Existence of positive solution and Hyers–Ulam stability for a nonlinear singular-delay-fractional differential equation

• Published in: Advances in difference equations 2019-03, Vol.2019 (1), p.1-13, Article 104
• Main Authors: Khan, Hasib, Abdeljawad, Thabet, Aslam, Muhammad, Khan, Rahmat Ali, Khan, Aziz
• Format: Article
• Language: English
• Subjects: Analysis; Caputo fractional derivative; Delay-Fractional differential equations with singularity; Difference and Functional Equations; Differential equations; EU of solution; Fixed points (mathematics); Functional Analysis; Mathematics; Mathematics and Statistics; Operators (mathematics); Ordinary Differential Equations; Partial Differential Equations; Riemann–Liouville derivative; Stability; Uniqueness
• ISSN: 1687-1847; 1687-1839; 1687-1847
• DOI: 10.1186/s13662-019-2054-z

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 ϕ p is more general than the one considered in (Khan et al. in Eur. Phys. J. Plus 133:26, 2018 )