Qualitative results and Lyapunov-type inequality for a new φ -ABC fractional boundary value problem

This paper investigates a new category of a fractional boundary value problem (BVP) involving the generalized Atangana-Baleanu-Caputo (ABC) derivatives of order belonging to \((2,3].\) First, the Green function with its properties for a proposed fractional \(\mathrm{BVP}\) are derived. Then, the theoretical results are established by various techniques. The existence and uniqueness of theorems of the solutions are proved by utilizing the Weissinger, the Aghajani fixed point theorems, the Meir-Keeler condensing operator, and the Kuratowski's measure of non-compactness techniques. The sufficient conditions of the existence and non-existence results of nontrivial solutions for the proposed problem are investigated by introducing the Lyapunov-type inequality (\(\mathbb{LTI}\)). Finally, our findings are compared with the existing results in the literature. The validity of the main outcomes is also tested by numerical examples with graphs and tables, and the strict minimum borders of eigenvalues for several fractional \(\mathrm{BVP}s\) are estimated. This work is the first to deal with \(\mathbb{LTI}\) for fractional \(\mathrm{BVP}\) in the sense of generalized ABC derivatives.