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The objective of the present paper is to investigate the solution of fullycoupled mean-field forward-backward stochastic differential equations (FBSDEsin short) and to study the stochastic control problems of mean-field type aswell as the mean-field stochastic game problems both in which state processesare described as FBSDEs. By combining classical FBSDEs methods introduced by Huand Peng [Y. Hu, S. Peng, Solution of forward-backward stochastic differentialequations, Probab. Theory Relat. Fields 103 (1995)] with specific arguments forfully coupled mean-field FBSDEs, we prove the existence and uniqueness of thesolution to this kind of fully coupled mean-field FBSDEs under a certain\textquotedblleft monotonicity" condition. Next, we are interested in optimalcontrol problems for (fully coupled respectively) FBSDEs of mean-field typewith a convex control domain. Note that the control problems are timeinconsistent in the sense that the Bellman optimality principle does not hold.The stochastic maximum principle (SMP) in integral form for mean-fieldcontrols, which is different from the classical one, is derived, specifying thenecessary conditions for optimality. Sufficient conditions for the optimalityof a control is also obtained under additional assumptions. Then we areconcerned the maximum principle for a new class of non-zero sum stochasticdifferential games. This game system differs from the existing literature inthe sense that the game systems here are characterized by (fully coupledrespectively) FBSDEs in the mean-field framework. Our paper deduces necessaryconditions as well as sufficient conditions in the form of maximum principlefor open equilibrium point of this class of games respectively.

Stochastic Maximum Principle for Mean-field Controls and Non-Zero Sum Mean-field Game Problems for Forward-Backward Systems