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Existence of solutions for a nonlinear fractional p-Laplacian boundary value problem I. Merzoug1,2 · A. Guezane-Lakoud1

· R. Khaldi1

Received: 2 May 2019 / Accepted: 1 October 2019 © Springer-Verlag Italia S.r.l., part of Springer Nature 2019

Abstract The main objective of this paper is to prove the existence of solutions for a fractional pLaplacian boundary value problem containing both left Riemann–Liouville and right Caputo fractional derivatives. The proofs are based on the upper and lower solutions method and Schauder’s fixed point theorem. The paper is ended by a numerical example. Keywords Fractional p-Laplacian · Boundary value problem · Method of upper and lower solutions · Existence of solutions Mathematics Subject Classification 34B15 · 34A08 · 26A33

1 Introduction Differential equations are an important tool for describing real problem models, such as modern physics, engineering, and various other scientific fields. Nowadays, the study of the qualitative and quantitative properties of solutions to the boundary value problems of fractional order has attracted the attention of many authors [1,5–11,15,22,26]. However, there are few studies dealing with fractional differential equations with the p-Laplacian operator, despite their importance in theory and their various applications in mathematics, analyzing mechanics, physics, dynamical systems …, [2–4,12–14,16,17,19–21,25–27]. To study turbulent flow in a porous medium, Leibenson [19] introduced for the first time a p-Laplacian differential equation, modeling this fundamental mechanics problem in the existence of solutions of the following p-Laplacian differential equation:

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A. Guezane-Lakoud [email protected] I. Merzoug [email protected] R. Khaldi [email protected]

1

Laboratory of Advanced Materials, Faculty of Sciences, Badji Mokhtar - Annaba University, P.O. Box 12, 23000 Annaba, Algeria

2

Laboratory of Numerical Analysis, Optimisation and Statistics, Badji Mokhtar - Annaba University, Annaba, Algeria

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I. Merzoug et al.



  φ p u  (t) = f (t, u (t)) , 0 ≤ t ≤ 1,

where φ p (s) is the p-Laplacian operator. In [27], Xu and O’Regan studied the existence of positive solutions to the following fractional p-Laplacian boundary value problem, by using the monotone iterative method and the fixed point index theory in cones:   β  D0+ φ p D0α+ u (t) = f (t, u (t)) , 0 ≤ t ≤ 1, u (0) = u  (0) = 0, u  (1) = au  (ξ ),

D0α+ u(0)

= 0, D0α+ u(1) = bD0α+ u(η). β

where 2 < α < 3, 1 < β < 2, D0α+ and D0+ are Riemann–Liouville fractional derivatives of order α and β respectively, φ p (s) = s |s| p−2 , p > 1, 0 < ξ, η < 1, 0 ≤ a < ξ 2−α , 0 ≤ 1−β

b < η p−1 and f ∈ C([0, 1] × R+ , R+ ). Chai [3] studied the existence and multiplicity of solutions for   β  D0+ φ p D0α+ u (t) + f (t, u (t)) = 0, 0 ≤ t ≤ 1, γ

u (0) = 0, u(1) + σ D0+ u(1) = 0,

D0α+ u(0) = 0,

by the help of a fixed point theorem on cones. Applying the lower and upper solutions method and the Schauder fixed point theorem, Khaldi and Guezane–Lakoud [16], proved the exi