A generalization of Hardy’s operator and an asymptotic Müntz–Szász Theorem

We prove an asymptotic version of the Muntz–Szasz theorem, and use it to prove that all monomial operators are vanishing preserving.     

On uniform propagation of chaos

In this paper we obtain a time-uniform propagation estimate for a system of interacting diffusion processes. Using a well defined metric function h , our result guarantees a time-uniform estimate for the convergence of a class of interacting stochastic differential equations towards their mean field limit, under conditions that ensure that the decay associated to the internal dynamics term dominates the interaction and noise terms. Our result should have diverse applications, particularly in neuroscience, and allows for models more elaborate than the one of Wilson and Cowan. In particular, the internal dynamics need not be that of linear decay.     

ALMOST PERIODIC SOLUTIONS TO STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS

By applying the properties of almost periodic function and exponential dichotomy of linear system as well as Banach fixed point theorem,we establish the conditions for the existence and uniqueness of square-mean almost periodic solution to some stochastic functional differential equations.     

带生长函数的非定常森林发展系统

讨论非定常森林发展系统的动力学模型,证明了方程解的存在唯一性,并用Lyapunov稳定性理论讨论了系统的稳定性.     

Weak attractors for the dissipative fractional Heisenberg equation

In this short paper, we consider the long time behaviors of the fractional Heisenberg equation and the existence of a global weak attractor is proved for the shift dynamics in the path space. The key ingredient is some new type of commutator structure introduced in this paper, which seems indispensable in proving the compactness of the dynamics. The technique introduced in this paper may also be useful to other fractional order partial differential equations.     

On the criteria for integrability of the Liénard equation

The Liénard equation is of a high importance from both mathematical and physical points of view. However a question about integrability of this equation has not been completely answered yet. Here we provide a new criterion for integrability of the Liénard equation using an approach based on nonlocal transformations. We also obtain some of the previously known criteria for integrability of the Liénard equation as a straightforward consequence of our approach’s application. We illustrate our results by several new examples of integrable Liénard equations.     

Asymptotic Behavior of Solutions to the Difference Equation Using the Second Differences of a Liapunov Function

We introduce the concept of the second Liapunov differences in difference equations and show that, in the autonomous case, the second variations can guarantee the existence of solutions tending to zero while others starting arbitrarily near 0 go to ∞. These are based on Yorke's results for ordinary differential equations.     

On the Dynamics of

We investigate the existence of unbounded solutions and the period-two convergence of solutions of the equation in the title with the parameter γ positive, the remaining parameters nonnegative, and with nonnegative initial conditions.     

Continuous dependence on data for bilocal difference equations

The continuous dependence on data is studied for a class of second order difference equations governed by a maximal monotone operator A in a Hilbert space. A nonhomogeneous term f appears in the equation and some bilocal boundary conditions a, b are added. One shows that the function which associates to {a, b, A, f} the solution of this boundary value problem is continuous in a specific sense. One uses the convergence of a sequence of operators in the sense of the resolvent. The problem studied here is the discrete variant of a problem from the continuous case.