| With the development of technology and studying of equations,people find that many resultsconcerning differential equations carry over quite easily to corresponding results for differenceequations, while other results seem to be completely different in nature from their continuouscounterparts .Then attention is paid to the problem :Can we find a new thing that equations built on it unify the differential equations and difference equations?After Stefan Hilger introducedthe theory of time scales in 1988,more and more interest is paid on dynamic equations on time scales.Because it harmonizes the continuous and the discrete ,also extends to normal case.Recently much attention is attracted by questions of oscillatory solutions for second order dynamic equations on time scales,stimulated by the work in this area.In this paper, we consider oscillation of solutions for the second order nonlinear neutral dynamic equations on time scales T(a(t)(x(t)-px(g(t)))△)△+f(t,x(τ(t)))=0We assume that,(H1) a(t) > 0,∫t∞1/a(s)△s=∞,a(t)∈Crd(T,R);(H2) p∈R+,τ(t), g(t)∈Crd(T,T), g(t)≤t,τ(t)≤t and g(t),τ(t) are nondecreasing,lim(?)g(t) = lim (?)τ(t) =∞;(H3) f∈Crd(T×R, R), and f(t,x)/x≥q(t)≥0, (x≠0),and q(t)is not always equal to zero;(H4) g2(t) = g(g(t)), gm(t) = g(gm-1(t)),g-1 is inverse function of g , and g-2(t) = g-1(g-1(t)), g-m(t) = g-1(g-m+1(t)).According to the contents,we divide this paper into several parts as follows: Part one is the perface, the background of dynamic equations on time scales is mainly introduced.Part two is the body of the passage .contains the following parts: (1)we introduce a survey to the basic notions on time scales.(2)related lemmas ,by Riccati substitution on equation(1.1),we get several Riccati inequations on time scales that nonoscillatory solutions must satisfy. (3)the theory of existence of oscillatory solutions. (4)verify the validity of the conclusion by examples. Part three is the conclusion.
|
PDF Full Text Request