The present paper concerns with optimal control problems allowing for time inconsistent utility functions of mean-field FBSDEs with mixed initial-terminal conditions. Moreover, the control variable enters the diffusion coefficient, and the control domain is with nonconvexity. Via extended Ekeland’s variational principle as well as the reduction method, a general stochastic maximum principle is established in the framework of mean-field theory. Finally, a linear-quadratic example is worked out to illustrate the application of the results.
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