Oscillation of certain partial difference equations

In this paper we consider a class of nonlinear delay partial difference equations and a class of linear delay partial difference equations with variable coefficients, which may change sign. We obtain oscillation criteria for these equations. There are no results for the oscillation of these equations up to now.     

Asymptotic stability criteria for linear systems of difference-differential equations of neutral type and their discrete analogues

Electrical networks containing lossless transmission lines are often modeled by difference-differential equations of neutral type. This paper finds sufficient conditions for asymptotic stability for linear systems of these equations. Also given is a modification of the direct method of Liapunov for difference equations. This method is applied to finding asymptotic stability criteria for the discrete analogs of the linear system of difference-differential equations.     

OSCILLATION CRITERIA FOR A FOURTH ORDER SUBLINEAR DYNAMIC EQUATION ON TIME SCALE

Some new criteria for the oscillation of a fourth order sublinear and/or linear dynamic equation on time scale are established. Our results are new for the corresponding fourth order differential equations as well as difference equations.     

On the mean square stability of linear difference equations

The mean square asymptotic stability of linear stochastic difference equations is studied. The Liapunov method of stability analysis is extended to stochastic difference equations, and several criteria for the mean square stability of the equilibrium state are established. Two examples of the application of the stability theorems are also considered.     

COMPARISON AND OSCILLATION THEOREMS FOR AN ADVANCED TYPE DIFFERENCE EQUATION

ＣＯＭＰＡＲＩＳＯＮ　ＡＮＤ　ＯＳＣＩＬＬＡＴＩＯＮ　ＴＨＥＯＲＥＭＳ　ＦＯＲ　ＡＮ　ＡＤＶＡＮＣＥＤ　ＴＹＰＥ　ＤＩＦＦＥＲＥＮＣＥ　ＥＱＵＡＴＩＯＮＣＯＭＰＡＲＩＳＯＮＡＮＤＯＳＣＩＬＬＡＴＩＯＮＴＨＥＯＲＥＭＳＦＯＲＡＮＡＤＶＡＮＣＥＤＴＹＰＥ...     

泛函微分方程安全全局渐近稳定性的一类判别准则

本利用分离变量型V函数，建立了泛函微分方程安全全局渐近稳定性的一类Razumikhin型定理，并对一类变时滞线性微分差分方程给出简明的安全全局渐近稳定性判别准则。     

Stability Criteria for Solutions of Systems of Linear Deterministic or Stochastic Delay Difference Equations with Continuous Time

We give spectral and algebraic coefficient criteria (necessary and sufficient conditions) as well as sufficient algebraic coefficient conditions for the Lyapunov asymptotic stability of solutions to systems of linear deterministic or stochastic delay difference equations with continuous time under white noise coefficient perturbations for the case in which all delay ratios are rational. For stochastic systems, mean-square asymptotic stability is studied. The Lyapunov function method is used. Our criteria on algebraic coefficients and our sufficient conditions are stated in terms of matrix Lyapunov equations (for deterministic systems) and matrix Sylvester equations (for stochastic systems).     

Relationship between spectral and coefficient criteria of mean-square stability for systems of linear stochastic differential and difference equations

We establish the relationship (equivalence) between the spectral and algebraic (coefficient) criteria (the latter is represented in terms of the Sylvester matrix algebraic equation) of mean-square asymptotic stability for three classes of systems of linear equations with varying random perturbations of coefficients, namely, the ltô differential stochastic equations, difference stochastic equations with discrete time, and difference stochastic equations with continuous time.     

On Bounded Solutions of Some Classes of Two-Parameter Difference Equations in a Banach Space

We obtain criteria for the existence of bounded solutions of some classes of linear two-parameter difference equations with operator coefficients in a Banach space.     

OSCILLATION OF CERTAIN NONLINEAR ADVANCED DIFFERENCE EQUATIONS

This paper deals with the oscillatory properties of a class of nonlinear advanced difference equations. Sufficient criteria in the form of infinite sum for the equation to be oscillatory are obtained. In the linear cases, our results coincide with those in the literature.     

GENERALIZED RICCATI TRANSFORMATION AND OSCILLATION FOR LINEAR DIFFERENTIAL EQUATIONS WITH DAMPING

Using generalized Riccati transformation, some new oscillation criteria for damped linear differential equations are established. These results improve and generalize some known oscillation criteria due to A.Wintner, I.V.Kamenev for the undamped linear differential equations, and Sobol, J.S.W.Wong for the damped linear differential equations.     

The behavior of the solutions of periodic linear neutral delay difference equations

Some new asymptotic, nonoscillation and stability criteria for linear neutral delay difference equations with periodic coefficients and constant delays are given. The results are obtained via a positive root (with suitable properties) of an associated equation which is, in a sense, the corresponding characteristic equation.     

