On Existence and Locally Attractivity Results for Fractional Order Nonlinear Random Integral Equation

In this paper, we investigate the existence and qualitative behaviour of solutions for a class of nonlinear random integral equation of fractional order in R_+=[0,∞). The analysis is carried out within the framework of Banach algebra, employing a hybrid fixed-point theorem as the principal tool. The problem is considered under the assumptions of Lipschitz continuity and Caratheodory conditions, which ensures the measurability and continuity properties required for the existence of random solutions. In addition to proving the existence of such solutions, we establish their local attractivity, thereby demonstrating the stability of the system in a probabilistic sense. Also we have proved existence of extremal solutions. The theoretical results presented in this work contribute to the growing field of fractional calculus and stochastic analysis by providing rigorous framework for studying fractional random integral equations. To illustrate the applicability of the main results, we provide a concrete example that verifies the theoretical findings and highlights the practical relevant of the proposed approach.