Mingzhou Guo and Hong-Rui Sun Lanzhou University and Lanzhou University

Abstract : In this paper, we are concerned with the following eigenvalue problems of $m$-point boundary value problem for $p$-Laplacian dynamic equation on time scales \[ \left( \varphi _p(u^\Delta (t))\right) ^\nabla +\lambda h(t)f(u(t))=0,\text{ }t\in (0,T), \] \[ u(0)=0,\text{ }\varphi _p\left( u^\Delta (T)\right) =\sum_{i=1}^{m-2}a_i\varphi _p\left( u^\Delta (\xi _i)\right) , \] where $ \varphi _p(u)=|u|^{p-2} u, p>1$ and $\lambda >0$ is a real parameter. Under certain assumptions, some new results on existence of one or two positive solution and nonexistence are obtained for $\lambda$ evaluated in different intervals. Our work develop and improve many known results in the literature even for the continual case. In doing so the usual restriction that $f_0=\lim_{u\rightarrow 0^{+}}{f(u)}/{\varphi _p(u)}$ and $f_\infty =\lim_{u\rightarrow {\infty } }{f(u)}/{\varphi _p(u)}$ exist is removed. As an applications, an example is given to illustrate the main results obtained.

Keywords : eigenvalue, time scale, positive solution, fixed point