Multiple positive solutions for nonlinear high-order Riemann\u2013Liouville fractional differential equations boundary value problems with p-Laplacian operator

Bibo Zhou,Emmanuel Addai,Lingling Zhang,Nan Zhang

doi:10.1186/s13661-020-01336-1

Abstract

In this paper, we study the existence of multiple positive solutions for boundary value problems of high-order Riemann–Liouville fractional differential equations involving the p-Laplacian operator. Not only new existence conclusions of two positive solutions are obtained by employing functional-type cone expansion-compression fixed point theorem, but also some sufficient conditions for existence of at least three positive solutions are established by applying the Leggett–Williams fixed point theorem. In addition, we demonstrate the effectiveness of the main result by using an example.

Highlights

1 Introduction In this paper, we study the high-order Riemann–Liouville fractional differential equations with p-Laplacian operator as follows:
To the best of our knowledge, there are few studies that consider the existence of multiple positive solutions on nonlinear Riemann–Liouville high-order fractional differential equations, especially with the p-Laplacian operator
Motivated greatly by the above mentioned excellent works and in order to fill this gap in the literature, in this paper, we investigate the multiple positive solutions of boundary value problems for high-order Riemann–Liouville fractional differential equations with p-Laplacian operator

Summary

Dαt is the standard

To the best of our knowledge, there are few studies that consider the existence of multiple positive solutions on nonlinear Riemann–Liouville high-order fractional differential equations, especially with the p-Laplacian operator. Motivated greatly by the above mentioned excellent works and in order to fill this gap in the literature, in this paper, we investigate the multiple positive solutions of boundary value problems for high-order Riemann–Liouville fractional differential equations with p-Laplacian operator. 3, by means of the properties of the Green’s function, Leggett–Williams fixed point theorem, and functional-type cone expansion-compression fixed point theorem, we investigate multiple positive solutions for boundary value problem of Riemann–Liouville fractional differential p-Laplacian equation systems on n – 1 < α ≤ n, our work establishes some novel results on a nonlinear Riemann–Liouville fractional-order boundary value problem.

For β

Dαt φp

Dαt u

Dαt uj

If u