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Existence of positive solutions of mixed fractional integral boundary value problem with p(t)-Laplacian operator Xiaosong Tang1

· Jieying Luo1 · Shan Zhou1 · Changyuan Yan1

Received: 10 July 2020 / Accepted: 14 October 2020 © Università degli Studi di Napoli "Federico II" 2020

Abstract In this paper, we investigate a mixed fractional integral boundary value problem with p(t)-Laplacian operator. Firstly, we derive the Green function through the direct computation and obtain the properties of Green function. For p(t) = constant, under the appropriate conditions of the nonlinear term, we establish the existence result of at least one positive solution of the above problem by means of the Leray–Schauder fixed point theorem. Meanwhile, we also obtain the positive extremal solutions and iterative schemes in view of applying a monotone iterative method. For p(t) = constant, by using Guo–Krasnoselskii fixed point theorem, we study the existence of positive solutions of the above problem. These results enrich the ones in the existing literatures. Finally, some examples are included to demonstrate our main results in this paper and we give out an open problem. Keywords Positive solution · Mixed fractional integral boundary value problem · p(t)-Laplacian operator · Fixed point theorem · Monotone iterative method Mathematics Subject Classification 26A33 · 34B10 · 34B18

1 Introduction Fractional differential equations have gained considerable attention, which can describe various phenomena in a variety of fields such as anomalous diffusion, physics, chemistry, mechanical, system control, biology, economics, and so on [1,2]. In recent years, we have to point out that there has been a significant theoretical development and application in fractional differential equations. Many scholars have studied the existence problem of solutions (or positive solutions), ultimate boundedness, oscillation properties, and stability and Hopf bifurcation for a variety types of fractional

B 1

Xiaosong Tang [email protected] School of Mathematics and Physics, Jinggangshan University, Ji’an 343009, Jiangxi, China

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X. Tang et al.

differential equations and obtained many meaningful results, see [3–13] and references cited therein. For example, Cabada and Wang [6] have discussed the following fractional integral boundary value problem: 

C D α u(t) + 0+

f (t, u(t)) = 0, 0 < t < 1, 1 u(0) = u (0) = 0, u(1) = λ 0 u(s)ds, 

where 2 < α ≤ 3, 0 < λ < 2, C D0α+ is a Caputo fractional derivative, f : [0, 1] × [0, +∞) → [0, +∞) is continuous. They obtained the existence of positive solutions for above problem by Guo–Krasnoselskii fixed point theorem. Liu and Jia [7] considered the following mixed fractional integral boundary value problem with p-Laplacian operator ⎧ α C β C β ⎪ ⎨ D0+ (ϕ p ( D0+ u(t))) = f (t, u(t), D0+ ), 0 < t < 1, C D β u(0) = u  (0) = 0, u(0) = 1 g (s)u(s)ds, 0 0 0+ ⎪ 1 ⎩ α−1 β β D0+ (ϕ p (C D0+ u(1))) = 0 g1 (s)ϕ p (C D0+ u(s))ds, β

where 1 < α, β ≤ 2, C D0+ is a Caputo fractional derivative and D0α+ is a Riemann– Li