A Coupled System of p-Laplacian Langevin Equations with ψ-Hilfer Fractional Derivatives and Lebesgue–Stieltjes Integral Boundary Conditions in Weighted Banach Spaces

Source abstract

We study a nonlinear system of two Langevin differential equations, each governed by ψ-Hilfer fractional derivatives acting through a p-Laplacian and coupled to one another both through the nonlinearities and through the boundary data. The latter consist of antiperiodic relations connecting an interior node η∈(a,b) with the endpoint b, alongside Lebesgue–Stieltjes integrals in which the terminal value of each unknown is expressed through the other. An integral relation, in which every constant generated by the successive inversion of the two ψ-Hilfer operators is computed explicitly, is derived and proved to be equivalent to the boundary value problem, leading to a coupled Hammerstein-type fixed-point formulation. Since ψ-Hilfer solutions may develop an algebraic singularity at the left endpoint, the natural functional framework is a product of weighted Banach spaces, and all estimates are performed in the associated weighted norms, with the convexity of the p-Laplacian entering the computations at several key points. Viewing the resulting solution operator as one completely continuous map, we obtain at least one solution from Schaefer’s fixed-point theorem, along with an explicit a priori radius. Two worked examples close the paper: one with p=2 and one with p=32. In each, the hypotheses of the main theorem are verified explicitly, and every constant they require is evaluated numerically.