An iterative algorithm for hierarchical fixed point problems for a finite family of nonexpansive mappings

Abstract

This paper aims to deal with an iterative algorithm for hierarchical fixed point problems for a finite family of nonexpansive mappings in the setting of real Hilbert spaces. We establish the strong convergence of the proposed method under some suitable conditions. Numerical examples are presented to illustrate the proposed method and convergence result. The algorithm and result presented in this paper extend and improve some well-known algorithm and results, respectively, in the literature.MSC: Primary 49J30; 47H09; secondary 47J20; 49J40

Mildly Inertial Subgradient Extragradient Method for Variational Inequalities Involving an Asymptotically Nonexpansive and Finitely Many Nonexpansive Mappings

“…In the case of θ n = 0 ∀n ≥ 1, one says that T is nonexpansive. Both the class of nonexpansive operators and asymptotically nonexpansive operators via various iterative techniques have been studied recently; see, e.g., the works by the authors of [1][2][3][4][5][6][7][8][9][10][11][12][13]. Let A : H → H be a self operator.…”

“…Consider the classical variational inequality problem (VIP) of consisting of u * ∈ C such that with τ ∈ (0, 1 L ). If problem (1) has a solution, then he showed the weak convergence of {u n } constructed by (2) to a solution of problem (1). Since then, Korpelevich's extragradient method and its variants have been paid great attention to by many scholars, who improved it in various techniques and approaches; see, e.g., the works by the authors of [23][24][25][26][27][28][29][30][31][32][33][34].…”

“…Let F be a κ-Lipschitzian, η-strongly monotone self-mapping on H, and f be a contractive map with constant δ ∈ (0, 1). In 2015, Bnouhachem et al [2] introduced an iterative algorithm for solving a hierarchical fixed point problem (HFPP) for a finite pool {T i } N i=1 , i.e., for arbitrarily given x 0 ∈ H, the sequence {x n } is constructed by…”

“…where T i,n = (1 − δ i,n )I + δ i,n T i and δ i,n ∈ (0, 1) for integer i ∈ {1, 2, ..., N}. Let the parameters satisfy 0 < µκ 2 < 2η and 0 ≤ ρτ < ν, with ν = µ(η − µκ 2 2 ). Also, suppose that the sequences {α n }, {β n }, {γ n } ⊂ (0, 1) satisfy the following requirements.…”

Multi-agent Finite-time Ideal Convergence Algorithm Based on Bilinear Manifold

“…In order to improve the controllability of the network, a lot of literatures at home and abroad put forward the architecture of new bilinear manifold network [1][2] at present. As the multi-agent system has the distribution, autonomy, sociality and other characteristics, the realization of network control by using the multi-agent system is conducive to reducing the complexity of the network control and improving the reliability and intelligence of the network control.…”