Oscillation Of Linear Autonomous Functional Differential Equations

In both the real world and the engineering-technical field there exist extensive phenomena of oscllation or surge, such as mechanical oscllation(swing of pendu-lums, vibration of elastic bodies etc), spread of sounds, electromagnetic oscillation, spread of electromagnetic waves and so on. Mathematical models for describing such phenomena are mostly various kinds of differential equations or differential systems, for instance, ordinary differential equations or systems, partial differential equations or systems, functional differential equations or systems and so on. The oscillation theory, originating from Newton's time in the 18-th century, is a subject of mathematics and physics to oscillation disciplines. Since the begining of the last century, from the research fields of both natural sciences and social sciences, such as bionomy, economics, genetics, epidemiology, specially, automatic control theory, a lot of questions of dynamical systems with delay has arised. The theory of functional differential equations is the subject of mathemetics to such questions of dynamical systems with delay. Oscillation theory of functional differential equations is both an important component of the oscillation theory and an important branch of the theory of functional differential equations.The oscillation of functional differential equations can be studied in different manners of classification. In this paper, we study oscillation problems of linear au-tonomous functional differential systems. In the recent half-century, in the research fields of oscillation of linear autonomous functional differential equations have been published many monographs and a great amount of scientific papers(see [1]-[18]). The research results of oscillation for linear autonomous functional differential sys-tems are, however, few, and entire results are much fewer. The complexity of study of oscillation for linear autonomous functional differential systems first meets with the fact that the definitions of oscillation for linear autonomous functional differ-ential systems is not yet commonly accepted. There exist two typical ones among these definitions. In Papers [19]-[22] one of them is used for discussing oscillation of functional differential systems; in Papers [23]-[27] another is used. We think that the definition of oscillation used in [19]-[22] is more reasonable and compatible. There- fore, we will adopt the former in the present paper. In [19], the authors obtain a necessary and sufficient condition for oscillation of a class of linear autonomous func-tional differential systems, which is a scarce entire work and of important theoretic significance but difficult to apply. In [20], the author establishes an easier to verify necessary and sufficient condition for oscillation of a more special class of linear au-tonomous functional differential systems with one delay and applying it, obtains an algebraic criterion for oscillation of a class of second order linear autonomous systems with one delay. In [21], the author gains an easier to verify necessary and sufficient condition for oscillation of a more special class of linear autonomous systems with several delays and applying it, obtains some algebraic criterions for oscillation of certain classes of special linear autonomous systems. In this paper, we will consider oscillation of second order linear autonomous differential difference systems of the form where (aij)2×2, (bij)×2 and (cij)2×2 are matrices of second order,τandσare positive constants, and where aij,bij (i, j= 1,2) are constants,τis a positive constant, establish necessary and sufficient conditions for oscillation of the first system and algebraic criterions for oscillation of the second one, which are easy to apply and verify, some of which are more short-cut, and other of which contain some of the above-mentioned results as special cases.