Multiple positive solutions for mixed fractional differential system with p-Laplacian operators

Ying Wang

doi:10.1186/s13661-019-1257-2

Abstract

This paper is focused on researching a class of mixed fractional differential system with p-Laplacian operators. Based on the properties of the corresponding Green’s function, different combinations of superlinearity or sublinearity for the nonlinearities and other appropriate conditions, the existence of multiple positive solutions are derived via the Guo–Krasnosel’skii fixed point theorem. An example is then given to illustrate the usability of the main results.

Highlights

1 Introduction In this paper, we investigate the following mixed fractional differential system:
Fractional differential equation with p-Laplacian operator can describe the nonlinear phenomena in non-Newtonian fluids and establishes complex process models; for some related articles, see [25,26,27,28,29,30,31]
T(u, v) = max T1(u, v), T2(u, v) ≥ r2 = (u, v), for any (u, v) ∈ ∂Kr2 . (3.14). It follows from the above discussion, (3.5), (3.9), (3.14), Lemmas 2.5, 2.6, that T has fixed points (u1, v1) ∈ K r2 \Kr, (u2, v2) ∈ K r\Kr1, that is to say, system (1.1) has at least two positive solutions (u1, v1), (u2, v2), satisfying 0 < (u1, v1) < r1 < (u2, v2)

Summary

Introduction

We investigate the following mixed fractional differential system:. where 1 < βi ≤ 2, 1 ≤ n – 1 < α1 ≤ n, 1 ≤ m – 1 < α2 ≤ m, n, m ≥ 2, Dβi is the Riemann–. Fractional differential equation with p-Laplacian operator can describe the nonlinear phenomena in non-Newtonian fluids and establishes complex process models; for some related articles, see [25,26,27,28,29,30,31]. Wu et al [33] researched the following fractional differential turbulent flow model and obtained the iterative solutions of the equation:. Inspired by the above articles, in this article we discuss the mixed fractional differential system with p-Laplacian operators under integral boundary value conditions. Lemma 2.6 ([45]) Let K be a positive cone in a Banach space E, Ω1 and Ω2 are bounded open sets in E, θ ∈ Ω1, Ω1 ⊂ Ω2, T : K ∩ Ω2\Ω1 → K is a completely continuous operator. Tx ≥ x , ∀x ∈ K ∩ ∂Ω1, Tx ≤ x , ∀x ∈ K ∩ ∂Ω2, T has at least one fixed point in K ∩ (Ω2\Ω1)

Main results Denote f10

An example