Existence of Three Positive Solutions to Some p-Laplacian Boundary Value Problems

e study of dynamic equations on time scales goes back to the 1989 Ph.D. thesis of Hilger [1, 2] and is currently an area of mathematics receiving considerable attention [3– 7]. Although the basic aim of the theory of time scales is to unify the study of differential and difference equations in one and the same subject, it also extends these classical domains to hybrid and in-between cases. A great deal of work has been done since the eighties of the XX century in unifying the theories of differential and difference equations by establishing more general results in the time scale setting [8–12]. Boundary value pp-Laplacian problems for differential equations and �nite difference equations have been studied extensively (see, e.g., [13] and references therein). Although many existence results for dynamic equations on time scales are available [14, 15], there are not many results concerning pp-Laplacian problems on time scales [16–19]. In this paper we prove new existence results for three classes of pp-Laplacian boundary value problems on time scales. In contrast with our previous works [17, 18], which make use of the �rasnoselskii �xed point theorem and the �xed point index theory, respectively, here we use the Leggett-Williams �xed point theorem [20, 21] obtainingmultiplicity of positive solutions.e application of the Leggett-Williams �xed point theorem for proving multiplicity of solutions for boundary value problems on time scaleswas �rst introduced byAgarwal andO’Regan [22] and is now recognized as an important tool to prove existence of positive solutions for boundary value problems on time scales [23–28]. e paper is organized as follows. In Section 2 we present some necessary results from the theory of time scales (Section 2.1) and the theory of cones in Banach spaces (Section 2.2). We end Section 2.2 with the Leggett-Williams �xed point theorem for a cone-preserving operator, which is our main tool in proving existence of positive solutions to the boundary value problems on time scales we consider in Section 3. e contribution of the paper is Section 3, which is divided into three parts.e purpose of the �rst part (Section 3.1) is to prove existence of positive solutions to the nonlocal pp-Laplacian dynamic equation on time scales