Three step algorithm for weighted resolvent average of a finite family of monotone operators

Malihe Bagheri, Mehdi Roohi
2018 Filomat  
In this paper, we introduce a composite iterative method for finding a common zero point of weighted resolvent average of a finite family of monotone operators. Furthermore, the strong convergence of the proposed iterative method is established. Finally, our results are illustrated by some numerical examples. A monotone operator A is called maximal monotone if there exists no monotone operator B such that gra A is a proper subset of gra B. The resolvent of A is the mapping J A = (A + Id) −1 .
more » ... A = (A + Id) −1 . Recall [2] that a map T : H → H is called nonexpansive if Tx − Ty ≤ x − y , ∀ x, y ∈ H. A point x ∈ H is said to be a fixed point of the operator T : H → H, if Tx = x. The set of all fixed points of T is denoted by Fix(T), i.e., Fix(T) = {x ∈ H : Tx = x}. Let us consider the zero point problem for monotone operator A on a real Hilbert space H, i.e., finding a point x ∈ dom A such that 0 ∈ A(x). It was first introduced by Martinet [12] in 1970. Rockafellar [16] defined the proximal point algorithm of Martinet by generalizing a sequence {x n } such that x n+1 = J s n A x n + e n , n ∈ N,
doi:10.2298/fil1817031b fatcat:kqr2ijeukzgz7lgfafl2fav53m