Nonlinear analysis for Hilfer fractional differential equations

In this paper, we discuss nonlinear Hilfer fractional differential equations with separated boundary conditions. Using the well-known Leggett–Williams theorem, we first explore the existence of multiple positive solutions for the nonlinear Hilfer fractional boundary value problem. Subsequently, we use nonlinear analytical methodologies to study the conditions for the Hilfer fractional differential equations involving separated boundary conditions with establishing four different types of Ulam’s stability: Ulam–Hyers stability, generalized Ulam–Hyers stability, Ulam–Hyers–Rassias stability, and generalized Ulam–Hyers–Rassias stability. To support our main results, we construct a numerical example.