Oscillation,Asymptotic Behavior Of Solutions And Boundary Value Problems For Nonlinear Dynamic Equations On Time Scales

Recently, more and more scholars are highly interested in the theory of dynamic equations on time scales .The theory of time scales was initiated by Stefan Hilger in his Ph.D. thesis in 1988 as a theory can contain both differential and difference calculus in a consistent way. The theory of dynamic equations on time scales can not only unify the differential equations 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. This paper introduces the historical background of problems which will be investigated and states the main results of this thesis, then focuses on the research of oscillation,asymptotic behavior of solutions and boundary value problems for dynamic equations on time scales.Firstly, this paper discusses oscillation criteria for second order forced dynamic equations with mixed nonlinearities on time scales. By Riccati technique and corresponding inequality, some sufficient conditions for oscillations of all solutions of equation are established. Meanwhile some examples are given to illustrate the main result.Secondly, consider the oscillation and asymptotic behavior of third order damped dynamic equations on time scales, some sufficient conditions to satisfy oscillation of all solutions of equation are obtained. Meanwhile some examples are given to illustrate the corresponding theorem.The existence of positive solutions of m point boundary value problems for second order dynamic equations on time scales and third order nonlinear p ?Laplacian three point boundary value problem on time scales are studied, and by using the Guo-Krasnosel'skii fixed point theorem, Leggett-Williams fixed point theorem and five functionals fixed point theorem, some results are obtained for the existence of positive solutions for the above problems. Meanwhile some examples are given to illustrate the result.Finally , this paper considers the existence of positive solutions for second order and third order p ? Laplacian functional dynamic equations on time scales, by means of Guo-Krasnosel'skii fixed point theorem, the Leggett-Williams fixed point theorem and five functionals fixed point theorem, some sufficient results are obtained for the existence of multiple positive solutions of the above problems. At the same time, some examples are given.