The authors are concerned with the second-order nonlinear dynamic equation on time scales
satisfying either the conjugate boundary conditions or the right focal boundary conditions , where and are positive. The number of positive solutions of the above boundary value problem for belonging to the half-line and the dependence of positive solutions of the problem on the parameter are discussed. It is proved that there exists a such that the problem has at least two, one and no positive solution(s) for and , respectively.
The main tool is a fixed-point index theorem on cones due to Guo-Lakshmikantham. Furthermore, by using the semi-order method on cones of Banach space, an existence and uniqueness criterion for a positive solution of the problem is established. In particular, such a positive solution of the problem depends continuously on the parameter , i.e., is nondecreasing in , and .