Delay effect in models of population growth

First, we systematize earlier results on the global stability of the model of population growth. Second, we investigate the effect of delay on the asymptotic behavior when the nonlinearity f is a unimodal function. Our results can be applied to several population models [Elements of Mathematical Ecology, 2001 [7]; Appl. Anal. 43 (1992) 109-124; Math. Comput. Modelling, in press; Funkt. Biol. Med. 256 (1982) 156-164; Math. Comput. Modelling 35 (2002) 719-731; Mat. Stos. 6 (1976) 25-40] because the function f does not need to be monotone or differentiable. Specifically, our results generalize earlier result of [Delay Differential Equations with Applications in Population Dynamics, 1993], since our function f may not be differentiable.     

Successive iteration and positive symmetric solution for a Sturm–Liouville-like four-point boundary value problem with a -Laplacian operator

In this paper, we consider the following Sturm–Liouville-like four-point p-Laplacian boundary value problem with the nonlinear term f depending on the first-order derivative subject to the boundary conditions By using a monotone iterative technique, the existence of symmetric positive solutions and corresponding iterative schemes are obtained.     

Persistence and global stability in discrete models of Lotka-Volterra type

In this paper, we establish new sufficient conditions for global asymptotic stability of the positive equilibrium in the following discrete models of Lotka-Volterra type:

     

Stability switches in a system of linear differential equations with diagonal delay

This paper deals with the stability problem of a delay differential system of the form x^′(t)=-ax(t-τ)-by(t), y^′(t)=-cx(t)-ay(t-τ), where a, b, and c are real numbers and τ is a positive number. We establish some necessary and sufficient conditions for the zero solution of the system to be asymptotically stable. In particular, as τ increases monotonously from 0, the zero solution of the system switches finite times from stability to instability to stability if ; and from instability to stability to instability if . As an application, we investigate the local asymptotic stability of a positive equilibrium of delayed Lotka-Volterra systems.     

A general iterative method for addressing mixed equilibrium problems and optimization problems

In this paper, we introduce a new general iterative method for finding a common element of the set of solutions of a mixed equilibrium problem (MEP), the set of fixed points of an infinite family of nonexpansive mappings and the set of solutions of variational inequalities for a ξ-inverse-strongly monotone mapping in Hilbert spaces. Furthermore, we establish the strong convergence theorem for the iterative sequence generated by the proposed iterative algorithm under some suitable conditions, which solves some optimization problems. Our results extend and improve the recent results of Yao et al. [Y. Yao, M.A. Noor, S. Zainab, Y.C. Liou, Mixed equilibrium problems and optimization problems, J. Math. Anal. Appl. 354 (2009) 319-329; Y. Yao, M. A. Noor, Y.C. Liou, On iterative methods for equilibrium problems, Nonlinear Anal. 70 (1) (2009) 479-509] and many others.     

A new hybrid iterative algorithm for variational inequalities

Let H be a real Hilbert space. Let F:H→H be a strongly monotone and Lipschitzian mapping. Let be an infinite family of non-expansive mappings with common fixed points set . We devise an iterative algorithm

     

Multifractal spectrum and generic properties of functions monotone in several variables

We study the singularity (multifractal) spectrum of continuous functions monotone in several variables. We find an upper bound valid for all functions of this type, and we prove that this upper bound is reached for generic functions monotone in several variables. Let be the set of points at which f has a pointwise exponent equal to h. For generic monotone functions f:d[0,1]→R, we have that for all h∈[0,1], and in addition, we obtain that the set is empty as soon as h>1. We also investigate the level set structure of such functions.     

Invariant manifolds around equilibria of Newtonian equations: Some pathological examples

Let the equation be periodic in time, and let the equilibrium x_∗≡0 be a periodic minimizer. If it is hyperbolic, then the set of asymptotic solutions is a smooth curve in the plane ; this is stated by the Stable Manifold Theorem. The result can be extended to nonhyperbolic minimizers provided only that they are isolated and the equation is analytic (Ureña, 2007 [6]). In this paper we provide an example showing that one cannot say the same for C² equations. Our example is pathological both in a global sense (the global stable manifold is not arcwise connected), and in a local sense (the local stable manifolds are not locally connected and have points which are not accessible from the exterior).     

The asymptotic lift of a completely positive map

Starting with a unit-preserving normal completely positive map acting on a von Neumann algebra—or more generally a dual operator system—we show that there is a unique reversible system (i.e., a complete order automorphism α of a dual operator system N) that captures all of the asymptotic behavior of L, called the asymptotic lift of L. This provides a noncommutative generalization of the Frobenius theorems that describe the asymptotic behavior of the sequence of powers of a stochastic n×n matrix. In cases where M is a von Neumann algebra, the asymptotic lift is shown to be a W^∗-dynamical system (N,Z), and we identify (N,Z) as the tail flow of the minimal dilation of L. We are also able to identify the Poisson boundary of L as the fixed algebra Nα. In general, we show the action of the asymptotic lift is trivial iff L is slowly oscillating in the sense that

     

Stability of equilibrium of conservative systems with two degrees of freedom

This article intends to study the Liapounof's stability of an equilibrium of conservative Lagrangian systems with two degrees of freedom.We consider an open neighborhood of the origin and the Lagrangian , where of class is the potential energy with a critical point at the origin and is the kinetic energy, of class .We assume that π has a jet of order k at the origin, and this jet shows that the potential energy does not have a minimum in 0. With these hypotheses we prove that (0;0) is an unstable equilibrium according to Liapounof for the Lagrange equations of . We achieve this by proving that there is an asymptotic trajectory to the origin.     

Oscillation and global asymptotic stability in a discrete epidemic model

In this paper, we study the oscillation, global asymptotic stability, and other properties of the positive solutions of the difference equation

     

The existence and uniqueness of the local solution for a Camassa-Holm type equation

A shallow water equation of Camassa-Holm type, containing nonlinear dissipative effect, is investigated. Using the techniques of the pseudoparabolic regularization and some prior estimates derived from the equation itself, we establish the existence and uniqueness of its local solution in Sobolev space H^s(R) with . Meanwhile, a new lemma and a sufficient condition which guarantee the existence of solutions of the equation in lower order Sobolev space H^s with are presented.     

The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations

In this paper, first, we introduce a new concept, called the norm-to-weak continuous semigroup in a Banach space, and give a technical theorem to verify this notion of continuity. Then we establish a general method which is necessary and sufficient to obtain the existence of the global attractor for this kind of semigroup. As an application, we obtain the existence of the global attractor for a nonlinear reaction-diffusion equation with a polynomial growth nonlinearity of arbitrary order and with some weak derivatives in the inhomogeneous term, the global attractors are obtained in L^p(Ω), and , respectively. A new a priori estimate method, called asymptotic a priori estimate, has been introduced. Since the solutions of the equation has no higher regularity and the semigroup associated the solutions is not continuous in L^p(Ω), and , the results in this part are new and appear to be optimal.