An efficient iterative procedure in hyperbolic space and application to non-linear delay integral equation

In the context of hyperbolic spaces, our study presents a novel iterative approach for approximating common fixed points satisfying general contractive condition involving a pair of mappings with weak compatibility. Also, we notice that our iterative procedure approximates to a point of coincidence if the weak compatibility condition is violated. We provide theorems to demonstrate the \(\Delta -\) $\mathrm{\Delta }-$ convergence, stability, and efficiency of this iteration process. Additionally, we provided some immediate corollaries that involve mappings with contractive condition, instead of general contractive condition. Furthermore, we demonstrate with examples and graphs that our iteration process is faster than all previous procedures, including those of Jungck-SP, Jungck-CR, and Jungck-DK, utilizing MATLAB software. Also, we compare the impact of the initial values and the parameters on the convergence behavior of the proposed iterative process with existing iterative schemes using an example. Finally, we focus on using our iterative technique to approximate the solution of a non-linear integral equation with two delays.