On Regularity, Transitivity, and Ergodic Principle for Quadratic Stochastic Volterra Operators

In this paper we showed an equivalence of notions of regularity, transitivity and Ergodic principle for quadratic stochastic Volterra operators acting on the finite dimensional simplex.Comment: 5 page

On infinite dimensional Volterra type operators

In this paper we study Volterra type operators on infinite dimensional simplex. It is provided a sufficient condition for Volterra type operators to be bijective. Furthermore it is shoved that the condition is not necessary.Comment: 10 page

On unification of the strong convergence theorems for a finite family of total asymptotically nonexpansive mappings in Banach spaces

In this paper, we unify all know iterative methods by introducing a new explicit iterative scheme for approximation of common fixed points of finite families of total asymptotically $I$-nonexpansive mappings. Note that such a scheme contains as a particular case of the method introduced in [C.E. Chidume, E.U. Ofoedu, \textit{Inter. J. Math. & Math. Sci.} \textbf{2009}(2009) Article ID 615107, 17p]. We construct examples of total asymptotically nonexpansive mappings which are not asymptotically nonexpansive. Note that no such kind of examples were known in the literature. We prove the strong convergence theorems for such iterative process to a common fixed point of the finite family of total asymptotically $I-$nonexpansive and total asymptotically nonexpansive mappings, defined on a nonempty closed convex subset of uniformly convex Banach spaces. Moreover, our results extend and unify all known results.Comment: 22 pages, Journal of Applied Mathematics (in press

Reaching a nonlinear consensus: polynomial stochastic operators

We provide a general nonlinear protocol for a structured time-varying and synchronous multi-agent system. We present an opinion sharing dynamics of the multi-agent system as a trajectory of a polynomial stochastic operator associated with a multidimensional stochastic hypermatrix. We show that the multi-agent system eventually reaches to a consensus if either one of the following two conditions is satisfied: (i) every member of the group people has a positive subjective opinion on the given task after some revision steps or (ii) all entries of a multidimensional stochastic hypermatrix are positive. Numerical results are also presented

Velocity of Long Gravity Waves in the Ocean

Multi-Agent Systems (MAS) have attracted more and more interest in recent years. Most researches in the study of discrete-time MAS, presented in the past few years, have considered linear cooperative rules. However, local interactions between agents are more likely to be governed by nonlinear rules. In this paper, we investigate the consensus of discrete-time MAS with time invariant nonlinear cooperative rules. Based on our presented nonlinear model, we show a consensus in the discrete-time MAS. Our model generalizes a classical time invariant De Groot model. It seems that, unlike a linear case, a consensus can be easily achieved a nonlinear case

Ergodicity of p-majorizing quadratic stochastic operators

A scrambling square stochastic matrix plays an important role in the theory of the classical Markov chain. One of the classical results states that a row-stochastic matrix is strongly ergodic if and only if its some power is a scrambling matrix. In this paper, we deal with the similar problem for a cubic stochastic matrix. We introduce a notion of p-majorizing quadratic stochastic operators and study the strong ergodicity of p-majorizing quadratic stochastic operators associated with scrambling, Sarymsakov, and Wolfowitz cubic stochastic matrices. © 2018 Polymat Ltd. All rights reserved