Viscoelastic Kelvin–Voigt model on Ulam–Hyer’s stability and T-controllability for a coupled integro fractional stochastic systems with integral boundary conditions via integral contractors

We discuss the existence, uniqueness, Ulam–Hyer’s stability, and Trajectory (T-) controllability for solutions of coupled nonlinear fractional order stochastic differential systems (FSDEs) with integral boundary conditions via integral contractors. Using Banach space, we obtain some relaxed conditions for existence and uniqueness for the mentioned problem via successive approximation techniques. Furthermore, to demonstrate the results, the concept of bounded integral contractors is combined with a fractional order coupled system using regularity conditions. The Green’s function is used to find the solution for the coupled system with boundary conditions. Through Integral Contractors (ICs), we examine the existence and uniqueness results for a higher-order nonlinear fractional coupled stochastic system with integral boundary conditions on Time Scales. Further, we develop some conditions for Ulam–Hyer’s stability for the non-linear fractional order coupled systems. To demonstrate our main result, we provide a proper example. The real-life application of a coupled fractional stochastic Kelvin–Voigt model on viscoelastic elastomer is investigated to justify the theoretical model with the numerical comparison with a single and a double fractional stochastic Kelvin–Voigt model.