Analytical Study of Impulsive Hilfer-Type Fractional p-Laplacian Problems Using Neural Networks and Finite-Difference Methods

We consider an impulsive BVP related to the Hilfer fractional derivatives and the nonlinear p-Laplacian operator. The type parameter ϑ∈[0,1] is kept unchanged in the formulation, and the Riemann–Liouville and Caputo cases are obtained as limiting cases of the formulation, not as separate cases. The variational functional is then built by adding the point-impulse contribution to the distributed potential and the use of an appropriate space of the Hilfer fractional derivative. Using variants of the fountain theorem, we prove the existence of two infinite sequences of weak solutions, one of which is of unbounded energy and another of which is of small energy and tends to zero from below. The weak residual based stability analysis is further developed, and local generalized Hyers–Ulam and Hyers–Ulam–Rassias stability estimates are obtained. Because of multiplicity of solutions, a uniqueness-based argument for stability, Ulam’s approach, is not possible and stability is instead achieved by providing residual-based arguments.The assumptions are verified through illustrative examples. Lastly, we examine the convergence behavior, residual decay, and effect of the Hilfer type parameter in conjunction with a Hilfer-type parameter neural surrogate with boundary constraints based on a discrete Hilfer scheme. The study, in general, proves a link between the solution multiplicity, residual stability, and the numerical realization in one impulsive fractional p-Laplacian framework.