Oscillation Of Dynamic Equations On Time Scales And Classification Of Nonoscillatory Solutions

With the rapid development of science and technology, many mathematical models which are described by differential equations and difference equations are applied in both natural science and edging fields such as physics, population dynamics, theory of control, biology, medicine, economics, etc. The theory of time scales,which has recently received a lot of attention ,unify continuous and discrete analysis and give basic methods to deal with continuous system and discrete system at the same time. The study of dynamic equations on time scales helps avoid proving results twice, once for differential equations and once for difference equations.According to the contents, this thesis is divided into five parts follows:Introduction, we introduce a survey to the background and the current development of dynamic equations on time scales.In section one, we introduce a survey to the basic notions to time scales.In section two, we consider the oscillation of higher order dynamic equations(x(t) - p(t)x(Ï„(t)))^â–³n = q(t)x(v(t)) (1.1)on time scales T. where n is an even integer, t∈[t₀,∞)_t, we all assume that:(1) p(t),q(t)∈C(T,R⁺)(2) v(t),Ï„(t)∈C(T,T),Ï„(t) < t,v(t) < t, (?) v(t) =∞, (?)Ï„(t) =∞, v(t)is nonde-creasing.We consider the following three situations respectively: 0≤p(t)≤p < 1, p(t)≥1, p(t) = l,and present some bounded oscillation criteria for equation (1.1).In section three, we consider the second-order self-adjoint neutral dynamic equations(a(t)|(x(t) - p(t)x(Ï„(t)))^â–³|^αsgn(x(t) - p(t)x(r(t)))^â–³)^â–³+ f(t, x(v(t))) = 0 (1.2)on time scales T. whereαis a positive constant, t∈[t₀,∞)_T, we all assume that:(i)Ï„(t),v(t)∈C(T,T),Ï„(t) < t, v(t) < t, (?) =∞, (?) =∞,Ï„(t)isnondecreasing.(ii) p(t),a(t)∈C(T,R⁺), 0≤p(t)≤λ<1(iii) A(t) = (?) <∞(iv) f(t,u)/u > 0, for t≥t₀,u≠0, f(t, u) is nondecreasing and continuous with respect to ufor fixed t. We present the classification of nonoscillatory solution of equation (1.2).Further we obtain some necessary and sufficient conditions of the existence for some kinds of nonoscillatory solutions of the equation (1.2).Conclusion,we introduce the main study results and the limitations of the thesis.