Existence Of Positive Solutions For A Class Of Multiple Points Boundary Value Problems In Banach Space

The boundary value problems are an important part in the field of differential equations, which have comprchensive application in the classical physics and electricity. On the base of the richful pratical and applied background, the cxistence of positive solutions for boundary valuc problems of nonlincar differential equtions in abstract space bccome cspccially important in the area of the research about differcatial equations. Classical boundary value problems (for example: Drichlet two-point boundary value problem, Robin two-point boundary value problem, and so on) had been studied widely for last thirty years and there are many excellent results. However, relatively few results about three-point boundary value problems (especially multiple points boundary value problem for high-order differential equations) had been obtained, for example, the results of reference [1]-[5], [13]-[16] and [18].On the basis of above discussions, this paper mainly deals with the existence of one solution and multiple solutions of some nonlinear differential equations with multiple-point boundary value problem in Banach space; because of the importance of scmipositone property, impulse and singularity, we study their effects on the solutions of nonlindar differential equations, and we obtain some useful results. So the content that we studied has important theoretical and applicable value.There are four chapters in the dissertation.In the first chapter, by using cone compression and expansion fixed point theorem of strict set contraction operator, we deal with the existence of multiple positive solutions of the following fourth-order three-point boundary value problem in Banach space andη∈(0, 1), and an example in finite-dimensional space is worked out to indicate our conditions are reasonable.In the second chapter, by using cone fixed point, theorem, we investigate the cxistcnce of positive solutions of the following semipositone boundary value problem and nonlinearity f(t, x) maybe negative at some t and x, and that nonlinearity satisfies Caratheodory conditions,λis a positive paramcter,η∈(0, 1), then an cxample is considcrcd to indicate our conditions are reasonable.In the third chapter, by using spedtrum thcory of bounded positive operator and fixed point index theorem of cone, this chapter deals with the existence of positive solutions for the following singular semipositone boundary value problem which respectively satisfies superlinear and sublinear in Banach space and cxamples are worked out to indicatc our conditions are reasonable, where f(t, x) maybe singular at t=0, t=1 and x=0, f(t, x) maybe negative at some t and x.In the last chapter, by using Monch fixed point theorem, we consider the cxistence of solutions of the following second-ordcr multiplc-point singular boundary value problem of impulsive ditferential equations in Banach space andβ∈R,η∈(0, 1). In the end, an example in infinite-dimensional space is worked out to indicatc our conditions are reasonablc.