Non-concave utility maximization problem with transaction costs and a given consistent price system

We investigate expected utility maximization problems from the terminal liquidation value in continuous time in markets with transaction costs and one fixed consistent price system, where a non-concave utility function is defined on the positive half real line. The sufficient conditions are given by the convex conjugate of the utility function, then the existence of the optimizer is proved by a maximizing sequence. Finally, we show that the value function of the envelope of the utility function and the concave envelope of the value function are coincide.