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In this article, we consider a monotone variational inequality with variational inequality constraint over the set of fixed points of a nonexpansive mapping, which is called a triple-hierarchical variational inequality (THVI). A relaxed hybrid steepest-descent method with variable parameters is introduced for solving the THVI. Strong convergence of the method to a unique solution of the THVI is studied under certain assumptions. We also investigate another monotone variational inequality with the variational inequality constraint over the set of common fixed points of a finite family of nonexpansive mappings, and present an iterative algorithm for solving such kind of problems. It is proven that under mild conditions the sequence generated by the proposed algorithm converges strongly to a unique solution of later problem. [ABSTRACT FROM AUTHOR]

We deal with a common fixed point problem for a family of quasinonexpansive mappings defined on a Hilbert space with a certain closedness assumption and obtain strongly convergent iterative sequences to a solution to this problem. We propose a new type of iterative scheme for this problem. A feature of this scheme is that we do not use any projections, which in general creates some difficulties in practical calculation of the iterative sequence. We also prove a strong convergence theorem by the shrinking projection method for a family of such mappings. These results can be applied to common zero point problems for families of monotone operators. [ABSTRACT FROM AUTHOR]

Let C be a nonempty closed convex subset of a real Hilbert space H. Let F : C → H be a κ-Lipschitzian and η-strongly monotone operator with constants κ, η > 0, V, T : C → C be nonexpansive mappings with Fix(T) ≠ ∅ where Fix(T) denotes the fixed-point set of T, and f : C → H be a ρ-contraction with coefficient ρ ∈ [0, 1). Let 0 < µ < 2η/κ² and 0 < γ ≤ τ, where τ = √1 - µ(2η - µκ²). For each s, t ∈ (0, 1), let xs,t be a unique solution of the fixed-point equation xs,t PC[sγf(xs,t) + (I - sµF)(tV + (1 - t)T)xs,t].We derive the following conclusions on the behavior of the net {xs,t} along the curve t = t(s): (i) if t(s) = O(s), as s → 0, then xs,t(s) → z∞ strongly, which is the unique solution of the variational inequality of finding z∞ ∈ Fix(T) such that <[(µF - γf) + l(I - V)]z∞, x - z∞> ≥ 0, for all x ∈ Fix(T) and ?(ii) if t(s)/s → ∞, as s → 0, then xs,t(s) → x∞ strongly, which is the unique solution of some hierarchical variational inequality problem. [ABSTRACT FROM AUTHOR]