Convergence results for cyclic-orbital contraction in a more generalized setting with application

In the realm of double-controlled metric-type spaces, we investigated obtaining fixed points using the application of cyclic orbital contractive conditions. Diverging from conventional approaches utilized in standard metric spaces, our technique took a unique route due to the unique features of our structure. We demonstrated the significance of our outcomes through exemplary cases, clarifying the breadth of our results through comprehensive investigations. Significantly, our work not only improved and broadened earlier findings in the literature, but also offered unique notions that were discussed in our explanatory notes. Towards the end of our inquiry, we used insights obtained from previous discoveries to develop a second-order differential equation. This equation was an effective tool for dealing with the second class of Fredholm integral problems. In conclusion, this investigation extended our examination of double-controlled metric type spaces by providing new insights on fixed point theory, expanding on prior debates and building a substantial road towards solving a class of integral equations.