The multiple positive solutions for p-Laplacian multipoint BVP with sign changing nonlinearity on time scales

In this paper, we consider the following p-Laplacian multipoint boundary value problem on time scales:(ϕp(uΔ(t)))∇+a(t)f(t,u(t))=0,t∈[0,T]T,ϕp(uΔ(0))=∑i=1n−2aiϕp(uΔ(ξi)),u(T)=∑i=1n−2biu(ξi),where ϕp(s)=|s|p−2sϕp(s)=|s|p−2s, p>1p>1, ξi∈[0,T]Tξi∈[0,T]T, 0<ξ1<ξ2<⋯<ξn−2<ρ(T)0<ξ1<ξ2<⋯<ξn−2<ρ(T). By using fixed point index, we provide some sufficient conditions for the existence of multiple positive solutions to the above boundary value problem. Especially, the nonlinear term f(t,u)f(t,u) is allowed to change sign. As an application, an example is given to demonstrate our result.