A generalized nonautonomous SIRVS model

A nonautonomous SIRVS model with time‐dependent parameters is considered. The global dynamics is investigated, and conditions for the permanence, extinction, and disease‐free equilibrium are studied. Substitute quantities that replace the basic reproduction number are presented. Numerical simulations illustrate the dynamic behavior of the proposed model. Copyright © 2012 John Wiley & Sons, Ltd.     

On the 1D viscous reactive and radiative gas with the first‐order Arrhenius kinetics

In this paper, we shall establish the global existence and exponential stability of solutions in Hⁱ(i = 1,2,4) for a Stefan‐Boltzmann model of a viscous, reactive, and radiative gas with the first‐order Arrhenius kinetics in a bounded interval.     

Global attractors for the viscous hyperelastic‐rod wave equation

We consider the viscous hyperelastic‐rod wave equation subject to an external force, where the viscous term is given by second order differential operator in divergence form. Under some mild assumptions on the viscous term, first, we establish the global well‐posedness in both the periodic case and the case of the whole line, afterwards, we show the existence of global attractors for the two cases, respectively. Copyright © 2012 John Wiley & Sons, Ltd.     

Global convergence of a modified limited memory BFGS method for non-convex minimization

In this paper, a modified limited memory BFGS method for solving large-scale unconstrained optimization problems is proposed. A remarkable feature of the proposed method is that it possesses global convergence property without convexity assumption on the objective function. Under some suitable conditions, the global convergence of the proposed method is proved. Some numerical results are reported which illustrate that the proposed method is efficient.     

Attractors for second order periodic lattices with nonlinear damping

We consider second order nonlinear lattices under the effect of nonlinear damping. The family we study is subject to cyclic boundary conditions and includes as distinguished examples the Fermi–Pasta–Ulam and sine-Gordon lattices. We prove global well posedness and existence of a global attractor.     

Coexistence in a discrete competition model with dispersal

We propose a discrete-time competition model between two populations to study the effects of dispersal upon population interactions. It is assumed that dispersal occurs after reproduction and in synchrony. We first analyse a two-patch single species population model with no interspecific competition. Based on these results, we derive sufficient conditions for population coexistence. It is proved that the system is uniformly persistent and possesses a unique coexisting equilibrium.     

A globally convergent BFGS method for nonlinear monotone equations without any merit functions

Since 1965, there has been significant progress in the theoretical study on quasi-Newton methods for solving nonlinear equations, especially in the local convergence analysis. However, the study on global convergence of quasi-Newton methods is relatively fewer, especially for the BFGS method. To ensure global convergence, some merit function such as the squared norm merit function is typically used. In this paper, we propose an algorithm for solving nonlinear monotone equations, which combines the BFGS method and the hyperplane projection method. We also prove that the proposed BFGS method converges globally if the equation is monotone and Lipschitz continuous without differentiability requirement on the equation, which makes it possible to solve some nonsmooth equations. An attractive property of the proposed method is that its global convergence is independent of any merit function.We also report some numerical results to show efficiency of the proposed method.