Positive Solutions of a Singular Fractional Boundary Value Problem with r-Laplacian Operators

Alexandru Tudorache,Rodica Luca

doi:10.3390/fractalfract6010018

Abstract

We investigate the existence and multiplicity of positive solutions for a system of Riemann–Liouville fractional differential equations with r-Laplacian operators and nonnegative singular nonlinearities depending on fractional integrals, supplemented with nonlocal uncoupled boundary conditions which contain Riemann–Stieltjes integrals and various fractional derivatives. In the proof of our main results we apply the Guo–Krasnosel’skii fixed point theorem of cone expansion and compression of norm type.

Highlights

Positive Solutions of a SingularWe consider the system of fractional differential equations with r1 -Laplacian and r2 -Laplacian operatorsD γ1 φr D δ1 u(τ )= f τ, u(τ ), v(τ ), I0σ+ u(τ ), I0σ+v(τ ), τ ∈ (0, 1), (1) ς2D γ2 φr D δ2 v(τ )= g τ, u(τ ), v(τ ), I0ς+ u(τ ), I0+
By using the Leggett–Williams fixed-point theorem, the authors studied in [8] the multiplicity of positive solutions for a Riemann–Liouville fractional differential equation with a p-Laplacian operator, subject to four-point boundary conditions
We present the Guo–Krasnosel’skii fixed point theorem, which we will use in the proofs of our main results

Summary

Introduction

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. By relying on the properties of the Kuratowski noncompactness measure and the Sadovskii fixed point theorem; in [6], the authors obtained new existence results for the solutions of a Riemann–Liouville fractional differential equation with a p-Laplacian operator in a Banach space, supplemented with multi-point boundary conditions with fractional derivatives. By using the Leggett–Williams fixed-point theorem, the authors studied in [8] the multiplicity of positive solutions for a Riemann–Liouville fractional differential equation with a p-Laplacian operator, subject to four-point boundary conditions. By applying the Guo–Krasnosel’skii fixed point theorem the authors investigated in [10] the existence, multiplicity and the nonexistence of positive solutions for a mixed fractional differential equation with a generalized p-Laplacian operator and a positive parameter, supplemented with two-point boundary conditions.

Preliminary Results

Existence of Positive Solutions

Examples

Conclusions