Narrow Results

The properties of Cournot mixed oligopoly consisting of one public firm and one or more than one private firms have mostly been analyzed for simple cases on the basis of numerical calculations of the equilibrium values for a linear market demand function and linear or quadratic cost functions. In this chapter, after proving the existence of a unique equilibrium in Cournot mixed oligopoly under general conditions on the market demand and each firm’s cost function, we derive conditions ensuring the existence of a unique Nash equilibrium for the mixed oligopoly where one public firm and at least one of the private firms are active in a general model of Cournot mixed oligopoly with one public firm and several private firms.

Whether quanta exist or not is a complex question that engages with a variety of issues in the realism/anti-realism debate and concerning the interpretation of quantum mechanics, ranging from the kinds of commitments and requirements needed to determine the existence of quantum particles to the sorts of empirical and instrumental control of the relevant phenomena. In this paper, I argue that one need not settle the issue of the existence of quanta to determine their objectivity, that is, to settle whether they are what they are independently of what one takes them to be. The objectivity of quanta is a separate issue from those concerning their existence and metaphysical specification of the kind of thing quanta are. Along the way, I discuss the similarities and differences between mathematical objectivity and the objectivity of quanta, and consider the role that the framework in which quanta are formulated plays in addressing the issue of their objectivity. In the end, the objectivity of quanta and their existence are importantly different. One can hold on to the former while being agnostic about the latter.

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.

Surfaces arising in amorphous thin-film-growth are often described by certain classes of stochastic PDEs. In this paper we address the question of existence of unique solutions. We obtain a gap of regularity between the unique local and the global solutions, which are necessary to define statistical quantities like mean interface width or correlation functions.

Sufficient conditions are obtained for the existence of periodic positive solution of an impulsive delay differential equation.

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 study the differential equations which govern the linear deformation of thin elastic shells made from a porous material. In our approach, we employ the theory of Cosserat elastic surfaces with voids. On the basis of inequalities of Korn's type for Cosserat surfaces, we prove existence and uniqueness results for the solution of the boundary-value problems, in the equilibrium theory. For the dynamic equations, we investigate the boundary-initial-value problems for porous Cosserat shells using the semigroup of linear operators theory and we establish the existence, uniqueness and continuous dependence of solution.

In this paper we study the existence and uniqueness of solutions for the initial-boundary value problem of a micropolar binary mixture. We use some results of the semigroup theory of linear operators to obtain an existence and uniqueness theorem for the initial value problem with homogeneous Dirichlet boundary conditions. Continuous dependence of the solutions upon the initial data and supply terms is also established.

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 aim of this paper is to establish the well-posedness of the global solution of an initial value problem for the damped Boussinesq equation

In this paper we consider an optimal control problem for a quasilinear elliptic obstacle variational inequality in which the obstacle is taken as the control. The cost functional of this optimal control problem is of Lagrange type in which the p-th power of Laplacian of the control appears. This feature leads that it is hard to derive the optimality system for the underlying problem. In this paper, the existence and optimality system for the optimal control are established.

This paper is concerned with the mathematical modeling of second harmonic generation of nonlinear optics in periodic structures. The governing equation is a system of nonlinear Maxwell's equations. The well-posedness of the model is studied. Preliminary results on L_p regularity of solutions for the model problems are presented. The results are expected to be useful in our current effort of solving the underlying optimal design problem. Some future directions are also highlighted.

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.

The existence is one of the ultimate concepts in the studies of western philosophy. Almost all well-known philosophers in history had their own understandings of this concept. As a new philosophical form in new era, the philosophy of information states that philosophy of information and information science should develop together. The existence in philosophy needs to be redefined. The segmentation of existential field should be the first step to tackle with the issue of existence and non-existence. In the view of the philosophy of information, there are four levels of the existence: A. objective direct existence, B. objective indirect existence, C. subjective for-itself existence and D. subjective regenerated existence. The existential levels of a certain thing cover four levels in part or in its totality. Consequently, the nonexistence, as the opposite concept of the existence, has also changed. The existence and non-existence can be transformed into each other, with five properties in the process, those are: continuity, developmental feature, contingency, retrospective feature and predictability.

In this paper we consider the problem: ∂_tu − Δu = f (u), u(0) = u₀ ∈ exp L^p (ℝ^N), where p > 1 and f : ℝ → ℝ having an exponential growth at infinity with f (0) = 0. We prove local well-posedness in $\exp L_0^p\left({ℝ}^N\right)$ for $f\left(u\right)\sim e^{{\left|u\right|}^q},0<q\le p,\left|u\right|\to \infty$. However, if for some λ > 0, $\underset{s\to \infty }{\lim }\inf \left(f\left(s\right)e^{-\lambda s^p}\right)>0$ then non-existence occurs in exp L^p (ℝ^N ). Under smallness condition on the initial data and for exponential nonlinearity f such that |f(u)| ∼ |u|^m as u → 0, $\frac{N\left(m-1\right)}{2}\ge p$, we show that the solution is global. In particular, p – 1 > 0 sufficiently small is allowed. Moreover, we obtain decay estimates in Lebesgue spaces for large time which depend on m.

In this paper, we shall be concerned with the existence result of the quasilinear elliptic equations of the form,

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.

In this paper we investigate linear systems in quaternions. We give conditions under which they have solutions.

This contribution is devoted to a review of some recent results on existence, symmetry and symmetry breaking of optimal functions for Caffarelli-Kohn-Nirenberg (CKN) and weighted logarithmic Hardy (WLH) inequalities. These results have been obtained in a series of papers [1–5] in collaboration with M. del Pino, S. Filippas, M. Loss, G. Tarantello and A. Tertikas and are presented from a new viewpoint.