Existence of multiple positive solutions for p-Laplacian multipoint boundary value problems on time scales

Abdulkadir Dogan

doi:10.1186/1687-1847-2013-238

Abstract

In this paper, we consider p-Laplacian multipoint boundary value problems on time scales. By using a generalization of the Leggett-Williams fixed point theorem due to Bai and Ge, we prove that a boundary value problem has at least three positive solutions. Moreover, we study existence of positive solutions of a multipoint boundary value problem for an increasing homeomorphism and homomorphism on time scales. By using fixed point index theory, sufficient conditions for the existence of at least two positive solutions are provided. Examples are given to illustrate the results.MSC:34B15, 34B16, 34B18, 39A10.

Highlights

The theory of dynamic equation on time scales was initiated by Stefan Hilger in his PhD thesis in [ ] as a means of unifying structure for the study of differential equations in the continuous case and study of finite difference equations in the discrete case
P-Laplacian equations for boundary value problems (BVPs) with nonlinearity depending on the first order derivative have been studied extensively, see [ – ] and references therein
There are few papers concerning p-Laplacian equations with nonlinearity depending on the first order derivative for BVPs on time scales, see [, ]

Summary

Introduction

The theory of dynamic equation on time scales (or measure chains) was initiated by Stefan Hilger in his PhD thesis in [ ] (supervised by Bernd Aulbach) as a means of unifying structure for the study of differential equations in the continuous case and study of finite difference equations in the discrete case. ( ) We consider the existence of at least three positive solutions to the following p-Laplacian multipoint boundary value problem (BVP) on time scales φp u (t) ∇ + a(t)f t, u(t), u (t) = , t ∈ ( , T)T, m–. Little work has been done on the existence of positive solutions for multipoint BVP on time scales when the nonlinear term is involved in the first order derivative explicitly, see [ ]. Motivated by all the works above, our main results will depend on an application of a generalization of the Leggett-Williams fixed point theorem due to Bai and Ge. Here, the emphasis is that the nonlinear term is involved explicitly with the first order derivative. Much attention has been paid to the existence of positive solutions for nonlinear boundary value problems on time scales, see [ – , – ] and the reference therein. F has at least three different fixed points u , u and u in P (α, r ; β, l ) with u ∈ P(α, r ; β, l ), u ∈ P (α, r ; β, l ; ψ, b) : ψ(u) > b ; and u ∈ P (α, r ; β, l ) \ P (α, r ; β, l ; ψ, b) ∪ P (α, r ; β, l )

Let the Banach space

Take α

Take a Consider the following