A General Framework for the Multiplicity of Positive Solutions to Higher-Order Caputo and Hadamard Fractional Functional Differential Coupled Laplacian Systems

This paper applies a general framework to explore the existence of multiple positive solutions for the fractional integral boundary value problem of high-order Caputo and Hadamard fractional coupled Laplacian systems with delayed or advanced arguments. We first focus on a generalized fractional homomorphic coupled boundary value problem with Hilfer fractional derivatives. Then we present the Green’s function corresponding to this Hilfer fractional system and its important properties. On this basis, by constructing a positive cone and applying a generalized cone fixed point theorem, we have established some novel criteria to ensure that the generalized fractional system has at least three positive solutions. As applications, we also obtain the multiplicity of the positive solutions of the Caputo and Hadamard fractional-order coupled Laplacian systems under two special Hilfer derivatives, respectively. Finally, we provide several examples to inspect the applicability of the main results.