Existence Of Positive Solutions For Three-Point Boundary Value Problem

Three-point boundary value problems for differential equations are that boundary value conditions of differential equations contain the value at interval endpoint and inside the interval. They arose in different fields of applicable mathematics and physics. Multi-point boundary value problems have wide background such as gas dynamics, Newtonian fluid mechanics. The theory of singular multi-point boundary value problems is used widely, so they have an important value of investigation in recent years.In 1973, D. Barr., T. Sherman firstly studied the multi-point boundary value problems. From the 1990s, C.P. Gupta. did lots of work on second-order three-point boundary value problems. After then there was lots of research was done in this aspect .Many domestic researchers also did lots of research on this aspect and made some achievements. But few papers study existence of positive solutions for three-point boundary value problems when nonlinearity may change sign and may be singular.This paper is divided into three chapters. By using Green's funct ion, upper and lower solution method and the fixed point theorem on cone and so on, this paper studies the existence of the positive solution for two singular nonlinear boundary value problem of second-order three-point.In Chapter 1, it gives the definition and some basic theorem of the nonlinear boundary value problem.In chapter 2, it gets a result of the existence of the positive solution, which improves and extends the proof of theorems in [7]. We consider the following singular nonlinear boundary value problem of second-order three-point.The existence theorem of continues positive solutions are given in this chapter.Suppose the homogeneous linear boundary value problem and the semi-homogeneous linear boundary value problem about boundary value problem (A) is respectivelyandFirst, the solution boundary value problem for (A') and (A") is given next. For Clarity, we list some conditions and lemma later as follows.(H1)α≥0 ,β≥0 ,η∈(0,1) is a constant, 00;(H2)b(t)∈C((0,1),[0,+∞)),and 0<(?)b(s)ds<+∞;(H3)f∈C([0,+∞),[0,+∞));Lemma 2.1.1 Let (H1) (H2) hold, then boundary value problem (A') has unique solutionLemma 2.1.2 Let (H1) hold, then Green's function for semi-homogeneous linear boundary value problem (A") is given by Lemma 2.1.3 Suppose b(t)≥0, then boundary value problem (A") has unique positive solution (1).Lemma 2.1.4 Let (H1) hold, then there is aδ= max{k(1-η)/(1-kη),1 +k} such thatLemma 2.1.5 Let (H1) (H2) hold, then the unique solution of boundary value problem(A") satisfieswhereγ=min{k(1-η)/(1-kη),kη,η}．LetThe P is a cone in Banach space E=C(I).Lemma 2.1.6 Let (H1) (H2) (H3) hold ,then A : P→E is a completely continuousoperator.Main Results In this chapter:Theorem 2.2.1 Assume that (H1) and(H4)a(t)∈C((0,1),[0,+∞)) and 0≤(?)a(s)ds<∞；(H5)f∈C([0,+∞),[0,+∞)) and the function f satisfies the following superlinear:f₀₊=0,f_+∞=+∞;hold, then the three-point boundary value problem (A) has a positive solution.Suppose that in Theorem 2.2.1 the conditions (H1) holds, the condition (H4) (H5) is become:(H6)a(t)∈C((0,1),[0,+∞)),0<(?)a(s)ds<+∞,and (?)t₀∈(0,1),such that a(t₀)>0;(H7)f∈C((0,+∞),[0,+∞)) and he function f satisfies the following superlinear:f₀₊=0,f_+∞=+∞;Corollary 2.3.1 Assume that (H1) (H6) (H7) and (H8)(?)c∈(0,+∞),hold, then the three-point boundary value problem (A) has a positive solution.In chapter 3, by using upper and lower solution method, and the properties of Green's function, we establish a existence criterion due to Schauder fixed point theorem for next three-point boundary value problemIn [17, 19, 20], Ruyun Ma,Qingliu Yao and other authors obtained the existence of positive solution for three-point boundary value problem (B) with Krasnosel'skii fixed point theorem when f is sublinear or superlinear but a(t) is not singular.The following conditions are satisfied throughout this chapter:(h1) f(t,u)∈C(J×R+,[0,+∞)), and f(t,u) is decreasing with respect to u;(h3)There exists a function a(t) such thata(t)≥kt(1-t),t∈I (k>0 is a certain real number) Theorem 3.1.1 Assume that (h1) (h2) (h3) hold, then the singular second-order three-point boundary value problem (B) has a C[0,1] positive solution w(t), and satisfiesWhen the condition (h2) is strengthened and then becomewe haveCorollary 3.3.1 Suppose the conditions (h1) (h3) (h4) holds, then the three-point boundary value problem (B) has a positive solution w(t)∈C¹(I)∩C²(J), which satisfiesCorollary 3.3.2 Let(h1')f(t,u):J×[0,+∞)→[0,+∞)is continuous and is decreasing with respect toThen the conclusions of Theorem 1.1 hold.Corollary 3.3.3 Suppose that in Corollary 3.2, (h1') holds, the condition (h2) isstrengthened and then becomeThen the positive solution w(t) of the problem (B) is a C¹(I) solution.Corollary 3.3.4 If f(t,u)∈C(I×[0,+∞),[0,+∞)) is decreasing with respectto u,and f(t,λ)≠0,(?)λ≥0.Then the problem (B) has a positive solution, and satisfies...