Existence and k-Mittag Leffler Ulam stabilities of implicit pantograph differential equation via \((k,\varrho )\)-Hilfer fractional derivative

In this paper, we aim to study the existence and two types of k-Mittag-Leffler Ulam stabilities for an implicit Pantograph fractional differential equation that incorporates the \((k,\varrho )\) $(k,\varrho )$ -Hilfer derivative. To establish the existence and uniqueness of solutions, we employ the Banach fixed-point theorem. Furthermore, we investigate the stability properties, specifically the k-Mittag-Leffler-Ulam-Hyers \(k-\mathcal {MLUH}\) $k-\mathcal{MLUH}$ stability and k-Mittag-Leffler-Ulam-Hyers-Rassias \(k-\mathcal {MLUHR}\) $k-\mathcal{MLUHR}$ stability, using the Grönwall-Bellman inequality. To validate our findings, we provide an illustrative example.