A simple interpolative contraction approach for analyzing existence and uniqueness of solutions in the fractional-order King Cobra model

This manuscript introduces advanced versions of interpolative type contractions together with a new class of simple ΘB functions that focus solely on the first property while excluding the aspects of the second and third properties to enrich the conceptual foundation. Our analysis includes several functions that belong to the class of simple ΘB contractions. Concrete examples with graphical visualization are presented that strengthen our main interpolative fixed point theorems. As an application, we examine the existence and uniqueness of the fractional-order King Cobra model solutions using the Atangana–Baleanu–Caputo (ABC) derivative through interpolative ΘB-Reich–Rus–Ćirić contraction. A combination of two-step Lagrange polynomial techniques along with ABC derivatives leads to numerical approximations, which are subsequently used to generate graphical simulations showing dynamic behaviors of fractional-order King Cobra models. The calculation of Lyapunov exponents reveals the chaotic nature by detecting positive exponents, which demonstrates initial condition sensitivity. This research employs fixed point theory, chaos theory, and fractional calculus to showcase that these approaches provide a central role in modeling and analyzing dynamical systems.