POSITIVE SOLUTIONS OF P-LAPLACIAN M-POINT BOUNDARY VALUE PROBLEMS ON TIME SCALES

Abstract

Let $\Bbb{T}$ be a time scale such that $0,T\in \Bbb{T},$ $a_i\geq 0$ for $% i=1,\ldots ,m-2.$ Let $\xi _i$ satisfy $0\lt \xi _1\lt \xi _2\lt \ldots \lt \xi _{m-2}\lt \rho (T)$ and $\sum_{i=1}^{m-2}a_i\lt 1.$ We consider the following $p$% -Laplacian $m$-point boundary value problem on time scales \begin{eqnarray*} \left( \varphi _p\left( u^\Delta (t)\right) \right) ^\nabla +a(t)f(t,u(t)) &\hspace{-0.2cm}=\hspace{-0.2cm}&0,t\in (0,T), \\ u^\Delta (0) &\hspace{-0.2cm}=\hspace{-0.2cm}&0,\text{ }u(T)=\sum_{i=1}^{m-2}a_iu(\xi _i), \end{eqnarray*} where $a\in C_{ld}[(0,T),\Bbb{[}0,\infty )]$ and $f\in C\left( (0,T)\times \Bbb{[}0,\infty ),\Bbb{[}0,\infty )\right) .$ Some new results are obtained for the existence of at least single, twin or triple positive solutions of the above problem by applying Krasnosel'skii's fixed point theorem, new fixed point theorem due to Avery and Henderson and Leggett-Williams fixed point theorem. In particular, our criteria extend and improve some known results.