Abstract
AbstractWe introduce a new hybrid extragradient viscosity approximation method for finding the common element of the set of equilibrium problems, the set of solutions of fixed points of an infinitely many nonexpansive mappings, and the set of solutions of the variational inequality problems for "Equation missing"-inverse-strongly monotone mapping in Hilbert spaces. Then, we prove the strong convergence of the proposed iterative scheme to the unique solution of variational inequality, which is the optimality condition for a minimization problem. Results obtained in this paper improve the previously known results in this area.
Highlights
Let H be a real Hilbert space, and let C be a nonempty closed convex subset of H
Let f be a contraction of H into itself, A a strongly positive bounded linear operator on H with coefficient γ > 0, and B a β-inverse-strongly monotone mapping of C into H; define sequences {xn}, {yn}, {kn}, and {un} recursively by x1 x ∈ C chosen arbitrary, 1 rn y − un, un − xn
We prove that the sequences {xn}, {yn}, {kn} and {un} generated by the above iterative scheme 1.25 converge strongly to a common element of the set of solutions of the equilibrium problem 1.13, the set of common fixed points of infinitely family nonexpansive mappings, and the set of solutions of variational inequality 1.1 for a β-inverse-strongly monotone mapping in Hilbert spaces
Summary
Let H be a real Hilbert space, and let C be a nonempty closed convex subset of H. Recall that a mapping T of H into itself is called nonexpansive see 1 if T x − T y ≤ x − y for all x, y ∈ H. We denote by F T {x ∈ C : T x x} the set of fixed points of T. Fixed Point Theory and Applications u − z, v − u ≥ 0, ∀v ∈ C, 1.2 if and only if u PCz. It is well known that PC is a nonexpansive mapping of H onto C and satisfies x − y, PCx − PCy ≥ PCx − PCy 2, ∀x, y ∈ H. One can see that the variational inequality 1.1 is equivalent to a fixed point problem. The variational inequality has been extensively studied in literature; see, for instance, 2– 6.
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