Oscillation,Existence Of Solutions And Boundary Value Problems For Several Classes Of Impulsive Dynamic Equations On Time Scales

In real life, when we handle various problems in the natural phenomena with mathematics method, not only run into the continuous problems, but also run into the discrete problems. Sometimes both the continuous and discrete components will be held in a single problem. It is even not clear for us whether this problem is the continuous one or the discrete one. This brought inconvenience into our research. The theory of time scales can unify continuous and discrete cases, which pioneers a new mathematical area. This theory not only unifies differential and difference equations, reveals the essence of continuous and discrete cases, avoids repeat study, but also contains many other cases. The prominent peculiarity of time scales is unification and generalization, so the study of time scales has important significance in theory and application. The oscillation,existence of solutions of impulsive dynamic equations and boundary value problems on time scales are researched.Firstly, oscillation of a class of second order nonlinear impulsive dynamic equations on time scales are discussed in this paper. Some sufficient conditions to satisfy oscillations of the equations are gained. At the same time, some examples are given.Secondly, existence of nonoscillatory solutions for a class of impulsive dynamic equations on time scales are considered. By employing Banach contraction mapping principle, some sufficient conditions for existence of solutions of the equation are established.A class of nonlinear delay impulsive dynamic equations four-point boundary value problems on time scales is considered. By employing Leggett-Williams fixed theorem, some sufficient conditions for existence of three positive solutions of the equation are established.Existence of solutions for a class of nonlinear impulsive dynamic equations periodic boundary value problems on time scales are studied. By employing the fixed theorem, some sufficient conditions for existence of solutions of the equation are established. At the same time, some examples are given.Finally, existence of extremal solutions for a class of impulsive dynamic equations periodic boundary value problems on time scales are studied. By employing the method of upper and lower solutions and the monotone iterative technique, some sufficient conditions for existence of maximum and minimum solutions of the periodic boundary value problems of impulsive dynamic equations on time scales are established.