Positive solutions for the nonhomogeneous three-point boundary value problem of second-order differential equations

This paper is concerned  with the existence of positive solutions to the nonhomogeneous three-point boundary value problem  of the second-order ordinary differential equation u″(t)+a(t)u′(t)+b(t)u(t)+h(t)f(u)=0u″(t)+a(t)u′(t)+b(t)u(t)+h(t)f(u)=0u(0)=0,u(1)−αu(η)=b where 0<η<10<η<1, 0<αη<10<αη<1 and b∈[0,∞)b∈[0,∞) are given. a(t)∈C[0,1]a(t)∈C[0,1], b(t)∈C([0,1],(−∞,0])b(t)∈C([0,1],(−∞,0]) and h(t)∈C([0,1],[0,∞))h(t)∈C([0,1],[0,∞)) satisfying that there exists x0∈[0,1]x0∈[0,1] such that h(x0)>0h(x0)>0, and f∈C([0,∞),[0,∞))f∈C([0,∞),[0,∞)). By applying Krasnosel’skii’s fixed-point theorem in Banach spaces, some sufficient conditions guaranteeing the existence of at least one positive solution if ff is either superlinear or sublinear are established for the above boundary value problem. The results obtained extend and complement some known results.