Positive solutions to semi-positone second-order three-point problems on time scales

Using a fixed point theorem of generalized cone expansion and compression we establish the existence of at least two positive solutions for the nonlinear semi-positone three-point boundary value problem on time scales u(Delta del)(t) + lambda f(t, u(t)) = 0, u(a) = 0, alpha u(eta) = u(T). Here t is an element of [a, T](T), where T is a time scale, alpha > 0, eta is an element of (a,rho(T))(T), alpha(eta - a) < T - a, and the parameter lambda > 0 belongs to a certain interval. These results are new for difference equations as well as for general time scales. An example is provided for differential, difference, and q-difference equations. (C) 2009 Elsevier Inc. All rights reserved.