Positive Solutions Of Some Class Of Dynamic Equations On Time Scales

This thesis mainly investigates the existence of positive solutions for some class of dynamic equations on time scales by using topological degree of nonlinear functional analysis. There are five chapters.Chapter 1 presents the research background and main results of this thesis.Chapter 2 is preliminaries, mainly including basic notions of time scale (?), "delta" derivative and its basic property, "delta" integral and its algorithm.Chapter 3 is concerned with the existence of positive solutions for some class of dynamic equations on finite time scale [a, b]. There are three sections in Chapter three.Section 3.1 mainly investigates the existence of positive solutions of boundary value problem (BVP in short)We consider general BVP (3.1.1) with weaker conditions than those of papers [23], [33], [36], [55], [59], [69]. The method of operator approximation is used to deal with difficulties caused by singular property. We obtain the existence of positive solutions of BVP (3.1.1) and the existence interval of positive solutions of BVP (3.1.2), by using fixed point index theorems. These results extend some main results of papers [23], [33], [36], [55], [59], [69].In Section 3.2 we go on to study BVP (3.1.2). The function f in this section does not necessarily satisfy conditions required in papers [23], [33], [36], [55], [59], [69], [72]. With some limit conditions and control conditions we get the existence of positive solutions of BVP (3.1.2). Its application is illustrated with a corresponding example.