Abstract
In this article, we consider a study of a general class of nonlinear singular fractional DEs with p-Laplacian for the existence and uniqueness (EU) of a positive solution and the Hyers–Ulam (HU) stability. To proceed, we use classical fixed point theorem and properties of a p-Laplacian operator. The fractional DE is converted into an integral alternative form with the help of the Green’s function. The Green’s function is analyzed as regards its nature and then, with the help of a fixed point approach, the existence of a positive solution and uniqueness are studied. After the EU of a positive solution, the HU-stability and an application are considered. The suggested singular fractional DE with phi _{p} is more general than the one considered in (Khan et al. in Eur. Phys. J. Plus 133:26, 2018)
Highlights
1 Introduction Fractional order models have attracted the attentions of researchers of various discipline, in the last two decades
For instance; Li [18] studied a fractional DE for the existence and uniqueness (EU) of positive solutions having integral boundary conditions and the nonlinear p-Laplacian operator
Proof By Theorem 3.2 and Definition 4.1, we assume that x(t) is a solution of the fractional DE with delay (3.1) and y(t) is an approximate solution and satisfying (4.2)
Summary
Fractional order models have attracted the attentions of researchers of various discipline, in the last two decades. Bai and Qiu [9] established EU of solutions for a nonlinear singular boundary value problem (BVP) of fractional DEs with the help of the Leray–Schauder and Krasnosel’skii’s fixed point theorems. They provided some applications to illuminate the results. For instance; Li [18] studied a fractional DE for the EU of positive solutions having integral boundary conditions and the nonlinear p-Laplacian operator. We are involved in the study of EU of solutions and stability for the fractional DEs with operator p-Laplacian relating time and space singularity and τ > 0 delay.
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