Abstract
The purpose of this paper is to introduce the extragradient methods for solving split feasibility problems, generalized equilibrium problems, and fixed point problems involved in nonexpansive mappings and pseudocontractive mappings. We establish the results of weak and strong convergence under appropriate conditions. As applications of our three main theorems, when the mappings and their domains take different types of cases, we can obtain nine iterative approximation theorems and corollas on fixed points, variational inequality solutions, and equilibrium points.
Highlights
Let H1 and H2 be two real Hilbert spaces, and let C and Q be two nonempty closed and convex subsets of H1 and H2, respectively
We denote the solution set of the split feasibility problem (SFP) by
Motivated and inspired by the above works, we will investigate the weak and strong convergence methods for solving the split feasibility problems, generalized equilibrium problems, and fixed point problems involved in nonexpansive mappings and pseudocontractive mappings
Summary
Let H1 and H2 be two real Hilbert spaces, and let C and Q be two nonempty closed and convex subsets of H1 and H2, respectively. E split feasibility problem (SFP) is to find a point x such that x ∈ C, Ax ∈ Q. To solve the SFP, Byrne [2, 7] first introduced the so-called CQ algorithm as follows:. E CQ algorithm can be viewed from two different but equivalent ways: optimization and fixed point [6]. From the view of optimization point, x∗ ∈ Ω in (2) if and only if x∗ is a solution of the following minimization problem with zero optimal value minx∈Cf(x) ≔ (1/2)‖Ax − PQAx‖2, where f is a differentiable convex function and has a Lipschitz gradient given by ∇f(x) A∗(I − PQ)A, with Lipschitz constant L ρ(A∗A). Motivated and inspired by the above works, we will investigate the weak and strong convergence methods for solving the split feasibility problems, generalized equilibrium problems, and fixed point problems involved in nonexpansive mappings and pseudocontractive mappings. Our results in this paper generalize and improve upon the corresponding modern results of many other authors
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