On Greenʹs functions and positive solutions for boundary value problems on time scales

In this paper we offer a form of self-adjoint differential equations on time scales so that the associated Greenʹs function is found symmetric in the usual sense. For this purpose together with the delta derivative we employ the nabla derivative as well. We introduce the concepts of Lebesgue delta and nabla integrals on time scales. Next, sign properties of the Greenʹs function are investigated and existence results for positive solutions of nonlinear boundary value problems are established. Upper and lower bounds for these positive solutions also are given.