Solutions For Several Kinds Of Boundary Value Problems Of Dynamic Equations On Time Scales

The theory of time scales was introduced by Stefan Hilger in1988. Afterthat more and more scholars take an interest in dynamic equations on time scales.There are two reasons. From the theoretical point of view, the time scales analysistheory can build bridges between continuous and dispersion. From the applicationpoint of view, dynamic equations on time scales has important applications. Forexample, the activities of insect populations in the diferent seasons and resting canbe described by the dynamic equations on time scales.This paper mainly discusses the existence of solutions for several kinds ofboundary value problems of nonlinear dynamic equations on time scales by usingLeray-Schauder nonlinear alternative, Avery-Perterson fixed point theorem, Legget-Williams fixed point theorem and the fixed point theorem of cone expansion andcompression. There are three chapters in this paper.Chaper1investigates the existence of nontrivial solutions of the followingsingular BVP whereTis a time scale. In [3] the authors considered the existence of positivesolutions of a second-order three-point boundary value problem on time scales byusing Krasnosel’skii fixed point theory. In [4] the author considered single solutionand multiple positive solutions for nonlinear three-point boundary value problem by using fixed point theorems in cones. But as far as we know, there are few papers to investigate the above boundary value problem on time scales as this paper does. Only [12] obtains one solution by using the shooting method when T=R. This paper considers the singular second-order three-point boundary value problem on time scales and obtains existence of nontrivial solution by using Leray-Schauder nonlinear alternative.Chapter2studies the existence of triple solutions for a second-order m point BVP of nonlinear p—Laplacian dynamic equations with derivative on time scales we denote the p—Laplacian operator by φp(u),i.e.,φp(u)=|u|P-2u, p1,(φp)-1=φq,1/p+1/q=1. In [6] the authors obtained triple positive solutions of three-point boundary value problems for p-Laplacian dynamic equations on time scales by using Avery-Perterson fixed point theorem. In [9] Yaslan considered the following dynamic equation on time scales u▽△(t)+h(t)f(t,u(t))=0with the analogous boundary value conditions and obtained at least one, two and three positive solutions by using the the fixed point theory of cone. However, the results of the above-mentioned literature [6],[9] were the existence of positive solutions with nonlinear terms not involving the derivative. We will face a lot of difficulties when discussing the existence of positive solutions with the nonlinear terms containing the derivative of the unknown function explicitly. The reason is that it is very difficult for us to control the nonlinear terms. Naturally, it is necessary for us to consider the existence of positive solutions to p-Laplacian boundary value problems when the nonlinear term is involved with the derivative explicitly. This chapter considers the problem and obtains at least three positive solutions by using Avery-Perterson fixed point theorem. Chapter3considers the existence of solution and the solution dependenceon the parameter for a system of dynamic equations three-point boundary valueproblem on time scales. In [10] the authors considered the existence of positivesolutions for the dynamic equations of two point of BVP with a single parameter,where nonlinear term required monotonic. This chapter does not require nonlinearterm monotonic and studies the existence of solution and the solution dependenceon the parameter for the following second order dynamic equations three-pointboundary value problemwith the boundary value conditions...