The two-level penalty finite element methods for Navier-Stokes equations with nonlinear slip boundary conditions are investigated in this paper, whose variational formulation is the Navier-Stokes type variational inequality problem of the second kind. The basic idea is to solve the Navier-Stokes type variational inequality problem on a coarse mesh with mesh size H in combining with solving a St...

For all Einstein-Podolsky-Rosen-type experiments on deterministic systems the Bell inequality holds, unless non-local interactions exist between certain parts of the setup. Here we show that in nonlinear systems the Bell inequality can be violated by non-local effects that are arbitrarily weak. Then we show that the quantum result of the existing Einstein-PodolskyRosen-type experiments can be r...

We consider the optimization problem of dividends and risk exposures of a firm in the diffusion model with linear costs. The variational inequality associated with this problem is given by the nonlinear form of elliptic type. Using the viscosity solutions technique, we solve the corresponding penalty equation and show the existence of a classical solution to the variational inequality. The opti...

We find that Bell’s inequality can be significantly violated (up to Tsirelson’s bound) with twomode entangled coherent states using only homodyne measurements. This requires Kerr nonlinear interactions for local operations on the entangled coherent states. Our example is a demonstration of Bell-inequality violations using classical measurements. We conclude that entangled coherent states with c...

We introduce and study a new system of generalized nonlinear relaxed cocoercive inequality problems and construct an iterative algorithm for approximating the solutions of the system of generalized relaxed cocoercive variational inequalities in Hilbert spaces. We prove the existence of the solutions for the system of generalized relaxed cocoercive variational inequality problems and the converg...

The alternating direction method is an attractive method for a class of variational inequality problems if the subproblems can be solved efficiently. However, solving the subproblems exactly is expensive even when the subproblem is strongly monotone or linear. To overcome this disadvantage, this paper develops a new alternating direction method for cocoercive nonlinear variational inequality pr...

Geometric rigidity states that a gradient field which is L-close to the set of proper rotations is necessarily L-close to a fixed rotation, and is one key estimate in nonlinear elasticity. In several applications, as for example in the theory of plasticity, energy densities with mixed growth appear. We show here that geometric rigidity holds also in L + L and in L interpolation spaces. As a fir...

This paper discusses the H∞ control problem for a class of nonlinear stochastic systems with Markovian jumps subjected to both stateand disturbance-dependent noise. We establish the equivalent conditions among Hamilton-Jacobi inequality (HJI), Hamilton-Jacobi equality (HJE), the dissipative inequality and L2-gain property for this class of systems. As to the infinite time horizon case, we synth...

This paper gives sufficient conditions for finite time stability and stabilization for perturbed nonlinear (non)homogeneous fractional order systems with time-varying delay. New stability criteria for this class of fractional order systems were derived using a recently obtained generalized Gronwall inequality as well as “classical” Bellman-Gronwall inequality. Last, a numerical example is given...

In this paper, we will try to find a new Lyapunov-type inequality for a class of nonlinear systems, special cases of which contain some well-known Hamiltonian system, EmdenFowler, half-linear and linear differential equations of second order. Our result extends the Lyapunov-type inequality given in [X. Wang, Stability criteria for linear periodic Hamiltonian systems, J. Math. Anal. Appl. 367 (2...