Constant-sign periodic and almost periodic solutions of a system of difference equations

We consider the following system of difference equations, ,where I is a subset of . Our aim is to establish criteria such that the above system has a constantsign periodic and almost periodic solution (u₁, u₂, …, u_n). The above problem is also extended to that on , .     

The periodic nature of the positive solutions of a nonlinear fuzzy max-difference equation

We investigate the periodic nature of the positive solutions of the fuzzy difference equation , where k, m are positive integers, A₀, A₁, are positive fuzzy numbers and the initial values x_i, i = −d, −d + 1, … , −1, d = max{k, m}, are positive fuzzy numbers. In addition, we give conditions so that the solutions of this equation are unbounded.     

Monotone properties of certain classes of solutions of second-order difference equations

The authors consider the difference equations

δ(a_nδx_n)=q_nx_n+1

and

δ(a_nδx_n)=q_nf(x_n+1),

where a_n > 0, q_n > 0, and f: R å R is continuous with uf(u) > 0 for u ≠ 0. They obtain necessary and sufficient conditions for the asymptotic behavior of certain types of nonoscillatory solutions of (*) and sufficient conditions for the asymptotic behavior of certain types of nonoscillatory solutions of (**). Sufficient conditions for the existence of these types of nonoscillatory solutions are also presented. Some examples illustrating the results and suggestions for further research are included.     

A further result on the oscillation of delay difference equations

In this paper, we are concerned with the delay difference equations of the form

(*)

y_n+1 − y_n + p_ny_n−k = 0, N = 0, 1, 2, …,

(*)where p_n ≥ 0 and k is a positive integer. We prove by using a new technique that

guarantees that all solutions of equation (*) oscillate, which improves many previous well-known results. In particular, our theorems also fit the case where Σⁿ⁻¹_i=n−kp_i ≤ k^k+1/(k + 1)^k+1. In addition, we present a nonoscillation sufficient condition for equation (*).     

闭环具有锯齿特性的PWM型DC-DC buck变换器周期解的存在性

应用周期方程方法研究具有锯齿波特性的闭环PWM型buck DC-DC变换器T周期解的存在性问题, 并给出了其在一个周期内仅有一次切换的T周期解存在的充分条件. 所给出的结果为闭环PWM型buck DC-DC变换器的控制器和锯齿波参数的设计提供了指导准则.     

Periodic solutions of a class of nonlinear difference equations via critical point method

In this paper, we obtain some new sufficient conditions for the existence of nontrivial m-periodic solutions of the following nonlinear difference equation

by using the critical point method, where f: Z × R → R is continuous in the second variable, m ≥ 2 is a given positive integer, p_n+m = p_n for any n  Z and f(t + m, z) = f(t, z) for any (t, z)  Z × R, (−1)^δ = −1 and δ > 0.     

New soliton-like solutions and compacton-like periodic wave solutions of parametric type to a nonlinear dispersive equation

In this paper, by using the integral bifurcation method, we study a nonlinear dispersive equation. Some new soliton-like solutions and some compacton-like periodic wave solutions are obtained. Their dynamic characters are investigated and the profiles are given by the mathematical software Maple. From the graphs of some soliton-like solutions, we find that their profiles are transformable.     

Asymptotic properties of solutions to discrete Coulomb equations

We construct two finite-difference models for the Coulomb differential equation which arises in the quantum mechanics analysis of the scattering of two charged point particles. These difference equations correspond to the standard and Mickens-Ramadhani schemes for the Coulomb equation. Our major goal is to determine the first two terms in the asymptotic solutions and compare them to the corresponding solutions of the Coulomb differential equation. In particular, the form of the anomalous phase term is examined.     

Subanalytic solutions of linear difference equations and multidimensional hypergeometric sequences

We consider linear difference equations with polynomial coefficients over C and their solutions in the form of sequences indexed by the integers (sequential solutions). We investigate the C-linear space of subanalytic solutions, i.e., those sequential solutions that are the restrictions to Z of some analytic solutions of the original equation. It is shown that this space coincides with the space of the restrictions to Z of entire solutions and that the dimension of this space is equal to the order of the original equation.We also consider d-dimensional (d≥1) hypergeometric sequences, i.e., sequential and subanalytic solutions of consistent systems of first-order difference equations for a single unknown function. We show that the dimension of the space of subanalytic solutions is always at most 1, and that this dimension may be equal to 0 for some systems (although the dimension of the space of all sequential solutions is always positive).Subanalytic solutions have applications in computer algebra. We show that some implementations of certain well-known summation algorithms in existing computer algebra systems work correctly when the input sequence is a subanalytic solution of an equation or a system, but can give incorrect results for some sequential solutions.     

Duality and eventually periodic systems

This paper employs semidefinite programming duality theory to develop new alternative linear matrix inequality (LMI) tools for eventually periodic systems. These tools are then utilized to rederive an important version of the Kalman–Yakubovich–Popov (KYP) Lemma for such systems, and further give new synthesis results. Copyright © 2005 John Wiley & Sons, Ltd.     

