < 1 . A : C → H is an α -inverse strongly monotone mapping, V : H → H is a γ ¯ -strongly monotone and L -Lipschitzian mapping with γ ¯ > 0 and L > 0 , T : C → C is a λ -hybrid mapping. In this paper, a general iterative scheme for approximating a point of F ( T ) ∩ ( A + B ) − 1 0 ∩ G − 1 0 ≠ ∅ is introduced, where F ( T ) is the set of fixed points of T , and a strong convergence theorem of the sequence generated by the iterative scheme is proved under suitable conditions. As applications of our strong convergence theorem, the related equilibrium and variational problems are also studied. MSC: 47H05, 47H10, 58E35.