Multivalued nonmonotone dynamic boundary condition

In this paper, we introduce a new class of hemivariational inequalities, called dynamic boundary hemivariational inequalities, reflecting the fact that the governing operator is also active on the boundary. In our context, it concerns the Laplace operator with Wentzell (dynamic) boundary conditions perturbed by a multivalued nonmonotone operator expressed in terms of Clarke subdifferentials. We show that one can reformulate the problem so that standard techniques can be applied. We use the well-established theory of boundary hemivariational inequalities to prove that under growth and general sign conditions, the dynamic boundary hemivariational inequality admits a weak solution. Moreover, in the situation where the functionals are expressed in terms of locally bounded integrands, a “filling in the gaps” procedure at the discontinuity points is used to characterize the subdifferential on the product space. Finally, we prove that, under a growth condition and eventually smallness conditions, the Faedo–Galerkin approximation sequence converges to a desired solution.

Symmetric Fuzzy Stochastic Differential Equations with Generalized Global Lipschitz Condition

The paper contains a discussion on solutions to symmetric type of fuzzy stochastic differential equations. The symmetric equations under study have drift and diffusion terms symmetrically on both sides of equations. We claim that such symmetric equations have unique solutions in the case that equations’ coefficients satisfy a certain generalized Lipschitz condition. To show this, we prove that an approximation sequence converges to the solution. Then, a study on stability of solution is given. Some inferences for symmetric set-valued stochastic differential equations end the paper.

Operator Fractional Brownian Motion and Martingale Differences

It is well known that martingale difference sequences are very useful in applications and theory. On the other hand, the operator fractional Brownian motion as an extension of the well-known fractional Brownian motion also plays an important role in both applications and theory. In this paper, we study the relation between them. We construct an approximation sequence of operator fractional Brownian motion based on a martingale difference sequence.

The Average Errors for the Grünwald Interpolation in the Wiener Space

We determine the weakly asymptotically orders for the average errors of the Grünwald interpolation sequences based on the Tchebycheff nodes in the Wiener space. By these results we know that for the -norm approximation, the -average error of some Grünwald interpolation sequences is weakly equivalent to the -average errors of the best polynomial approximation sequence.

OPTIMAL CONTROL OF CHAOS IN NONLINEAR DRIVEN OSCILLATORS VIA LINEAR TIME-VARYING APPROXIMATIONS

An optimal chaos control procedure is proposed. The aim of using this method is to stabilize the chaotic behavior of forced continuous-time nonlinear systems by using an approximation sequence technique and linear optimal control. The idea of the approximation technique is to use a sequence of linear, time-varying equations to approximate the solution of nonlinear systems. In each of these equations, the linear-quadratic optimal tracking control is applied. The purpose is to find a linear time-varying feedback controller which produces an optimized trajectory that converges to a desired signal. This controller is then used in the original nonlinear system. As an example, the procedure is proved to work in the Duffing oscillator and the chaotic pendulum, in which the goal is to direct chaotic trajectories of these systems to a period-n orbit.

ON GAMES OF PERFECT INFORMATION: EQUILIBRIA, ε–EQUILIBRIA AND APPROXIMATION BY SIMPLE GAMES

We show that every bounded, continuous at infinity game of perfect information has an ε–perfect equilibrium. Our method consists of approximating the payoff function of each player by a sequence of simple functions, and to consider the corresponding sequence of games, each differing from the original game only on the payoff function. In addition, this approach yields a new characterization of perfect equilibria: A strategy f is a perfect equilibrium in such a game G if and only if it is an 1/n–perfect equilibrium in Gn for all n, where {Gn} stands for our approximation sequence.