非对称NNV方程新的广义孤波解和周期解

指数函数法是求解非线性发展方程的一种简单有效的方法,利用该方法并借助Mathematica软件的符号计算功能求解了非对称NNV方程,得到了新的孤波解和周期解。     

Oscillation in nonlinear difference equations

In this paper, we shall discuss oscillatory behavior of the solutions of difference equations, including the self-adjoint second-order linear equation and the discrete version of the nonlinear wave equation. Our work is to give sufficient conditions such that every nontrivial solution of the equations oscillates.     

Some new solitonary solutions of the modified Benjamin-Bona-Mahony equation

In this paper, we use the exp-function method to construct some new soliton solutions of the Benjamin-Bona-Mahony and modified Benjamin-Bona-Mahony equations. These equations have important and fundamental applications in mathematical physics and engineering sciences. The exp-function method is used to find the soliton solution of a wide class of nonlinear evolution equations with symbolic computation. This method provides the concise and straightforward solution in a very easier way. The results obtained in this paper can be viewed as a refinement and improvement of the previously known results.     

Differential periodic Riccati equations: Existence and uniqueness of nonnegative definite solutions

In this paper we consider the differential periodic Riccati equation. All the periodic nonnegative definite solutions are characterized in the more general case, providing a method for constructing them. The method is obtained from the study of the invariant subspaces of the monodromy matrix of the associated Hamiltonian system, and from the relations between these invariant subspaces and the controllability and unobservability subspaces. Finally, the method is applied to obtain necessary and sufficient conditions for the existence of any periodic nonnegative definite solution and to study the existence and uniqueness of minimal, maximal, stabilizing, and strong solutions.This work has been partially supported by Spanish DGICYT Grant No. PB91-O535.     

Cyclomonotonicity and stabilizability properties of solutions ofthe difference periodic Riccati equation

The Kalman filter associated with a discrete-time linear T-periodic system is tested. The problem considered is that of selecting an initial covariance matrix such that the periodic filter based on the first T values of the Kalman filter gain is stabilizing. Sufficient conditions are given that hinge on the cyclomonotonicity of the solution of the periodic Riccati equation. Potential applications are found in filter design, quasi-linearization techniques for the periodic Riccati equation, and the design of receding-horizon control strategies for periodic and multirate systems. When specialized to time-invariant systems, the results give rise to new sufficient conditions for the cyclomonotonicity of the solutions of the time-invariant Riccati equation and the existence of periodic stabilizing feedback     

Positive periodic solutions of second-order nonlinear differential systems with two parameters

By employing the Deimling fixed point index theory, we consider a class of second-order nonlinear differential systems with two parameters . We show that there exist three nonempty subsets of : Γ, Δ₁ and Δ₂ such that and the system has at least two positive periodic solutions for (λ,μ)Δ₁, one positive periodic solution for (λ,μ)Γ and no positive periodic solutions for (λ,μ)Δ₂. Meanwhile, we find two straight lines L₁ and L₂ such that Γ lies between L₁ and L₂.     

Asymptotic and oscillatory behavior of solutions of general nonlinear difference equations of second order

The asymptotic and oscillatory behavior of solutions of some general second-order nonlinear difference equations of the form

δ(a_nh(y_n+1)δy_n)+p_nδy_n+q_n+1f(y_σ(n+1))=0 nZ,

is studied. Oscillation criteria for their solutions, when “p_n” is of constant sign, are established. Results are also presented for the damped-forced equation

δ(a_nh(y_n+1)δy_n)+p_nδy_n+q_n+1f(y_σ(n+1))=e_n nZ.

Examples are inserted in the text for illustrative purposes.     

Variable-coefficient auxiliary equation method for exact solutions of non-linear evolution equations

In this paper, a variable-coefficient auxiliary equation method is proposed to seek more general exact solutions of non-linear evolution equations. Being concise and straightforward, this method is applied to the Kawahara equation, Sawada–Kotera equation and (2+1)-dimensional Korteweg–de Vries equations. As a result, many new and more general exact solutions are obtained including Jacobi elliptic, hyperbolic and trigonometric function solutions. It is shown that the proposed method provides a straightforward and effective method for non-linear evolution equations in mathematical physics.     

Existence of periodic solutions to automatic control system with relay nonlinearity and sinusoidal external influence

We consider an automatic control system with relay nonlinearity and sinusoidal external influence. Sufficient conditions for the existence of periodic solutions in the case when the characteristic equation of the system has complex roots are obtained. We use an approach that permits to investigate the system analytically and prove the existence of at least one periodic solution (including asymptotically stable solution) with the period multiple to the period of external influence and two switching points on the phase plane. Copyright © 2016 John Wiley & Sons, Ltd.