Boundary Value Problems For Dynamic Equations On Time Scales

An important branch of dynamic equations on time scales is boundary value problems (BVPs), due to their striking applications to almost all areas of science, engineering and technology. By researching BVPs for dynamic equations on time scales the results not only unify the theory of differential and difference equations and understand deeply the essential difference between them, but also provide accurate information of phenomena that manifest themselves partly in continuous time and partly in discrete time. At the same time, many difficulties occur when considering BVPs for dynamic equations on time scales. For example, basic tools from calculus such as Fermat theorem, Rolle theorem and the intermediate value theorem may not necessarily hold and fundamental concepts such as chain and product rules and certain smoothness properties all need to be modified.In this PhD thesis, we first study the following BVP for first-order dynamic equation on time scale (?)The corresponding integral operator is constructed and its completely continuity is proved. For the caseβ= 1, we obtain the existence and multiplicity results of solutions and positive solutions by using Schaefer, Guo—Krasnoselskii, Schauder and Leggett—Williams fixed-point theorems and fixed-point index theory. For the case 0 <β< 1, we obtain the existence results of at least two positive solutions by using Avery—Henderson two-fixed-point theorem.Next, we are concerned with the BVPs for second-order dynamic equations on time scale (?), including Dirichlet and Focal BVPs. For the Dirichlet BVP, we establish some existence criteria of at least n solutions or positive solutions and of at least one positive solution under local and global conditions, respectively, and our main tools are the Guo—Krasnoselskii fixed-point theorem and the fixed-point index theory. Compared with the known results, our nonlinear term here is semipositone. For the Focal BVP, we first consider the Right—Focal BVP. An existence result is obtained by using a fixed-point theorem due to Krasnoselskii and Zabreiko. Our conditions imposed on the nonlinear term are very easy to verify. And then, we consider the system of Left—Focal BVPs. Different from the known results in essence, theσ2(T), which is the right end-point of domain of solutions, is not fixed. By using the fixed-point index theory, we investigate the effect ofσ~2(T) on the existence and nonexistence of positive solution for the system in sublinear cases.Finally, we discuss the existence of solutions and positive solutions to a class of BVP for third-order dynamic equations on time scale (?) by using the Schauder fixed-point theorem. It is worth pointing out that this method can also be applied to study some other BVPs for higher-order dynamic equations on time scales.