Vibration Theory Of Linear Hamiltonian Systems With The Asymptotic Theory

As we all know, The comparison and separation theory of zeros distribution for second order homogeneous linear differential equations established by G.Sturm lays a foundation of oscillation theory for differential equations. During one and a half century, oscillation theory of differential equations has developed quickly and played an important role in qualitative theory of differential equations and theory of boundary values problem. With the further study of this direction, the studying contents and methods continuously get abundant. A lots of results have been obtained in twenty years whether for linear equations or for nonlinear equations. The studying direction is also extended to corresponding fields, such as matrix differential systems, functional differential equations, difference equations, partial differential equations and so on.This thesis is divided into four parts.Part I. Consider the second order self-adjoint matrix differential equationswhere P(t), Q(t), X(t} are n x n real continuous matrix functions, P(t) > 0 is positive definite,Q(t] is symmetric.In Chapter 2, we study the oscillatory behavior of (1) by using averaging functions and a generalized Riccati transformation. The results obtained in this paper generalize and improve some previous results. What's more, most known oscillation criteria for (1) are given on some real half-line and hence require good integral behavior of P(t), Q(t) on [t_o,∞). However, from the definition of oscillation, the oscillation of system (1) is only an interval behavior. Therefore, it is natural to think that we can study the oscillatory behavior of system (1) only on a sequence of subiiitervals of [t₀,∞), whichweakens the restriction to P(t),Q(t) to a great extent. In Chapter 2. we also establishmany interval criteria for oscillation of (1). The results obtained in this paper are different from previous results in the sense that they are based on the information only on a sequence of subintervals of [t₀,∞). Our results are sharper than some previous results and can be applied to extreme cases such asrtPart II We study the linear Hamiltonian systemwhere A(t}, B(t), C(t) are n x n matrix functions, B, C are Hermitian, if B*(t) = B(t], C*(t) = C(t). By M* we mean the conjugate transpose of the matrix M. We also study the oscillation of the corresponding matrix equation of the formX' = A(t)X + B(t)YAt present, there are few oscillation results of Hamiltonian system (3). In Chapter 3, by using Riccati technique, some oscillation criteria are established, which extend and improve some known oscillation criteria. On the other hand, we obtain some new interval oscillation criteria which are different from most known results in the sense that they are based on the information only on a sequence of subintervals of [t₀,∞), rather than on the whole half-line.Part III Consider the second order differential equationwhere r(t) > 0 is absolutely continuous on [a, ∞), p(t) , q₁(t) , q₂(t) , f (t) are partly integrable real functions defined on [a,∞). Equation (4) is said to belong to limitcircle type if all solutions of equation (4) belong to L² (simply denoted by L.C.) Equation (4) is said to belong to Lagrange stable if all solutions of equation (4) belong to (simply denoted by L.S.).In Chapter 4, we study criteria for the linear nonhomogeneous differential equation belonging to the limit circle type. By introducing an important skill inequality, we establish some criteria for equation (4) belonging to L.S. or L.S.L.C, and hence resolve such problems. Because we improve the inequality and use a class of generalized auxiliary functions, the results obtained in this paper are sharper than known results for the case when f(t)= 0.Part IV We study the oscillation of second order quasi-linear neutral delay differential equations of the formwhere t ≥ t₀, a > 0, T ≥ 0, and ≥ 0 are constants, a,p,q ∈ C([t₀, ∞); R), f ∈ C(R;R).By using averaging functions and a generali...