On the asymptotic period of powers of a fuzzy matrix

AbstractIn our prior study, we have examined in depth the notion of an asymptotic period of the power sequence of an n×n fuzzy matrix with max-Archimedean-t-norms, and established a characterization for the power sequence of an n×n fuzzy matrix with an asymptotic period using analytical-decomposition methods. In this paper, by using graph-theoretical tools, we further give an alternative proof for this characterization. With the notion of an asymptotic period using graph-theoretical tools, we additionally show a new characterization for the limit behaviour, and then derive some results for the power sequence of an n×n fuzzy matrix with an asymptotic period

Asymptotic stability and generalized Gelfand spectral radius formula

AbstractLet ∑ be a set of n × n complex matrices. For m = 1, 2, …, let ∑m be the set of all products of matrices in ∑ of length m. Denote by ∑′ the multiplicative semigroup generated by ∑. ∑ is said to be asymptotically stable (in the sense of dynamical systems) if there is 0 < α < 1 such that there are bounded neighborhoods U, V ⊂ Cn of the origin for which AV ⊂ αmU for all A ∈ ∑m, m = 1, 2, …. For a bounded set ∑ of n × n complex matrices, it is shown that the following conditions are mutually equivalent:(i) ∑ is asymptotically stable; (ii) \̂g9(∑) = lim supm → ∞[supA ∈ ∑m ‖ A ‖]1/m < 1; (iii) ϱ(∑) = lim supm → ∞[supA ∈ ∑m ϱ(A)]1/m < 1, where ϱ(A) stands for the spectral radius of A; and (iv) there exists a positive number α such that ϱ(A) ⩽ α < 1 for all A ∈ ∑′. This fact answers an open question raised by Brayton and Tong. The generalized Gelfand spectral radius formula, that is, ϱ(∑) = \̂g9(∑), conjectured by Daubechies and Lagarias and proved by Berger and Wang using advanced tools from ring theory and then by Elsner using analytic-geometric tools, follows immediately form the above asymptotic stability theorem

The Hierarchical Minimax Inequalities for Set-Valued Mappings

We study the minimax inequalities for set-valued mappings with hierarchical process and propose two versions of minimax inequalities in topological vector spaces settings. As applications, we discuss the existent results of solutions for set equilibrium problems. Some examples are given to illustrate the established results

Multistep Hybrid Extragradient Method for Triple Hierarchical Variational Inequalities

We consider a triple hierarchical variational inequality problem (THVIP), that is, a variational inequality problem defined over the set of solutions of another variational inequality problem which is defined over the intersection of the fixed point set of a strict pseudocontractive mapping and the solution set of the classical variational inequality problem. Moreover, we propose a multistep hybrid extragradient method to compute the approximate solutions of the THVIP and present the convergence analysis of the sequence generated by the proposed method. We also derive a solution method for solving a system of hierarchical variational inequalities (SHVI), that is, a system of variational inequalities defined over the intersection of the fixed point set of a strict pseudocontractive mapping and the solution set of the classical variational inequality problem. Under very mild conditions, it is proven that the sequence generated by the proposed method converges strongly to a unique solution of the SHVI

Hierarchical Fixed Point Problems in Uniformly Smooth Banach Spaces

We propose some relaxed implicit and explicit viscosity approximation methods for hierarchical fixed point problems for a countable family of nonexpansive mappings in uniformly smooth Banach spaces. These relaxed viscosity approximation methods are based on the well-known viscosity approximation method and hybrid steepest-descent method. We obtain some strong convergence theorems under mild conditions

Multistep Hybrid Iterations for Systems of Generalized Equilibria with Constraints of Several Problems

We first introduce and analyze one multistep iterative algorithm by hybrid shrinking projection method for finding a solution of the system of generalized equilibria with constraints of several problems: the generalized mixed equilibrium problem, finitely many variational inclusions, the minimization problem for a convex and continuously Fréchet differentiable functional, and the fixed-point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. We prove strong convergence theorem for the iterative algorithm under suitable conditions. On the other hand, we also propose another multistep iterative algorithm involving no shrinking projection method and derive its weak convergence under mild assumptions

Triple Hierarchical Variational Inequalities with Constraints of Mixed Equilibria, Variational Inequalities, Convex Minimization, and Hierarchical Fixed Point Problems

We introduce and analyze a hybrid iterative algorithm by virtue of Korpelevich's extragradient method, viscosity approximation method, hybrid steepest-descent method, and averaged mapping approach to the gradient-projection algorithm. It is proven that under appropriate assumptions, the proposed algorithm converges strongly to a common element of the fixed point set of infinitely many nonexpansive mappings, the solution set of finitely many generalized mixed equilibrium problems (GMEPs), the solution set of finitely many variational inequality problems (VIPs), the solution set of general system of variational inequalities (GSVI), and the set of minimizers of convex minimization problem (CMP), which is just a unique solution of a triple hierarchical variational inequality (THVI) in a real Hilbert space. In addition, we also consider the application of the proposed algorithm to solve a hierarchical fixed point problem with constraints of finitely many GMEPs, finitely many VIPs, GSVI, and CMP. The results obtained in this paper improve and extend the corresponding results announced by many others

Strong and Weak Convergence Criteria of Composite Iterative Algorithms for Systems of Generalized Equilibria

We first introduce and analyze one iterative algorithm by using the composite shrinking projection method for finding a solution of the system of generalized equilibria with constraints of several problems: a generalized mixed equilibrium problem, finitely many variational inequalities, and the common fixed point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense and infinitely many nonexpansive mappings in a real Hilbert space. We prove a strong convergence theorem for the iterative algorithm under suitable conditions. On the other hand, we also propose another iterative algorithm involving no shrinking projection method and derive its weak convergence under mild assumptions. Our results improve and extend the corresponding results in the earlier and recent literature