Existence and Local Stability Results for First-Order Nonlinear Random Differential Equations via Random Fixed-Point Theorems

The study of random differential equations plays a crucial role in modeling dynamic systems influenced by uncertainty. Classical fixed point theorems have been extended to random settings to analyze the behavior of such systems. In this paper, we investigate the existence and local stability (attractivity) of solutions for a class of nonlinear first-order random differential equations defined on unbounded intervals of the real line. Using the framework of random fixed point theory in separable Banach spaces, we establish sufficient conditions that guarantee the existence of random solutions under appropriate measurability, continuity, and boundedness assumptions. We further demonstrate that these random solutions are locally attractive when certain contraction-type conditions are satisfied. The analysis unifies and generalizes several known results in the literature related to deterministic and random operator equations. The findings presented here provide a foundation for studying higher-order and perturbed random differential equations and may be applied to a wide range of problems in applied mathematics and engineering.