## Multiplicity of positive solutions for Sturm-Liouville boundary value problems of fractional differential equations with

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Multiplicity of positive solutions for Sturm-Liouville boundary value problems of fractional differential equations with p-Laplacian Hongling Lu, Zhenlai Han* and Shurong Sun * Correspondence: [email protected] School of Mathematical Sciences, University of Jinan, Jinan, Shandong 250022, P.R. China

Abstract In this article, we investigate the Sturm-Liouville boundary value problems of fractional diﬀerential equations with p-Laplacian

β

D0+ (φp (Dα0+ u(t))) + f (t, u(t)) = 0, 0 < t < 1, ξ u(0) – ηu (0) = 0, γ u(1) + δ u (1) = 0,

Dα0+ u(0) = 0,

β

where 1 < α ≤ 2, 0 < β ≤ 1, Dα0+ , D0+ are the standard Caputo fractional derivatives, φp (s) = |s|p–2 s, p > 1, φp–1 = φq , 1/p + 1/q = 1, ξ , η, γ , δ ≥ 0, ρ := ξ γ + ξ δ + ηγ > 0, and f : [0, 1] × [0, +∞) → [0, +∞) is continuous. By means of the properties of the Green’s function, Leggett-Williams ﬁxed-point theorems, and ﬁxed-point index theory, several new suﬃcient conditions for the existence of at least two or at least three positive solutions are obtained. As an application, an example is given to demonstrate the main result. MSC: 34A08; 34B18; 35J05 Keywords: Sturm-Liouville boundary value problem; positive solution of fractional diﬀerential equation; Leggett-Williams ﬁxed-point theorem; ﬁxed-point index theory; p-Laplacian operator

1 Introduction During the past decades, much attention has been focused on the study of equations with p-Laplacian diﬀerential operator. The motivation for those works stems from the applications in the modeling of diﬀerent physical and natural phenomena: non-Newtonian mechanics [], system of Monge-Kantorovich partial diﬀerential equations [], population biology [], nonlinear ﬂow laws [], combustion theory []. There exist a very large number of papers devoted to the existence of solutions for the equation with p-Laplacian operator. The ordinary diﬀerential equation with p-Laplacian operator φp u (t) + f t, u(t) = ,

< t < ,

subject to various boundary conditions, has been studied by many authors, see [, ] and the references therein. ©2014 Lu et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Lu et al. Boundary Value Problems 2014, 2014:26 http://www.boundaryvalueproblems.com/content/2014/1/26

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The existence of positive solutions of the diﬀerential equation with p-Laplacian operator φp u (t) + q(t)f t, u(t), u (t) = ,

< t < ,

satisfying diﬀerent boundary conditions have been established by using ﬁxed-point theorems and monotone iterative technique, see [, ] and the references therein. In [], Hai considered the existence of positive solutions for the boundary value problem ⎧ ⎨(q(t)φ(u )) + λf (t, u(t)) = , r < t < R, ⎩au(r) – bφ – (q(r))u (r) = , cu(R) + dφ – (q(R))u (R) = , where φ(u

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