Triple positive solutions for some second-order boundary value problem on a measure chain

In this paper, we study the existence of three positive solutions for the second-order two-point boundary value problem on a measure chain, xΔΔ(t)+p(t)f(t,x(σ(t)),xΔ(t))=0,t∈[t1,t2],a1x(t1)−a2xΔ(t1)=0,a3x(σ(t2))+a4xΔ(σ(t2))=0, where f:[t1,σ(t2)]×[0,∞)×R→[0,∞)f:[t1,σ(t2)]×[0,∞)×R→[0,∞) is continuous and p:[t1,σ(t2)]→[0,∞)p:[t1,σ(t2)]→[0,∞) a nonnegative function that is allowed to vanish on some subintervals of [t1,σ(t2)][t1,σ(t2)] of the measure chain. The method involves applications of a new fixed-point theorem due to Bai and Ge [Z.B. Bai, W.G. Ge, Existence of three positive solutions for some second order boundary-value problems, Comput. Math. Appl. 48 (2004) 699–707]. The emphasis is put on the nonlinear term ff involved with the first order delta derivative xΔ(t)xΔ(t).