POSITIVE HOMOCLINIC ORBITS FOR A CLASS OF ASYMPTOTICALLY PERIODIC SECOND ORDER DIFFERENTIAL EQUATIONS

This note studies the existence of positive homoclinic orbits of the second order equation-u″+α(x)u=β(x)u q+γ(x)u p, x∈R,where 1<q<p.Assume that the coefficient functions α(x),β(x) and γ(x) are asymptotically periodic and satisfy0<a≤α(x), 0<γ(x)≤B, -M≤β(x)≤M.A positive homoclinic orbit of the equation is obtained by means of variational methods.

Non-Nehari manifold method for asymptotically periodic Schrodinger equations

We consider the semilinear Schrdinger equation-△u + V(x)u = f(x, u), x ∈ RN,u ∈ H 1(RN),where f is a superlinear, subcritical nonlinearity. We mainly study the case where V(x) = V0(x) + V1(x),V0∈ C(RN), V0(x) is 1-periodic in each of x1, x2,..., x N and sup[σ(-△ + V0) ∩(-∞, 0)] < 0 < inf[σ(-△ +V0)∩(0, ∞)], V1∈ C(RN) and lim|x|→∞V1(x) = 0. Inspired by previous work of Li et al.(2006), Pankov(2005)and Szulkin and Weth(2009), we develop a more direct approach to generalize the main result of Szulkin and Weth(2009) by removing the "strictly increasing" condition in the Nehari type assumption on f(x, t)/|t|. Unlike the Nahari manifold method, the main idea of our approach lies on finding a minimizing Cerami sequence for the energy functional outside the Nehari-Pankov manifold N0 by using the diagonal method.

Existence of asymptotically almost periodic solutions and stability properties for functional difference equations

For functional difference equations with unbounded delay,we characterized the existence of totally stable and asymptotically almost periodic solution by using stability properties of a bounded solution in a certain limiting equation.

THE ASYMPTOTIC PERIODICITY OF LORENZ MAPS

A Lorenz map f : I --> I is a one dimensional piecewise monotone map with a single discontinuity c. Let [GRAPHICS] be the collection of all the preimsges of c. Authors prove that if C'(f) is countable then there exists M such that Card(omega(x)) less than or equal to M for all x is an element of I. If C'(f) is uncountable then omega(x) is uncountable for some x is an element of I. So f is asymptotically periodic if and only if C'(f) is countable.

THE ASYMPTOTIC PERIODICITY OF PIECEWISE MONOTONE SELFMAPS OF THE INTERVAL

Let f:I→I be a piecewise monotone interval map. The critical point set C(f) of f consists of all the preimages of its turning points. It is proved that the complex dynamical behaviors of f are all concentrated on the derived set of C(f). f is asymptotically periodic if and only if the derived set of C(f) is countable.

ASYMPTOTICALLY MEAN ALMOST PERIODIC SOLUTIONS TO RANDOM DIFFERENCE EQUATIONS

In this paper,we first investigate some basic properties of asymptotically mean almost periodic random sequences on Z + and then show some properties of asymptotically mean almost periodic solutions to random difference equations.

Impulse Spatial-Temporal Domains in Semiconductor Laser with Feedback

An initial value boundary problem for system of diffusion equations with delay arguments and dynamic nonlinear boundary conditions is considered. The problem describes evolution of the carrier density and the radiation density in the semiconductor laser or laser diodes with “memory” and with feedback. It is shown that the boundary problem can be reduced to a system of difference equations with continuous time. For large times, solutions of these equations tend to piecewise constant asymptotic periodic wave functions which represent chain of shock waves with finite or infinite points of discontinuities on a period. Applications to the optical systems with linear media and nonlinear surface optical properties with feedback have been done. The results are compared with the experiment.

ASYMPTOTICALLY ALMOST PERIODIC FUNCTIONS IN PROBABILITY

In this paper, we first study the properties of asymptotically almost periodic functions in probability and then prove the existence of almost periodic solutions in probability to some differential equations with random terms.

ON THE PERSISTENCE AND POSITIVE PERIODIC SOLUTION FOR PLANAR COMPETING LOTKA-VOLTERRA SYSTEMS

We study the qualitative property of solutions of planar periodic competing otka-Volterra systems New criteria of uniform persistence of solutins and existence, uniqueness and globally asymptotical stability of positive periodic solution are established. The results results in [1 -6] are summarized and improved in this paper.

LINEARIZED OSCILLATIONS FOR ODD-ORDERNEUTRAL DELAY DIFFERENTIAL EQUATIONSWITH ASYMPTOTIC PERIODIC COEFFICIENTS

In this paper, we prove that, under appropriate hypotheses, the odd-order nonlinear neutral delay differential equation has the same oscillatory character as the associated linear equation with periodic coefficients Our method is based on the establishment of a new comparison theorem for the oscillation of Eq.(E). Hence we prove that the parameter set such that Eq.(E) has a nonoscillatory solution is closed in certain metric space, and avoid the difficulty to set the necessary and sufficient condition for the oscillation of Eq.(E).

ALMOST PERIODIC SOLUTIONS OF SINGLE POPULATION MODELS WITH ALMOST PERIODIC ENVIRONMENT

In this paper we investigate the existence and stability of almost periodic solutions of some single population models with almost periodic environment. We obtain some results about the existence and stability of the positive almost periodic solutions.

EXISTENCE UNIQUENESS AND ASYMPTOTIC STABILITY OF PERIODIC SOLUTIONS OF A NONLINEAR EQUATION IN PHASE LOCKED TECHNOLOGY

In [1] and [2], the authors made a deep qualitative analysis of the equationwith the character of tangent detected phase and they mathematically provided atheoretical basis of why the phase looked loop has no look--losing point. However,according to many practical experts, it is rather difficult to put such a phaselooked loop into practice, though it has fine properties. W. C. Lindsey [3] made a

Composite quantile regression estimation for P-GARCH processes

We consider the periodic generalized autoregressive conditional heteroskedasticity(P-GARCH) process and propose a robust estimator by composite quantile regression. We study some useful properties about the P-GARCH model. Under some mild conditions, we establish the asymptotic results of proposed estimator.The Monte Carlo simulation is presented to assess the performance of proposed estimator. Numerical study results show that our proposed estimation outperforms other existing methods for heavy tailed distributions.The proposed methodology is also illustrated by Va R on stock price data.