Abstract
We present an iterative method for fixed point problems, generalized mixed equilibrium problems, and variational inequality problems. Our method is based on the so-called viscosity hybrid steepest descent method. Using this method, we can find the common element of the set of fixed points of a nonexpansive mapping, the set of solutions of generalized mixed equilibrium problems, and the set of solutions of variational inequality problems for a relaxed cocoercive mapping in a real Hilbert space. Then, we prove the strong convergence of the proposed iterative scheme to the unique solution of variational inequality. The results presented in this paper generalize and extend some well-known strong convergence theorems in the literature.
Highlights
Throughout this paper, unless otherwise specified, we consider H to be a real Hilbert space with inner product ·, · and its induced norm ·
Marino and Xu 13 introduced a new iterative scheme by the viscosity approximation method: xn 1 nγ f xn 1 − nA Sxn. They proved that the sequence {xn} generated by 1.20 converges strongly to the unique solution of the variational inequality: γfz − Az, x − z ≤ 0, ∀x ∈ F S, 1.21 which is the optimality condition for the minimization problem: min x∈F S
We introduce an iterative scheme by using a viscosity hybrid steepest descent method for finding a common element of the set of solutions of a generalized mixed equilibrium problem, the set of fixed points of a nonexpansive mapping, and the set of solutions of variational inequality problem for a relaxed cocoercive mapping in a real Hilbert space
Summary
Throughout this paper, unless otherwise specified, we consider H to be a real Hilbert space with inner product ·, · and its induced norm ·.
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