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 p-Laplacian dynamic equation on time scales (phi(P)(u(Delta)(t)))(V) a(t)f (t, u Delta(t), (t)) = 0, t is an element of (0,T)(T), subject to the multipoint boundary condition, u(0) = Sigma(m-2)(i=1) alpha iu, (xi(i)) = u Delta(T) = 0, where phi(p)(S) = vertical bar s vertical bar p(-2) s, p > 1, xi i is an element of [0, T](T,) 0 < xi 1 < xi 2 < . . . < xi m-2 < p(T). By using fixed point theorems, we prove the existence of at least three nonnegatvie 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.