Existence Of Positive Solutions For Second Order Dynamic Equations On Time Scales

Many important dynamical system are described by differential equation and difference equation. The differential equation and difference equation can be regard-ed as two special cases of the dynamic equation on time scales. It has opened up a new field of mathematical research. In1988, Stefan Hilger[1]first introduced the time scales theory and laid the foundation for continuous and discrete equations. The time scales theory has wide application background, and has a solid practical foundation. It arose from ecological models, neural network model, heat-conduction model, economic models[2] and so on. Many scholars are engaged in the new math-ematics study and obtain a series of outstanding achievements, see [3]-[10] and references therein. In addition to the application of biology, the mathematical tools have been used to improve the calculation mode of the stock market.This paper mainly discusses the existence of solutions for several kinds of second-order boundary value problems of dynamic equations on time scales by using Leggett-Williams fixed point theorem and the fixed point index theorem of cone. There are three chapters in this paper.Chapter1investigates the existence of triple solutions for a second-order BVP of p-Laplacian dynamic equations on time scales where T is a time scales.We denote the p-Laplacian operator by (?)p(u), i.e.,(?)p(u)=|u|p-2/u, p>1,((?)>p)-1=(?)q,1/p+1/q=1. In [11] the authors considered the above dynamic equation on time scales with the analogous boundary value conditions and obtained at least two positive solutions by using the fixed point index theo-rem. In [19], the authors obtained triple positive solutions of a second order three-point boundary value problems for p-Laplacian dynamic equations on time scales by using Avery-Perterson fixed point theorem. This chapter considers the problem and obtains at least three positive solutions by using Leggett-Williams fixed point theorem.Chapter2studies the existence of positive solutions for a second-order BVP of dynamic equation with integral boundary value conditions on time scales. In [12], when T=R, the authors obtained at least one positive solution by using Krasnosel’skii fixed point theory. In [18], when T=R, the authors considered the second-order BVP with integral boundary value conditions by using Leggett-Williams fixed point theorem and the fixed point theorem of cone expansion and compression and obtained the existence of positive solutions and obtained the existence of multiple positive solutions. This chapter considers the problem and obtains at least one and two positive solutions by using fixed point index theorem on time scales.Chapter3considers the existence of positive solutions for a system of the following second order dynamic equations with the boundary value conditions In [20], the authors considered the existence of positive solutions for the dynamic equations of two point of BVP with a single parameter. In [19], without the mono- tone of the nonlinear term, the authors studied the existence of solution for second order dynamic equations three-point BVP. This chapter obtains triple positive so-lutions of boundary value problem on time scales by using Leggett-Williams fixed point theorem. An example is given to illustrate the result of the application.