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

In this paper, we define a new operator which improves and generalizes a $p$-Laplacian operator for some $p>1$. By using this operator, we consider the existence of triple positive solutions of $m$-point boundary value problems for some dynamic equations on time scales $\begin{array}{c}{\phi \left(p\left(t\right)u^{\Delta }\left(t\right)\right)]}^{?}+a\left(t\right)f\left(u\left(t\right)\right)=0\text{,}t\in {0,T]}_{T^{\kappa }\cap T_k}\text{,}\hfill \\ u\left(0\right)?B_0\left(\sum _{i=1}^{m?2}{\alpha }_i{u}^{\Delta }\left({\xi }_i\right)\right)=0\text{,}{u}^{\Delta }\left(T\right)=0\text{,}\hfill \end{array}$ where $\phi :R\to R$ is an increasing homeomorphism and positive homomorphism and $\phi \left(0\right)=0$. 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.