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.

S-asymptotically ω-periodic Solutions of R-L Fractional Derivative-Integral Equation

The aim of this paper is to study the S-asymptotically ω-periodic solutions of R-L fractional derivative-integral equation:v′(t)=∫t0(t-s)α-2/Γ(α-1)Av(s)ds+∫+∞-∞e-|τ|f(u(t-τ))dτ,(1)v(0)=u0∈X,(2)where 1 <α <2, A:D(A)X→X is a linear densely defined operator of sectorial type on a completed Banach space X, f is a continuous function satisfying a suitable Lipschitz type condition. We will use the contraction mapping theory to prove problem(1) and(2) has a unique S-asymptoticallyω-periodic solution if the function f satisfies Lipshcitz condition.

Asymptotic periodic solutions of some generalized Burgers equations

In this paper, we construct asymptotic periodic solutions of some generalized Burgers equations using a perturbative approach. These large time asymptotics(constructed) are compared with relevant numerical solutions obtained by a finite difference scheme.

Binary Bell Polynomials,Bilinear Approach to Exact Periodic Wave Solutions of(2+l)-Dimensional Nonlinear Evolution Equations

In the present letter,we get the appropriate bilinear forms of(2 + l)-dimensionaI KdV equation,extended (2+1)-dimensional shallow water wave equation and(2 +1)-dimensional Sawada-Kotera equation in a quick and natural manner,namely by appling the binary Bell polynomials.Then the Hirota direct method and Riemann theta function are combined to construct the periodic wave solutions of the three types nonlinear evolution equations.And the corresponding figures of the periodic wave solutions are given.Furthermore,the asymptotic properties of the periodic wave solutions indicate that the soliton solutions can be derived from the periodic wave solutions.

Existence and Stability Property of Almost Periodic Solutions in Discrete Almost Periodic Systems

In this paper, we consider an almost periodic system which includes a system of the type , where k is a positive integer, aij are almost periodic in n and satisfy aij(n)≥0 for i≠j,? for 1≤j≤m. In the special case where aij(n) are constant functions, above system is a mathematical model of gas dynamics and was treated by T. Carleman and R. D. Jenks for differential systems. In the main theorem, we show that if the m X m matrix (aij(n)) is irreducible, then there exists a positive almost periodic solution which is unique and has some stability. Moreover, we can see that this result gives R. D. Jenks’ result for differential model in the case where aij(n) are constant functions. In Section 3, we consider the linear system with variable cofficients . Even in nonlinear problems, this linear system plays an important role, as their variational equations, and it is requested to determine the uniform asymptotically stability of the zero solution from the information about A(n). In order to obtain the existence of almost periodic solutions of both linear and nonlinear almost periodic discrete systems: above linear system and? for 1≤i≤m, respectively, we shall consider between certain stability properties, which are referred to as uniformly asymptotically stable, and the diagonal dominance matrix condition.

The Problem of Periodic Solutions for a Class of Dynamical Systems (Ⅰ)

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The Problem of Periodic Solutions fora Class of Dynamical System(Ⅱ)

In this paper, we consider the dynamical systems which are from a kind of Hamilton systems under a disturbance. We use theories in Liapunov stability,and show that there are not any periodic solutions in some a neibourhood of the equilibrium points of the dynamical systems.

Periodic Solitary Wave Solutions of the (2 + 1)-Dimensional Variable-Coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada Equation

In this paper, through symbolic computations, we obtain two exact solitary wave solitons of the (2 + l)-dimensional variable-coefficient Caudrey-Dodd- Gibbon-Kotera-Sawada equation. We study basic properties of l-periodic solitary wave solution and interactional properties of 2-periodic solitary wave solution by using asymptotic analysis.

An asymptotic solving method for the periodic solution of a class of disturbed nonlinear evolution equation

A class of disturbed evolution equation is considered using a simple and valid technique. We first introduce the periodic traveling-wave solution of a corresponding typical evolution equation. Then the approximate solution for an original disturbed evolution equation is obtained using the asymptotic method. We point out that the series of approximate solution is convergent and the accuracy of the asymptotic solution is studied using the fixed point theorem for the functional analysis.

POSITIVE PERIODIC SOLUTION FOR A NONAUTONOMOUS LOGISTIC MODEL WITH LINEAR FEEDBACK REGULATION

A nonautonomous delayed logistic model with linear feedback regulation is proposed in this paper. Sufficient conditions are derived for the existence, uniqueness and global asymptotic stability of positive periodic solution of the model

PERTURBED PERIODIC SOLUTIONFOR BOUSSINESQ EQUATION

We consider the solution of the good Boussinesq equation Utt -Uxx ＋ Uxxxx = （U2）xx, -∞ 〈 x 〈 ∞, t ≥ 0, with periodic initial value U（x, 0） = ε（μ ＋ φ（x））, Ut（x, 0） = εψ（x）, -∞ 〈 x 〈 ∞, where μ = 0, φ（x） and ψ（x） are 2π-periodic functions with 0-average value in [0, 2π], and ε is small. A two parameter Bcklund transformation is found and provide infinite conservation laws for the good Boussinesq equation. The periodic solution is then shown to be uniformly bounded for all small ε, and the H1-norm is uniformly bounded and thus guarantees the global existence. In the case when the initial data is in the simplest form φ（x） = μ＋a sin kx, ψ（x） = b cos kx, an approximation to the solution containing two terms is constructed via the method of multiple scales. By using the energy method, we show that for any given number T 〉 0, the difference between the true solution u（x, t; ε） and the N-th partial sum of the asymptotic series is bounded by εN＋1 multiplied by a constant depending on T and N, for all -∞ 〈 x 〈 ∞, 0 ≤ ｜ε｜t ≤ T and 0 ≤ ｜ε｜≤ε0.

