MULTIPLE POSITIVE SOLUTIONS FOR p-LAPLACIAN FUNCTIONAL DYNAMIC EQUATIONS ON TIME SCALES

Abstract

In this paper we consider the following boundary value problems for $p$-Laplacian functional dynamic equations on time scales $$\begin{array}{rl} & \left[ \Phi _p(u^{\bigtriangleup }(t))\right]^{\bigtriangledown }+a(t)f(u(t),u(\mu (t)))=0,t\in \left( 0,T\right) _{\mathbf{T}},\\ u_0(t)& =\varphi (t), \, \, t\in \left[ -r,0\right] _{\mathbf{T}},\, \, u(0)-B_0(u^{\bigtriangleup }(\eta ))=0,\, \, u^{\bigtriangleup }(T)=0, \mbox{ or}\\ u_0(t)& =\varphi (t),\, \, t\in \left[ -r,0\right] _{\mathbf{T}},\, \, u^{\bigtriangleup }(0)=0,u(T)+B_1(u^{\bigtriangleup }(\eta ))=0. \end{array} $$ Some existence criteria of at least three positive solutions are established by using the well-known Leggett-Williams fixed-point theorem. An example is also given to illustrate the main results.