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The hybrid steepest-descent method introduced by Yamada (2001) is an algorithmic solution to the variational inequality problem over the fixed point set of nonlinear mapping and applicable to a broad range of convexly constrained nonlinear inverse problems in real Hilbert spaces. Lehdili and Moudafi (1996) introduced the new prox-Tikhonov regularization method for proximal point algorithm to generate a strongly convergent sequence and established a convergence property for it by using the technique of variational distance in Hilbert spaces. In this paper, motivated by Yamada's hybrid steepest-descent and Lehdili and Moudafi's algorithms, a generalized hybrid steepest-descent algorithm for computing the solutions of the variational inequality problem over the common fixed point set of sequence of nonexpansive-type mappings in the framework of Banach space is proposed. The strong convergence for the proposed algorithm to the solution is guaranteed under some assumptions. Our strong convergence theorems extend and improve certain corresponding results in the recent literature. [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]

We propose a relaxed composite implicit iteration process for finding approximate common fixed points of a finite family of strictly pseudocontractive mappings in Banach spaces. Several convergence results for this process are established. [ABSTRACT FROM AUTHOR]