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The present paper concerns with a near-optimal control problem for systems governed by mean-field forward-backward stochastic differential equations (FBSDEs) with mixed initial-terminal conditions. Utilizing Ekeland’s variational principle as well as the reduction method, the necessary and sufficient near-optimality conditions are established in the form of Pontryagin’s type. The results are obtained under restriction on the convexity of the control domain. As an application, a linear-quadratic stochastic control problem is solved explicitly.

By means of the method of quasi-lower and quasi-upper solutions and monotone iterative technique, we consider the nonlinear boundary value problems with Caputo fractional derivative and introduce two well-defined monotone sequences of quasi-lower and quasi-upper solutions which converge uniformly to the actual solution of the problem, and then the existence results of the solution for the problems are established. A numerical iterative scheme is introduced to obtain an accurate approximate solution and to give one example to demonstrate the accuracy and efficiency of the new approach.

In this paper, a new abstract result is given to verify the continuity of exponential attractors with respect to a parameter for the underlying semigroup. We do not impose any compact embedding on the main assumptions in the abstract result which is different from the corresponding result established by Efendiev et al. in 2004. Consequently, it can be used for equations whose solutions have no higher regularity. As an application, we prove the continuity of exponential attractors in H01 for a class of nonclassical diffusion equations with initial datum in H01.