Positive Solutions For A Fourth-order Three-point BVP With Sign-changing Green’s Function

Boundary value problems of fourth-order ordinary differential equations are mathematical models that are closely related to people’s life.For example,the d-ifferent stress conditions can be described by the different fourth-order two-point boundary value problems at both ends of the elastic beams in equilibrium.Peo-ple have paid much attention to the research of the fourth-order boundary value problems and have achieved many outstanding achievements.However,the sign of Green’s functions involved are almost changeless in existing work.This paper primarily studies the nonlinear fourth-order three-point boundary value problemwhere f:[0,1]x[0,+∞)→[0,+∞))is continuous and satisfies the following condi-tions:(Ⅰ)for any x ∈[0,+∞),the mappingt→f(t,x)is decreasing;(Ⅱ)for any t ∈[0,1],the mapping x∈f(t,x)is increasing.Although the corresponding Green’s function is sign-changing,we will still acquire the existence of positive solutions to the above boundary value problem if we set some suitable assumptions on f and η.In chapter 2,for arbitrary integer n(>2),we give a proof that the above problem has at least n—1 decreasing positive solutions,which is based on the fixed point index theory.In this chapter,η∈[1/3,1),which is the optimal condition that ensures the above boundary value problem has positive solutions on η.f need to satisfy the requirement:(Ⅲ)there exist n(≥2)positive constants ri,i=1,2,...,n with r1<r2<...<rn such that either(a)f(0,r2i-1)<(r2i-1)/H i=1,…,[(n+1)/2]and f(θ,θr2i)>r2i/Q,i=1,…,[n/2],or(b)f(θ,θr2i-1)>(r2i-1)/Q,i=1,...,[(n+1)/2]and f(0,r2i)<r2i/H i=1,...,[m/2].In chapter 3,we respectively define two iterative sequences that begin with zero function and positive constant function and discuss the existence of positive solutions to the above problem by applying to the fixed point theory in cones and iterative technique.In this chapter,η∈((2=3√2-3√4)/3.1),f requires to fulfill the assumptions:(Ⅳ)there exists positive constant r such that f(0,r)≤ 6r;(Ⅴ)there exist positive constants σ and μ with σ<μ≤ση3/2(1-η)3 such thatσ(u2-u1)≤ f(t,u2)-f(t,u1)≤μ(u2-u1),0≤t≤1,0≤u1≤u2≤r.