Oscillation Of Second-order Matrix Differential Systems

The theory of matrix differential system is one of important branches of differential equations. In the field of modern applied mathematics, it has made considerable headway in recent years, because all the structures of its emergence have deep physical background and realistic mathematical models. The oscillation theory is one of important branches of the theory of matrix differential system. Many scholars take on the research of this field, they have achieved many good results. In very resent years, great changes of this field have taken place. Especially, the second order matrix differential system has been paid more attentions and investigated in various classes (see [1]-[33]).The present paper employs a generalized Riccati transformation, integral average technique arid matrix inequality to investigate the oscillation criteria for some class of matrix differential systems, and gets some new results.The thesis is divided into three sections according to contents.In chapter 1, Preface. we introduce the main contents of this paper.In chapter 2, the chapter is divided into two sections to investigate the oscillation criteria for the second order linear matrix differential system with damped term(P(t)Y′)′+ R(t)Y′+ Q(t)Y = 0, t≥t₀, (2.1.1)where P(t) = P^*(t) > 0 (i.e., P(t) is positive definite), Q(t) = Q*(t) and R(t)=R*(t) are n x n matrices of real valued continuous functions on the interval [t₀,∞). By M^* we mean the transpose of the matrix M.In the first section, we introduce a positive linear functional g and use generalized Riccati transformation and integral average technique to establish some new interval criteria. The superiority of the criteria is they are given by the behavior of P(t), R(t), Q(t) only on a sequence of subintervals of [t₀,∞). Especially by choosing the appropriate positive linear functional g and average function, we shall present several easily verifiable oscillation criteria. So our main results improve and extend some recent results and enrich the existing Kamenev type oscillation criteria.In the second section we use comparison theorem to obtain the relation of oscillation between system (2.1.1) and a second order scalar equation, which enables one to consider the question of the oscillation of system (2.1.1) in terms of the oscillation of a corresponding second order scalar differential equation. Thus the very large number of well-known criteria for second order scalar differential equation can be used to determine associated oscillation criteria for system (2.1.1).In chapter 3, the chapter is divided into two sections to investigate the oscillation criteria for two class of second order nonlinear matrix differential systems. We state the main results as follows:Firstly, consider the second order nonlinear matrix differential system of the form(P(t)Y′)′+ R(t)Y′+ F(Y) = 0, t≥t₀, (3.1.1)where P(t) = P^*(t) > 0 (i.e., P(t) is positive definite), R(t) = R^*(t)∈C¹([t₀,∞),Rⁿ²) and F(Y) are n×n real continuous matrix functions on the interval [t₀,∞). By M^* we mean the transpose of the matrix M.We use generalized Riccati transformation and consider the function H(t,s)k(s) (which partial derivative with respect to s is uncertain non-positive). We establish some new oscillation criteria, our main results improve and extend some results of [9] and other known criteria.Secondly, consider the second order nonlinear matrix differential system of the form(P(t)Y′)′+ R(t)G(Y)Y′+ Q(t)F(Y′)Y = 0, t≥t₀, (3.2.1)where P(t) = P^*(t) > 0 (i.e., P(t) is positive definite), R(t) = R^*(t) and F(Y) are n×n real continuous matrix functions on the interval [t₀,∞). By M^* we mean the transpose of the matrix M.In this section, using the method of integral average technique and defining a new prepared solution, we present some new oscillation criteria of system (3.2.1), which improve and extend some recent results of [12]. Furthermore, we give another form of interval criteria for the system (2.1.1).