The Existence Of Positive Solutions To Boundary-Value Problem Of Ordinary Differential Systems

This paper is a survey on the studies of BVP of Ordinary Differential Systems. In this paper. We focus on the existence of positive solutions to boundary-value problem of Ordinary Differential Systems as well as some specific applications and the implications of the research . The paper is divided into three chapters. The main contents are as follows:In Chapter one, we mainly introduce the basic application background and the general situation of domestic and foreign research, Finally, We also give the outline to this paper and state some fundamental lemmas and definitions we will introdue.In Chapter two, we introduce the existence of positive solutions to Boundary-Value problem of first-order and higher-order Ordinary Differential Systems. Firstly. we consider that the following problemsThe main results are Theorem 1. Suppose (?),andIf one of the following conditions is satisied:then Problem (1) has at least one positive solution .Next, we introduce some results including the existence of positive solutions to boundary-value problem of ordinary differential systems by applying the degree theory, fixed point index theory in a cone, theorems for calculus of variations. we will provide the basis for the main results of the following problems:where F_i :(?) ,i=1, 2 are continuous.Theorem 2. If one of the following conditions holds: then Problem, (2)has at least one positive solution.In general.we discuss the boundary value problem for systems of nonlinear ordinary differential equations by using a topological method.Futher,we list some results obtained by fixed point index theory in a cone and theorems for calculus of variations, the problem we will consider is: (H₁) There exists p∈(0,1], such thatFor all t∈[0,1];(H₂) There exists q∈(0, +∞), such thatFor all t∈[0,1];(H₃) There exists r∈(0. +∞), such that(H₄) There exists r∈(0.1), such thatFor all t∈[0,1];(H₅) f(t,u) and g(t, u) are increasing with u,and there exists a positive constant N such thatFor above problem, the main results are: Theorem 3. suppose (H₁)and(H₂)hold, then Problem(3)has at least one strict positive solutionTheorem 4. suppose (H₃)and(H ₄)hold, then problem(3) has at least one strict positive solutionTheorem 5. suppose (H₁), (H₄ ) and (H₅) hold, then problem (3) has at least two strict positive solutionsThereafter, we state some existence results on periodic solutions for second order quasilinear ordinary differential systems of boundary value problems. we consider the problemwhere p(t)∈C([0. T], R¹), T > 0, F : [0, T]×R^N→R¹,For above problem, the main results are:Theorem 6. Suppose F satisfies the following conditions:(A₁) F(t,x) = F(t +T,x), (?)F(t,x) is continous for each t∈[0,T] andx∈R^N, F(0. 0) = 0,(?) for all :r∈R^N;(A₂) there existβ>p⁺ and (?), such that(?) for |x| >γ₁, where (?, ?) is the usual inner product of R^N and p⁺ =(?);(A₃) there existμ> p⁺ and g∈C([0.T],R¹) such that(A₄) (?), for all t∈R¹ then Problem, (4) has at least one periodic solution.Theorem 7. In addition to (A₁) - (A₄)in Theorem.6, we suppose that F satisfies F(t,x) = F(t,-x), for all (t,x)∈R¹Ã—R^N , then Problem (4) has infinite periodic solutionFinally,we list some results on positive solution for a coupled system of second and fourth order ordinary differential equations:where f,g satisfy the following conditions : (F₁) f(t), g(t) : R→R arc continous: (F₂) f(0),g(0)≥0,t(t),g(t) arc nondescending in [0,+∞).The results of the existence of positive solution are :Theorem 8. Suppose f,g satisfy (F₁),(F₂), and the following conditionshold: then Problem(5) has at least one positive solution. Theorem 9. Suppose f, g satisfy (F₁),(F₂), and the following conditions hold: (1) g(t) is nonincreasing in ( -∞,0]; (2) for all c > 0,(?) ; (3) g(0) > 0,and f(t) > 0, for t > 0; then Problem(5)has at least one positive solution.In Chapter three, we state some specific applications in agriculture, chemistry,physics, archeology, architecture and so on. At last, we propose that the existence of positive solutions to boundary value problems for higher order differential systems should be considered to be a very interesting topics and with great prospects .