P-Laplacian Boundary Value Problems And Periodic Solutions Of Hamiltonian Systems On Time Scales

The theory of dynamic equations on time scales cannot only unify 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. Hence, the studying of dynamic equations on time scales is worth with theoretical and practical values.As considering the difference equations and differential equations, it is very difficult and important in studying the singular boundary value problems on time scales with the nonlinear term changing sign. Hence, we investigate singular m-point p-Laplacian boundary value problems on time scales with the sign changing nonlinearity. By using the well-known Schauder fixed point theorem and upper and lower solution method, we obtain some new existence criteria for positive solution of generalized Dirichlet, generalized Robin and nonlinear Robin boundary value problems.For nonsingular boundary value problems, we firstly deal with p-Laplacian multi-point generalized Neumann boundary value problem on time scales T. By using Krasnosel'skii's fixed point theorem, the generalized Avery and Henderson fixed point theorem and Avery-Peterson fixed point theorem. Some new sufficient conditions are obtained for the existence of at least single, twin, triple and arbitrary odd positive solutions of the above generalized Neumann problem. For considering the character of solutions, we concern with a class of p-Laplacian two-point boundary value problem on time scales T. By using symmetry technique and a five functionals fixed-point theorem, we prove that the boundary value problem has at least three positive symmetric solutions. We also prove that another p-Laplacian three-point boundary value problem on time scales has at least three positive pseudo-symmetric solutions in view of pseudo-symmetric technique and a five functionals fixed-point theorem.Variational theory is a very important method in nonlinear functional analysis. Does the variational theory exert analogous strength to study the problem in analysis on time scales? This problem arrests many people's attention. However, just as Ahlbrandt (MR1962542) reviewed for the reference (Buhner M., Peterson A., Advances in Dynamic Equation on Time Scales, Birkh(a|¨)user. Boston, 2003.), the Hilger's integral is based solely on antiderivatives. The so-called "delta integral" and "nabla integral" are defined by a sort of Darboux integral and as a limit of a modified Riemann sum, respectively. The absence of these integrals has hindered the development of variational theory on time scales. Therefore, Rynne defined a new integral on time scales T (JMAA, 2007). This new integral got over the absence of Hilger's integral. In view of variational methods and critical theory, we discuss two classes of second order Hamiltonian systems on time scales T and establish existence results for periodic solutions of the above-mentioned second order Hamiltonian systems on time scales T. This is probably the first time the existence of periodic solutions for second order Hamiltonian system on time scales has been studied by using variational methods and critical theory.