Abstract
In this paper, we suggest and analyze an iterative algorithm to approximate a common solution of a hierarchical fixed point problem for nonexpansive mappings, a system of variational inequalities, and a split equilibrium problem in Hilbert spaces. Under some suitable conditions imposed on the sequences of parameters, we prove that the sequence generated by the proposed iterative method converges strongly to a common element of the solution set of these three kinds of problems. The results obtained here extend and improve the corresponding results of the relevant literature.
Highlights
Let H1 and H2 be two real Hilbert spaces, whose inner product and norm are denoted by ·, · and ·
Recall that the mapping T : C1 → C1 is nonexpansive if Tx – Ty ≤ x – y for all x, y ∈ C1
We denote the fixed point set of T by Fix(T) = {x ∈ C1 : x = Tx}
Summary
Let H1 and H2 be two real Hilbert spaces, whose inner product and norm are denoted by ·, · and ·. They added a nonexpansive mapping in the algorithm and proved that the generated sequence converged strongly to a common element of the fixed point set of a nonexpansive mapping and the zero point set of the sum of monotone operators They applied their main result both to equilibrium problems and convex programming. Ceng et al [9] introduced a hybrid viscosity extragradient method for finding the common elements of the solution set of a general system of variational inequalities and the common fixed point set of a countable family of nonexpansive mappings and zero points of an accretive operator in real smooth Banach spaces.
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