Unbounded positive solutions for m-point time-scale boundary value problems on infinite intervals

In this work, we consider the following second-order m-point boundary value problem on time scales $$\left\{\begin{array}{@{}l}(\phi_{p}(u^{\triangle}(t)))^{\nabla}+h(t)f(t,u(t),u^{\triangle }(t))=0,\quad t\in(0,+\infty)_{\mathbb{T}},\\4pt]\displaystyle u(0)=\sum_{i=1}^{m-2}\alpha_{i}u(\eta_{i}),\qquad u^{\triangle}(+\infty)=\sum_{i=1}^{m-2}\beta_{i}u^{\triangle}(\eta_{i}).\end{array}\right.$$ We establish new criteria for the existence of at least three unbounded positive solutions. Our results are new even for the corresponding differential $({\mathbb{T}}={\mathbb{R}})$ , difference equation $({\mathbb{T}}={\mathbb{Z}})$ and for the general time-scale setting. An example is given to illustrate our results.