EIGENVALUE PROBLEMS FOR p-LAPLACIAN DYNAMIC EQUATIONS ON TIME SCALES

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 $(\varphi_p(u^{\Delta}(t)))^\nabla+{\lambda}h(t)f(u(t))=0,\;t\in(0,T)$, $u(0)=0,\varphi_p(u^{\Delta}(T))=\sum\limits_{i=1}^{m-2}a_i\varphi_p(u^{\Delta}(\xi_i))$, 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.