Existence of positive solutions for p-Laplacian an m-point boundary value problem involving the derivative on time scales

Abstract

We are interested in the existence of positive solutions for the -Laplacian dynamic equation on time scales, $$ (\phi_p(u^\Delta(t)))^\nabla+a(t)f(t,u(t),u^\Delta(t))=0,\quad t\in(0,T)_{\mathbb{T}}, $$ subject to the multipoint boundary condition, $$ u(0)=\sum_{i=1}^{m-2}\alpha_i u(\xi_i), \quad u^\Delta(T)=0, $$ where $\phi_p(s)=|s|^{p-2} s$, $p>1$, $\xi_i\in [0,T]_{\mathbb{T}}$, $ 0<\xi_1<\xi_2<\dots<\xi_{m-2}<\rho(T)$. By using fixed point theorems, we prove the existence of at least three non-negatvie solutions, two of them positive, to the above boundary value problem. The interesting point is the nonlinear term $f$ is involved with the first order derivative explicitly. An example is given to illustrate the main result.

Keywords

• Time scales
• boundary value problem
• p-Laplacian
• positive solution
• fixed point theorem