Positive Solutions of Boundary Value Problems

At the time of this book, there is considerable research devoted to positive solutions for boundary value problems in each of the areas of partial differential equations, ordinary differential equations, and finite difference equations. In view of the unification theory brought forth by dynamic equations on time scales, it is natural that some research has emerged in terms of positive solutions for boundary value problems for dynamic equations on time scales. In this chapter, we present a synopsis of some of this recent work in a number of venues, including positive solutions that arise as eigenfunctions in a positive cone associated with eigenvalue comparison results, existence of at least one positive solution for a nonlinear dynamic equation as an application of the Guo-Krasnosel’skii fixed point theorem, existence of at least two positive solutions as dual applications of the Guo-Krasnosel’ski fixed point theorem or in some cases of the Avery-Henderson fixed point theorem. Finally, we conclude with results concerning the existence of at least three positive solutions via applications of either the Leggett-Williams fixed point theorem or a generalization of this theorem due to Avery.