自治差分方程的稳定性

本文首先给出了文献[2]中关于一次近似系统不稳定性定理的一个反例,然后给出了关于自治差分方程利用其一次近似系统的不稳定性来判别原系统不稳定性的判别定理。     

Recent development in oscillatory properties of certain differential equations

In this paper, we summarize some recent oscillation criteria for second order nonlinear differential equations and systems of differential equations, some known oscillation criteria for second order linear differential equations are also involved, and we point out the origin of theses criteria.     

Strong limit point criteria for a class of singular discrete linear Hamiltonian systems

This paper is concerned with the limit point case for a class of singular discrete linear Hamiltonian systems. The limit point case is divided into the strong and the weak limit point cases. Several sufficient conditions for the strong limit point case are established. In consequence, two criteria of the strong limit point case for second-order formally self-adjoint vector difference equations are obtained.     

Stability of highly nonlinear hybrid stochastic integro-differential delay equations

For the past few decades, the stability criteria for the stochastic differential delay equations (SDDEs) have been studied intensively. Most of these criteria can only be applied to delay equations where their coefficients are either linear or nonlinear but bounded by linear functions. Recently, the stability criterion for highly nonlinear hybrid stochastic differential equations is investigated in Fei et al. (2017). In this paper, we investigate a class of highly nonlinear hybrid stochastic integro-differential delay equations (SIDDEs). First, we establish the stability and boundedness of hybrid stochastic integro-differential delay equations. Then the delay-dependent criteria of the stability and boundedness of solutions to SIDDEs are studied. Finally, an illustrative example is provided.     

Hille and Nehari type criteria for third-order dynamic equations

In this paper, we extend the oscillation criteria that have been established by Hille [E. Hille, Non-oscillation theorems, Trans. Amer. Math. Soc. 64 (1948) 234-252] and Nehari [Z. Nehari, Oscillation criteria for second-order linear differential equations, Trans. Amer. Math. Soc. 85 (1957) 428-445] for second-order differential equations to third-order dynamic equations on an arbitrary time scale T, which is unbounded above. Our results are essentially new even for third-order differential and difference equations, i.e., when T=R and T=N. We consider several examples to illustrate our results.     

Oscillation criteria for a forced mixed type Emden-Fowler equation with impulses

Some oscillation criteria for a forced mixed type Emden-Fowler equation with impulses are given. When the impulses are dropped, our results extend those of Sun and Meng [Y.G. Sun, F.W. Meng, Interval criteria for oscillation of second-order differential equations with mixed nonlinearities, Appl. Math. Comput. 15 (2008) 375-381], Sun and Wong [Y.G. Sun, J.S.W. Wong, Oscillation criteria for second order forced ordinary differential equations with mixed nonlinearities, J. Math. Anal. Appl. 334 (2007) 549-560] for second-order forced ordinary differential equation with mixed nonlinearities, Nasr [A.H. Nasr, Sufficient conditions for the oscillation of forced superlinear second order differential equations with oscillatory potential, Proc. Am. Math. Soc. 126 (1998) 123-125], Yang [Q. Yang, Interval oscillation criteria for a forced second order nonlinear ordinary differential equations with oscillatory potential, Appl. Math. Comput. 135 (2003) 49-64] for forced superlinear Emden-Fowler equation, Kong [Q. Kong, Interval criteria for oscillation of second-order linear differential equations, J. Math. Anal. Appl. 229 (1999) 483-492] for unforced second order linear differential equations, and Wong [J.S.W. Wong, Oscillation criteria for a forced second order linear differential equation, J. Math. Anal. Appl. 231 (1999) 235-240] for forced second order linear differential equation.     

Stability analysis of highly nonlinear hybrid multiple-delay stochastic differential equations

Stability criteria for stochastic differential delay equations (SDDEs) have been studied intensively for the past few decades. However, most of these criteria can only be applied to delay equations where their coefficients are either linear or nonlinear but bounded by linear functions. Recently, the stability of highly nonlinear hybrid stochastic differential equations with a single delay is investigated in [Fei, Hu, Mao and Shen, Automatica, 2017], whose work, in this paper, is extended to highly nonlinear hybrid stochastic differential equations with variable multiple delays. In other words, this paper establishes the stability criteria of highly nonlinear hybrid variable multiple-delay stochastic differential equations. We also discuss an example to illustrate our results.     

Oscillation and Nonoscillation in Delay or Advanced Differential Equations and in Integrodifferential Equations

Some new oscillation and nonoscillation criteria are given for linear delay or advanced differential equations with variable coefficients and not (necessarily) constant delays or advanced arguments. Moreover, some new results on the existence and the nonexistence of positive solutions for linear integrodifferential equations are obtained.