Eigenvalue problems for second-order nonlinear dynamic equations on time scales

Abstract: This paper is concerned with the existence and nonexistence of positive solutions of the second-order nonlinear dynamic equation , , satisfying either the conjugate boundary conditions or the right focal boundary conditions , where a and f are positive. We show that there exists a such that the above boundary value problem has at least two, one and no positive solutions for , and , respectively. Furthermore, by using the semiorder method on cones of the Banach space, we establish an existence and uniqueness criterion for positive solution of the problem. In particular, such a positive solution of the problem depends continuously on the parameter λ, i.e., is nondecreasing in λ, and . [Copyright &y& Elsevier]