Multiple and unique nontrivial solutions for fractional differential equations with singular property and derivatives contained in the nonlinear term

In this article, a class of p -Laplacian fractional differential equations with multiple derivative terms in the nonlinearity is studied. First, the Green’s function is derived and its key properties are analyzed, enabling us to construct a suitable cone for further analysis. Subsequently, using the Avery-Peterson fixed point theorem, the existence theorem of multiple positive solutions is established by examining the action of the Green’s function on the defined cone. The presence of multiple derivative terms in the nonlinearity introduces significant analytical challenges, particularly in proving the continuity and equicontinuity of the associated operator in the space C i [0, 1], which we have successfully overcome. Additionally, the nonlinearity exhibits singularity, and this issue is addressed by imposing appropriate conditions. To further explore solution existence, the Leray-Schauder nonlinear alternative and the Schauder fixed point theorem are employed to ensure the existence of nontrivial solutions. Moreover, under a generalized Lipschitz condition, the uniqueness of the solution is proved via the Banach fixed point theorem. Finally, two illustrative examples are provided to demonstrate the validity of the theoretical results presented.