Abstract
In this paper, modifying the set of variational inequality and extending the nonexpansive mapping of hybrid steepest descent method to nonexpansive semigroups, we introduce a new iterative scheme by using the viscosity hybrid steepest descent method for finding a common element of the set of solutions of a system of equilibrium problems, the set of fixed points of an infinite family of strictly pseudocontractive mappings, the set of solutions of fixed points for nonexpansive semigroups, and the sets of solutions of variational inequality problems with relaxed cocoercive mapping in a real Hilbert space. We prove that the sequence converges strongly to a common element of the above sets under some mild conditions. The results shown in this paper improve and extend the recent ones announced by many others.
Highlights
Let H be a real Hilbert space with inner product ⟨⋅, ⋅⟩ and induced norm ‖ ⋅ ‖
Where T is a self-nonexpansive mapping on H, f is an αcontraction of H into itself (i.e., ‖f(x) − f(y)‖ ≤ α‖x − y‖, ∀x, y ∈ H and α ∈ (0, 1)), {αn} ⊂ (0, 1) satisfies certain conditions, and B is strongly positive bounded linear operator on H and converges strongly to fixed point x∗ of T which is the unique solution to the following variational inequality:
Takahashi [23] introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of solution (1) and the set of fixed points of a nonexpansive mapping in a Hilbert space
Summary
In this paper, modifying the set of variational inequality and extending the nonexpansive mapping of hybrid steepest descent method to nonexpansive semigroups, we introduce a new iterative scheme by using the viscosity hybrid steepest descent method for finding a common element of the set of solutions of a system of equilibrium problems, the set of fixed points of an infinite family of strictly pseudocontractive mappings, the set of solutions of fixed points for nonexpansive semigroups, and the sets of solutions of variational inequality problems with relaxed cocoercive mapping in a real Hilbert space. We prove that the sequence converges strongly to a common element of the above sets under some mild conditions. The results shown in this paper improve and extend the recent ones announced by many others
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