Exploring the solutions of Hilfer delayed Duffing problem on the positive real line

In this article, we focus on studying the Duffing problem with the time delay of pantograph type via the Hilfer fractional derivatives on the infinite interval $(0,\infty )$ . An appropriate Banach space supported with the Bielecki norm in the Mittag–Leffler function sense is introduced for new and convenient analysis. The existence and uniqueness ( $\mathbf{E\&U}$ ) of the solutions are proved by utilizing the classical fixed point theorems (FPTs). Moreover, the Hyers–Ulam (HU) stability is discussed for our Hilfer fractional Duffing pantograph system (HFDPS). Ultimately, our results are enhanced by providing numerical examples with graphics simulations to check the validity of the main outcomes.