Existence and uniqueness of equilibria in general-sum fractional differential games

Establish existence and uniqueness of equilibria for general-sum differential games in which the state evolution is described by Caputo fractional differential equations, including a precise specification of strategy spaces and equilibrium concepts compatible with memory.

Background

The survey extends dynamic programming and optimality principles to fractional systems and notes that many results are currently established for zero-sum settings.

General-sum games with memory raise additional challenges for equilibrium existence and uniqueness because the value functionals are path dependent and the Isaacs-type structure does not apply directly.

References

Open questions. Refinements are still needed on: (a) Isaacs–type conditions under memory; (b) existence/uniqueness for general–sum FO games; (c) verifiable sufficient conditions for Stackelberg equilibria in FDGs; and (d) scalable algorithms with a priori/a posteriori certificates that couple fractional–approximation error with game–theoretic suboptimality.

— Fractional Calculus in Optimal Control and Game Theory: Theory, Numerics, and Applications -- A Survey (2512.12111 - Mojahed et al., 13 Dec 2025) in Section 4, Subsection “Fractional Differential Games (FDGs)”, Open questions paragraph