Existence And Oscillation Of Solutions For Some Classes Of Dynamic Equations And Dynamic Inclusions On Time Scales

Time scales theory, introduced by Hilger, not only establishes a bridge be-tween the continuous and the discrete analysis, but also understands deeply the essential between them. The theorem on time scales describes more accurately the phenomena which happen sometimes in continuous time and sometimes in discrete time. In order to develop time scales theorem and application further, our attention focus on the properties of solutions for dynamic equations and dy-namic inclusions,including the three aspects of the time scales theory:Existence of solutions for dynamic equations, existence of solutions for dynamic inclusions, oscillation for dynamic equations.In Chapter 1, we recall the background of our topics of this thesis, show the outline and give the main tools for dealing the existence and oscillations of solutions for dynamic equations and dynamic inclusions and some basic calculus on time scales related with this thesis.Chapter 2 is dealing with the existence of solutions for three classes of dy-namic equations on time scales. Thus, by means of fixed-point theorems due to Krasnoselskii-Zabreiko, and Schaefer, we present the existence of solutions for fourth-order four-point boundary problems, a class of first-order impulsive dy-namic equations and a class of second-order dynamic equations on time scales, respectively. It is worth pointing out for the Forth-order dynamic equation, the Green function is new and this method can be applied to study some other boundary value problems for high-order dynamic equations on time scales. For the first-order or second-order impulsive dynamic equations, the obtained result complement and extend some known results.In Chapter 3, combining fixed-point theorems on multi-valued mappings with continuous selection theorems for upper (or lower) semi-continuous, we in-vestigate existence of solutions for two classes of dynamic inclusions with bound-ary conditions on time scales. Some new criteria are given. We also provide examples to illustrate our main results. The obtained results fill the gap that there are only a few results about second-order dynamic inclusions on time scales in the literatures.Chapter 4 is devoted to the oscillation of solutions for dynamic equations. We discuss the problems in two subsections. In section 1, sufficient criteria of oscillation for a half-linear second-order dynamic equation is established. In sec-tion 2, we are concerned with the oscillation of a nonlinear second-order dynamic equation. The main tools include Riccati transformation and some inequalities. The results in Chapter 3 generalize and extend some known results in the liter-atures.