Abstract
In this paper, we study the implicit and inertial-type viscosity approximation method for approximating a solution to the hierarchical variational inequality problem. Under some mild conditions on the parameters, we prove that the sequence generated by the proposed methods converges strongly to a solution of the above-mentioned problem in $q$-uniformly smooth Banach spaces. The results obtained in this paper generalize and improve many recent results in this direction.
Highlights
Let E be a real Banach space and K be a nonempty, closed and convex subset of E
Under some assumptions on the parameters, they proved that the sequence {xn} generated by (9) converges strongly to the hierarchical fixed point of T with respect to the mapping S which is a unique solution to the variational inequality (8)
Motivated and inspired by ongoing research in this direction, our purpose in this study is to provide an affirmative answer to the question mentioned above by introducing an inertial-type viscosity approximation method for solving hierarchical variational inequality problem in q−uniformly smooth
Summary
Let E be a real Banach space and K be a nonempty, closed and convex subset of E. By combining (4) and (5), Tian [26] introduced the following general viscosity method for approximating fixed point of nonexpansive mapping in real Hilbert spaces: xn+1 = αnγf (xn) + (1 − μαnG)T xn, n ≥ 0, (6)
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