Narrow Results

The realization problem for positive, continuous-time linear single-input, single-output systems with delays is formulated and solved. Sufficient conditions for the existence of positive realizations of a given proper transfer function are established. A procedure for computation of positive minimal realizations is presented and illustrated by an example.

In this paper, we introduce and study a class of new generalized nonlinear mixed quasi-variational inclusions involving non-monotone set-valued mappings with noncompact values and construct a new iterative algorithm. And proved the existence of solutions for this kind of generalized nonlinear mixed quasi-variational inclusions and the convergence of the iterative sequence generated by the algorithm. Those results improve and extend the corresponding results in the recently literatures.

In this paper, existence and uniqueness results for a class of dynamic and quasi-static problems with elastic-plastic systems are recalled, and a stability result is obtained for the quasi-static paths of those systems. The studied elastic-plastic systems are continuum 1D (bar) systems that have linear hardening, and the concept of stability of quasi-static paths used here takes into account the existence of fast (dynamic) and slow (quasi-static) times scales in the system. That concept is essentially a continuity property relatively to the size of the initial perturbations (as in Lyapunov stability) and relatively to the smallness of the rate of application of the forces (which plays here the role of the small parameter in singular perturbation problems).

We discuss existence, uniqueness, regularity, and homogenization results for some nonlinear time-dependent material models. One of the methods for proving existence and uniqueness is the so-called energetic formulation, based on a global stability condition and on an energy balance. As for the two-scale homogenization we use the recently developed method of periodic unfolding and periodic folding. We also take advantage of the abstract Γ-convergence theory for rate-independent evolutionary problems.

We review the long-standing issue of regularity of solutions to the basic problem in the calculus of variations, in both the one-dimensional and the multidimensional settings. It is shown how certain recent results fit in with the classical ones, in particular the theories of De Giorgi and Hilbert-Haar.

The system of reaction-diffusion equations with zero-flux and periodic boundary conditions, which has stable steady-states, will induce the cell reproduction pattern by basic analysis of Turing Principle. For models with constant coeffcients, their cell division patterns do not relate to the time factor. Time-related process of cell division is called aberrance of cell division, whose patterns have significant meaning to cancer pathology, especially, for the mechanism of split of cancer cell. Unlike normal cells, cancer cells do not carry on maturing once they have been made. In fact, the cells in a cancer can become even less mature over time. With all the reproducing, it is not surprising that more of the genetic information in the cell can become lost. So the cells become more and more primitive and tend to reproduce more quickly and even more haphazardly. As a result, to explain cancer cells reproduction, one has to study time-related patterns. Therefore, we extend Predator-Prey model with constant coefficients to the model with coefficients to be positive T-periodic functions (we call such a model EP-P model). The goal of this research is to study the pattern formations of EP-P model and simulate the process of tumor forming.

In this paper, by employing the powerful and effective coincidence degree method, we show the existence of T-periodic solutions of ESP-P model in , where is a strictly positively invariant region. Furthermore, Floquet theory is provided to show that the T-periodic solution x₀(t) of ESP-P model is unique in and locally uniformly asymptotically stable. This establishes a solid foundation for studying the patterns of EP-P model.

This note discusses the recent development on the existence and stability of two-dimensional and three-dimensional gravity-capillary waves on water of finite-depth. The Korteweg-de Vries (KdV) equation is derived formally from the exact governing equations and the solitary-wave solutions are obtained. The recent results on the existence of solutions of the exact equations near the solitary-wave solutions of the KdV equation are presented and various two-and three-dimensional wave solutions are given. Then, the stability of these solutions using exact equations is discussed. The ideas and methods to obtain these results are briefly mentioned.

Recently, fractional differential equations have drawn more and more attention due to their wide applications in different research areas and engineering, which leads to intensive development of the theory of fractional calculus. More and more results about the existence and uniqueness of solutions appear. In this paper, a brief overview on the recent existence and uniqueness results of fractional differential equations are provided. These equations include initial value problem for fractional differential equations, boundary value problem for fractional differential equations, fractional differential equations with time-delay.