The existence of three positive solutions of mm-point boundary value problems for some dynamic equations on time scales

In this paper, we define a new operator which improves and generalizes a pp-Laplacian operator for some p>1p>1. By using this operator, we consider the existence of triple positive solutions of mm-point boundary value problems for some dynamic equations on time scales [φ(p(t)uΔ(t))]∇+a(t)f(u(t))=0,t∈[0,T]Tκ∩Tk,u(0)−B0(∑i=1m−2αiuΔ(ξi))=0,uΔ(T)=0, where φ:R→Rφ:R→R is an increasing homeomorphism and positive homomorphism and φ(0)=0φ(0)=0. We show the existence of at least three positive solutions with suitable growth conditions imposed on the nonlinear term by using a new fixed-point theorem.