Existence, uniqueness, and stability analysis of fractional order singular boundary value problems

This paper is concerned with a class of singular multi-point boundary value problems (BVPs) subject to integral Riemann–Stieltjes boundary conditions, involving the $\varpi$-Caputo fractional derivative and a nonlinear $p$-Laplacian operator. The analysis is performed in the range $1<p\le 2$. To support the theoretical framework, suitable estimates for the Green's functions arising in the integral formulation of the problem are derived. The existence of at least one solution is further obtained through the application of Schaefer's fixed-point (FP) theorem. The uniqueness of solutions to the associated nonlinear $\varpi$-Caputo fractional differential equation is established by means of the Banach contraction principle. In addition, the stability of solutions is investigated in the sense of both Ulam–Hyers and Ulam–Hyers–Rassias. As a result, the study provides a comprehensive treatment of the existence, uniqueness, and stability properties for the considered singular multi-point fractional BVP with integral Riemann–Stieltjes conditions. The theoretical results are complemented by two examples that illustrate the applicability of the developed framework.