EXISTENCE AND UNIQUENESS OF ALMOST PERIODIC SOLUTION TO A CLASS OF NONAUTONOMOUS SYSTEM

Almost periodic oscillations appearing in high-tension electricity network are considered in this paper. By utilization of Liapunov function, the foreboding conditions that result in almost periodic oscillations are obtained and thus the possibility of making precautions is presented.

RESEARCH OF THE PERIODIC SOLUTION FOR A CLASS OF NONLINEAR DIFFERENTIAL EQUATIONS

In this paper,we study the existence,uniqueness and asymptotic stabgility of the periodic solution for a class of the most,universal fourth-order nonlinear nonautonomous periodic systems.We give the relevant Liapunov function by using the method of analogical slowly changing coefficients.We also give a considerably accurate estimation of the slowly changing coefficients and obtain the sufficient conditions which guarantee the existence,uniqueness and asymptotic Stability of the periodci solutions.

ASYMPTOTIC BEHAVIOR OF SOLUTIONS FOR THE CHAFEE-INFANTE EQUATION

In higher dimension,there are many interesting and challenging problems about the dynamics of non-autonomous Chafee-Infante equation.This article is concerned with the asymptotic behavior of solutions for the non-autonomous Chafee-Infante equation∂u∂t−Δu=λ(t)(u−u3)in higher dimension,whereλ(t)∈C1[0,T]andλ(t)is a positive,periodic function.We denoteλ1 as the first eigenvalue of−Δφ=λφ,x∈Ω;φ=0,x∈∂Ω.For any spatial dimension N≥1,we prove that ifλ(t)≤λ1,then the nontrivial solutions converge to zero,namely,limt→+∞u(x,t)=0,x∈Ω;ifλ(t)>λ1 as t→+∞,then the positive solutions are"attracted"by positive periodic solutions.Specially,ifλ(t)is independent of t,then the positive solutions converge to positive solutions of−ΔU=λ(U−U^3).Furthermore,numerical simulations are presented to verify our results.

The Problem of Periodic Solution for Dynam icalSystem s (Ⅱ)

In this paper,we consider the dynamical system which are from general Hemilton systems under a disturbance,we use theories in Liapunov stability and show that there are not any periodic solutions in some a neighborhood of the equilibrium points of the dynamical systems.

Periodic solutions of a semilinear variable coefficient wave equation under asymptotic nonresonance conditions

We consider the periodic solutions of a semilinear variable coefficient wave equation arising from the forced vibrations of a nonhomogeneous string and the propagation of seismic waves in nonisotropic media.The variable coefficient characterizes the inhomogeneity of media and its presence usually leads to the destruction of the compactness of the inverse of the linear wave operator with periodic-Dirichlet boundary conditions on its range.In the pioneering work of Barbu and Pavel(1997),they gave the existence and regularity of the periodic solution for Lipschitz,nonresonant and monotone nonlinearity under the assumptionηu>0(see Section 2 for its definition)on the coefficient u(x)and left the caseηu=0 as an open problem.In this paper,by developing the invariant subspace method and using the complete reduction technique and Leray-Schauder theory,we obtain the existence of periodic solutions for such a problem when the nonlinear term satisfies the asymptotic nonresonance conditions.Our result needs neither requirements on the coefficient except the natural positivity assumption(i.e.,u(x)>0)nor the monotonicity assumption on the nonlinearity.In particular,when the nonlinear term is an odd function and satisfies the global nonresonance conditions,there is only one(trivial)solution to this problem in the invariant subspace.

Multiplicity of Periodic Solutions for Second Order Hamiltonian Systems with Asymptotically Quadratic Conditions

A class of second order non-autonomous Hamiltonian systems with asymptotically quadratic conditions is considered in this paper.Using Fountain Theorem,one multiplicity result of periodic solutions is obtained,which improves some previous results.

Pseudo asymptotically periodic solutions for fractional integro-differential neutral equations

In this paper, we study the existence and uniqueness of pseudo S-asymptotically ω-periodic mild solutions of class r for fractional integro-differential neutral equations. An example is presented to illustrate the application of the abstract results.

CONVERGENCE OF SOLUTIONS FOR RLC-NONLINEAR NETWORKS WITH TIME-VARYING ELEMENTS

This paper studies the asymptotic behavior of solutions for nonlinear RLCnetwoiks which have the following form The function p i,v,t),called tile mired potential function,call be used to construct Liapunov functions to prove the convergence of solutions under certain conditiolts. Under the assumption that every element value involving voltage source is asymptotically constallt, we establish four creteria for all solutiolls of such a system to converge to the set of equilibria of its limiting equations via LaSalle invariant principle.We also present two theorems on the existence of periodic solutions for periodically excited uonliltear circuits.This results generalize those of Brayton and Moser[1,2].

Weighted Pseudo Asymptotically Periodic Mild Solutions of Evolution Equations

In this paper, we propose a new class of functions called weighted pseudo S-asymptotically periodic function in the Stepanov sense and explore its properties in Banach space including composition theorems. Furthermore, the existence, uniqueness of the weighted pseudo S-asymptotically periodic mild solutions to partial evolution equations and nonautonomous semilinear evolution equations are investigated. Some interesting examples are presented to illustrate the main findings.