【作者】 刘静；

【导师】 孙太祥；

【作者基本信息】 广西大学 ， 基础数学， 2016， 硕士

【摘要】 为了统一连续型与离散型分析,Hilger于1988年创立了时标动力方程理论.近年来,人们对时标动力方程的Lyapunov不等式进行了深入研究,得到了许多有意义的不等式.本文主要是研究时标上的几类动力方程及系统的Lyapunov不等式.在第1章,我们介绍了时标动力方程的基础理论及Lyapunov不等式的国内外研究现状.在第2章和第3章,我们分别研究了时标T上的Hamiltonian系统xΔ(t)=-A(t)x(σ(t))-B(t)y(t), yΔ(t)=C(t)x(σ(t))+AT(t)y(t),和quasi-Hamiltonian系统xΔ(t)=-A(t)x(σ(t))-B(t)|y(t)|p-2y(t), yΔ(t)=C(t)|x(σ(t)) |q-2x(σ(t))+AT(t)y(t),在一定条件下得到了上述系统的若干Lyapunov不等式.其中p,q∈(1,+∞)且1/p+1/q=1,A(t)是T上的n阶实矩阵值函数且,+μ(t)A(t)可逆,B(t)和c(t)是T上的n阶实对称矩阵值函数且B(t)是正定的,x(t),y(t)是T上的两个n维实向量值函数.在第4章,我们研究了时标T上的高阶动力方程SnΔ (t,x(t)) +φ(t)xβ(t) = 0在一定条件下得到了上述方程的Lyapunov不等式.其中n是正整数,β(≥1)是两个正奇数的比值,S0(t,x(t))=x(t), Sk(t,x(t))=ak(t)Sk-1Δ(t,x(t)) (1≤k≤n-1), Sn(t,x(t))=an(t)[Sn-1Δ(t,x(t))]β,且ak∈Crd(T,(0,∞)) (1≤k≤n),φ(t)∈Crd(T,R).在第5章,我们研究了时标T上的高阶动力方程|SnΔ(t,X(t))|p-2SnΔ(t,X(t))+B(t)|X(t)|p-2X(t)=0在反周期边界条件下得到了上述方程的Lyapunov不等式.其中n是正整数,p∈(1,+∞),X(t)是T上的n维实向量值函数,且S0(t,X(t))=X(t), Sk(t,X(t))= Ak(t)Sk-1Δ(t,X(t))(1≤k≤n), Ak(t) (1≤k≤n)是T上的n阶实正定矩阵值函数,B(t)是T上的n阶实矩阵值函数且I+μ(t)B(t)可逆.更多还原

【Abstract】 The theory of dynamic equations on time scale was initiated by Hilger in 1990 in order to create a theory that can unify discrete and continuous analysis. During the last few years, some Lyapunov inequalities for dynamic equations on time scales have been obtained by many authors. In this thesis we study some Lyapunov inequalities of dynamic equations (or systems) on time scales.In Chapter one, we introduce the basc theory of dynamic equations and the current situation for the Lyapunov inequalities of dynamic equations.In Chapter two and three, we investigate Hamiltonian systems xΔ(t)=A(t)x(a(t))-B(t)y(t), yΔ(t)＝C(t)x(σ(t))＋AT(t)Y(t), and quasi-Hamiltonian systems =-A(t)x(σ(t))-B(t)|y(t)|p-2y(t), = C(t)|x(σ)|q-2x(σ(t))+AT(t)y(t), on time scale T respectively, and obtain several Lyapunov inequalities of those systems under some conditions, where p, q ∈ (1,+∞) satisfying 1/p+1/q= 1, A(t) is a real n×n matrix-valued function on T and I+μ(t)A(t) is invertible, B(t) and C(t) are two real n×n symmetric matrix-valued functions on T and B(t) is positive definite, and x(t),y(t) are two real n-dimensional vector-valued functions on T.In Chapter four, we investigate the following higher-order dynamic equation SnΔ(t,x(t)+φ(t)(t)=0 on time scale T, and obtain some Lyapunov inequalities of that equation under some con-ditions, where n is a positive integer, β(≥1) is a quotient of two odd positive integers, S0(t,x(t))= x(t) (k= 0), Sk(t,x(t))=ak(t)Sk-1Δ(t,x(t) (1≤k≤n-1), Sn(t,x(t))= an(t)[Sn-1Δ(t,x(t_)]β, and ak ∈ Crd(T, (0,∞>)) 1≤k≤n),φ(t)∈Crd(T,R).In Chapter five, we investigate the following higher-order dynamic equation |SnΔ(t,X(t))|p-2SnΔ(t,X(t))+B(t)|X(t)p-2X(t)=0 on time scale T, and obtain a Lyapunov inequality of that equation under anti-periodic bound-ary conditions, where n is a positive integer, p ∈ (1,+∞), X(t) is a rea n-dimensional vector-valued functions on T, S0(t,X(t))= X(t) and Sk(t,X(t))= Ak(t)Sk-1Δ(t,X(t))(1≤k≤n), Ak(t) (1≤k≤n) are real n×n positive definite matrix-valued functions on T and B(t) is a real n×n matrix-valued function on T with I+μ(t)B(t) being invertible.更多还原