Abstract
We present a new iterative method for finding a common element of the set of fixed points of a nonexpansive mapping, the set of solutions to an equilibrium problem, and the set of zeros of the sum of maximal monotone operators and prove the strong convergence theorems in the Hilbert spaces. We also apply our results to variational inequality and optimization problems.
Highlights
Let C be a nonempty closed convex subset of a real Hilbert space H
We present a new iterative method for finding a common element of the set of fixed points of a nonexpansive mapping, the set of solutions to an equilibrium problem, and the set of zeros of the sum of maximal monotone operators and prove the strong convergence theorems in the Hilbert spaces
A mapping S : C → C is nonexpansive if ‖Sx − Sy‖ ≤ ‖x − y‖ for all x, y ∈ C
Summary
Let C be a nonempty closed convex subset of a real Hilbert space H. There are two iterative methods for approximating fixed points of a nonexpansive mapping. The iteration procedure of Mann’s type for approximating fixed points of a nonexpansive mapping S is the following: x1 ∈ C and xn+1 = αnxn + (1 − αn) Sxn,. Lin and Takahashi [12] introduced an iterative sequence that converges strongly to an element of (A + B)−10 ∩ F−10, where F is another maximal monotone operator. Motivated by the above results, in this paper, we introduce a new iterative algorithm for finding a common element of the set of fixed points of a nonexpansive mapping, the set of solutions to an equilibrium problem, and the set of zeros of the sum of maximal monotone operators and prove the strong convergence theorems in the Hilbert spaces. We give the applications to the variational inequality and optimization problems
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