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

In order to unify continuous and discrete analysis, German mathematician Hilger published a paper of Analysis on measure chains-a unified approach to con-tinuous and discrete calculus ([1]) to introduce the concept of dynamic equation on time scales, which received extensive attention, and became an important area of research, such as insect breeding model, virus spread and so on.The existence and uniqeness of positive solutions for differential equations have been considered extensively since twenty years ago ([2]-[40]). This paper discusses the existence of solutions for several kinds of boundary value problems of nolinear differential equations on time scales by using the fixed point theorem of cone expan-sion and compression, the fixed point index theory, Avery-Henderson fixed point theorem and Legget-Williams fixed point theorem.There are three chapters in this paper. In chaper l,we investigate the existence of positive solutions for second-order multiple point boundary value problem on time scalesIn [6] the authors considered the existence of positive solutions of a second-order nolinear differential equations on time scales by using the theory of cone and the fixed point index theory. In [7] the author considered a first-order nolinear boundary value problem by using Avery-Henderson fixed point theorem. But as far as we know.there are few papers to investigate multiple points boundary value problem on time scales as this paper does. In this chapter by using fixed point theorem in cones of A very-Henderson.we prove the existence of two positive solutions for above problem on time scales.In chapter2, we investigate the existence of positive solutions to the m-point boundary value problem where f(t, x) is superlinear and semipositone.In [15],the author was concerned with a superlinear semipositone on time scales with a parameter, and obtained the existence of positive solutions for the semipositone BVP with the parameter belonging to an explicit open interval.In [16], the author proved the existence of two positive solutions for the fol-lowing problem on time scales In this paper we firstly obtain two positive solutions by using fixed point theorem, and secondly get three positive solutions by using the fixed point theorem of Legget-Williams.In chapter3, we consider the existence of multiple solutions which depend on the parameter for m-point boundary value problem with p-Laplace operator on on time scales. In [31] the authors considered the following m-point boundary value problem with p-Laplace operator Activated by this paper,we study the existence of positive solutions of the following BVP by using the fixed point index theory where φp(s)=|s|p-2s,p>1,φp-1=φq,1/p+1/q=1,0<ξ1<ξ2<…<ξm-2<1.