Comparison theorems for the oscillation of higher order difference equations with deviating arguments

The difference equation Δy + δp(k)f (y (g (k))) = 0, where p(k) is positive, is classified into four cases according to is odd or even and δ is 1 or −1. In each case, we shall offer comparison theorems for the oscillation of the difference equation. Examples are also included to illustrate the importance of the results obtained.     

On the oscillation of functional differential equations

In this paper we present certain criteria for the oscillation of functional differential equations of the form where δ = ±1, p, g: [t₀, ∞) → IR, H: [t₀,∞) × IR → IR are continuous, p(t) ≥ 0 for t ≥ t₀ and lim_t → ∞ g(t) — ∞. We like to point out that condition of the form will not be employed.     

Global asymptotic stability and oscillation of a family of difference equations

We consider the family of difference equations of the form

     

On oscillation of nonlinear second-order differential equations with damping term

The paper is concerned with oscillation of a novel class of nonlinear differential equations with a damping term. First it is demonstrated how known oscillation results for another intensively studied class of equations can be translated to the one in question, and vice versa. Advantages and drawbacks of such translation are carefully examined. Then an oscillation criterion for the new class of equations is established. The principal result of the paper is compared to those reported in the literature, and an illustrative example to which known oscillation criteria fail to apply is provided.     

Two non-probabilistic set-theoretical models to predict the transient vibrations of cross-ply plates with uncertainty

In practical engineering and scientific researches, all engineering analysis and design problems involve uncertainties to various degrees. Dynamic loads acting on a structure are usually with uncertain nature due to the difficulty of predicting the magnitudes of such loads. In this paper, a non-probabilistic and set-theoretical model named interval analysis method is developed to predict the transient vibrations of cross-ply plates with uncertain excitations. The dynamic loads involve deterministic and uncertain components of force function and initial conditions. Uncertainties in these functions are required to be bounded on the L₂ norm and expressed by finite eigenmodes. Analyzed by a numerical example, the width of the upper and lower bounds of the critical buckling loads that calculated by the interval analysis method is sharper than those are obtained by convex models. Moreover, the interval analysis has less computational cost than convex models. Considering specific cases, the effect of various parameters and the level of uncertainty on the response of the cross-ply plates are different.     

An extension of Sharkovsky's theorem to periodic difference equations

We present an extension of Sharkovsky's theorem and its converse to periodic difference equations. In addition, we provide a simple method for constructing a p-periodic difference equation having an r-periodic geometric cycle with or without stability properties.     

Solutions to a family of matrix equations by using the Kronecker matrix polynomials

Closed form solutions to a family of generalized Sylvester matrix equation in form of are given by using the so-called Kronecker matrix polynomials. It is found that the structure of the solutions is independent of the orders ?,ψ and φ. This type of uniform closed form solutions includes our early results as special cases. The results provide great convenience to the computation and analysis of the solutions to this class of equations, and can perform important functions in many analysis and design problems in linear systems.     

拓扑定理及其在超线性脉冲方程中的应用

本文研究一类含脉冲的非保守超线性方程的周期振动问题.通过详细刻画跳跃映射的角度函数,并应用相平面分析方法,本文证明方程的Poincaré映射有无穷多个不动点,而这些不动点对应了方程的无穷多个2π周期解.由于本文考虑的Poincaré映射不是同胚,所以,本文重新构造并证明了一个拓扑定理用于替代经典的Poincaré-Birkhoff扭转不动点定理.     

On the behavior of the solutions to periodic linear delay differential and difference equations

Some new results on the behavior of the solutions to periodic linear delay differential equations as well as to periodic linear delay difference equations are given. These results are obtained by the use of two distinct roots of the corresponding (so called) characteristic equation.     

Existence and asymptotic behaviour of solutions to first- and second-order difference equations with periodic forcing

By using previous results of Djafari Rouhani for non-expansive sequences in Refs (Djafari Rouhani, Ergodic theorems for nonexpansive sequences in Hilbert spaces and related problems, Ph.D. Thesis, Yale University, Part I (1981), pp. 1–76; Djafari Rouhani, J. Math. Anal. Appl. 147 (1990), pp. 465–476; Djafari Rouhani, J. Math. Anal. Appl. 151 (1990), pp. 226–235), we study the existence and asymptotic behaviour of solutions to first-order as well as second-order difference equations of monotone type with periodic forcing. In the first-order case, our result extends to general maximal monotone operators, the discrete analogue of a result of Baillon and Haraux (Rat. Mech. Anal. 67 (1977), 101–109) proved for subdifferential operators. In the second-order case, our results extend among other things, previous results of Apreutesei (J. Math. Anal. Appl. 288 (2003), 833–851) to the non-homogeneous case, and show the asymptotic convergence of every bounded solution to a periodic solution.     

A continuation theorem with applications to periodically forced Liénard equations in the presence of a separatrix

In this paper we prove a result on the existence of periodic motions for the periodically forced Liénard differential equation x ^″+f(x)x ^′+g(x)=e(t) in a situation where the phase portrait of the associated autonomous equation is similar to that of a centre limited by an unbounded separatrix. The existence result, which is based on a degree theoretic continuation theorem, enables us to treat some interesting cases not previously considered in the literature. Received: April 27, 2000 Published online: December 19, 2001     

A modified extended method to find a series of exact solutions for a system of complex coupled KdV equations

In this paper an algebraic method is devised to uniformly construct a series of complete new exact solutions for general nonlinear equations. For illustration, we apply the modified proposed method to revisit a complex coupled KdV system and successfully construct a series of new exact solutions including the soliton solutions and elliptic doubly periodic solutions.