On deformable fractional order implicit differential equations involving orthogonal super metric spaces

In this manuscript, we use the notion of orthogonality in a super metric space and prove some fixed point theorems that focus on orthogonal contractions and orthogonal $\mathcal{F}$-contractions, which are types of mappings exhibiting particular contraction properties while satisfying orthogonality. To illustrate and support our theoretical results, we provide concrete examples that demonstrate the application of these findings. Furthermore, we show how our results can be utilized in practical scenarios by presenting a solution to a deformable implicit differential equation, highlighting the relevance of our work in both theoretical and applied contexts.