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.