Analysis on Fractional-order Asymmetric games with varying interaction rates

Introduction:

Complex adaptive systems (CASs) consist of interacting adaptive agents whose collective behavior leads to emergent properties not found in individual components and not caused by external forces. Examples include the brain, immune system, and economy. Understanding such systems requires a holistic approach rather than traditional reductionism. Due to their memory and nonlocal behavior, fractional differential equations are well-suited for modeling CASs. However, studies on fractional-order CASs remain limited. In this paper, we investigate the existence, uniqueness, equilibrium points, and uniform stability of fractional-order differential equations in games with non-uniform interaction rates.

Method and results: Let exist and be bounded on D. Condition (2) implies that the functions fi satisfy the Lipschitz condition where and .

Theorem 1: (Existence and uniqueness) Let the assumptions (1)-(2) be satisfied. Then, the initial value problem has a unique solution .

Theorem 2: (Games with non-uniform interaction rates) Let . Then, the initial value problem has a unique solution .

Theorem 3: (Asymmetric games) The initial value problem with the initial data has a unique solution .

Conclusion: The results confirm solution reliability for diverse fractional game models with complex interaction structures.