The existence of countably many positive solutions for nonlinear singular m-point boundary value problems on time scales

In this paper, by introducing a new operator, improving and generating a p-Laplace operator for some p>1, we study the existence of countably many positive solutions for nonlinear boundary value problems on time scales [φ(uΔ)]∇+a(t)f(u(t))=0,t∈[0,T]T,u(0)=∑i=1m−2αiu(ξi),uΔ(T)=0, where φ:R→R is the increasing homeomorphism and positive homomorphism and φ(0)=0. We show the sufficient conditions for the existence of countably many positive solutions by using the fixed-point index theory and a new fixed-point theorem in cones.