Class Of Differential Equations Boundary Value Problem

This paper investigates the existence of positive solutions for differential boundary value problems, and investigates the existence of positive periodic solutions of a class of bio-mathematical model.It consists of four chapters.In chapter one,the background and present situation of differential boundary value problems are introduced, and the development of population ecology is described.In chapter two,we study a kind of two-point boundary value problem x'"(t)＋f(x(t))＝0,0≤t≤1, x(0)=x'(0)=0,αx'(1)+βx"(1)=0, whereα,β≥0 andα+β>0.Using Leggett-Williams fixed point theorem, sufficient conditions for existence of triple positive solutions are established.In chapter three,we study a kind of three order boundary value problem x'"m(t)=g(t)f(t,x(t)) (0≤t≤1), x(0)=x'(λ)=0,αx(1)+βx"(1)=0, where f:[0,1]×[0, +∞)�'[0,+∞),g:[0,1]�'[0,+∞) are continuous,α>0,β>0, We will not require the existence of lower and upper solutions.By applying monotone iterative techniques,we not only obtain the existence of positive solutions,but also establish iterative schemes for approximating the solutions.In chapter four,we study a kind of nonlinear four order boundary value problem u(4)(t)＝g(t)f(ut)(0≤t≤1), u(t)＝φ(t),-τ≤t≤0, u(0)=u'(0)＝u'(1)＝u'"(1)＝0, where ut(s)=u(t+s),-τ≤s≤0,0<τ<1,φ(t)∈C([-τ,0],R+)andφ(0)=0.Using fixed-point theorem of cone expansion and compression of norm type, we study the existence of positive solutions for this boundary value.In chapter five,a more realistic impulsive predator-prey model with mutual interference and Beddington-DeAngelis functional response is studied. By using the continuation theorem of coincidence degree theory,we establish a set of easily verifiable sufficient conditions for the existence of at least one positive periodic solution for the model,the example shows that our criteria are easily verifiable.