Novel results for Wardowski contraction principle in triple-controlled orthogonal $\mathcal{S} $-metric space with applications to fractional equations

We introduce triple-controlled orthogonal \(\mathcal{S} \) -metric type spaces involve triple auxiliary control functions β , μ , & γ , extending the classical controlled \(\mathcal{S} \) -metric type sitting. Furthermore, we formulate \((\alpha _{s},\bot ) \) -admissible function & strengthen Wardowski’s contraction principle, designing \((\alpha _{s}-\mathcal{A},\bot ) \) -contractive function fitted to the triple-controlled orthogonal structure. On complete spaces of this kind, we establish fixed-point theorems that ensure existence & uniqueness under natural conditions on the control functions and admissibility. As an application, we show that the main result guarantees a unique solution to a class of fractional differential equations.