In the paper, we study a differential inclusion with a given continuous convex multivalued mapping. For a given finite time interval, it is required to construct a solution of the differential inclusion, that satisfies the given initial condition or both the initial and final conditions. With the help of support functions, the original problem is reduced to the problem of global minimization of some functions in the space of piecewise continuous functions. In the case of continuous differentiability of the support function of a multivalued mapping with respect to the phase variables, this functional is Gateaux differentiable. In the paper, Gateaux gradient is found, necessary and (in some particular cases) sufficient conditions for the global minimum of the given functions are obtained. On the basis of these conditions, the method of steepest descent is applied to the original problem. Numerical examples illustrate the method realization.