The mesoscopic approach in the dynamics of polymer melts is based on the equation of the dynamics of a single macromolecule in the entangled system. The general linear form of the dynamic equation of a macromolecule allows different hypotheses about the decay law of memory function to be tested. To obtain consistency with experimental evidence for strongly entangled polymer melts, the exponential-law decay with a single correlation time ought to be assumed. The resulting picture of the thermal motion of a macromolecule is consistent with the common idea about localisation of the macromolecule. Intermediate length can be introduced, which has the sense of tube diameter and/or length of a macromolecule between adjacent entanglement. Two sets of relaxation times appear: the first relaxation branch corresponds to the slow motion of macromolecule (conformational branch); the second corresponds to the fast motion (of the macromolecule in the tube). Self-consistency of the theory is kept as the identity of the introduced correlation time and the calculated relaxation time of macroviscoelasticity. However, non-linear terms should be added to the linear dynamic equation to describe the mobility of very long macromolecules during the longest observation times. The main part of the paper is devoted to discussion of viscoelastic properties and constitutive equations for polymer melts considered by the mesoscopic approach. To calculate the stress tensor of a polymer melt as a suspension of coupled Brownian particles, the theory developed for liquids is used. The set of constitutive equations consists of an expression of stress tensor through two sets of internal variables, which characterise: (1) the size and form of the macromolecule coil in the system; and (2) internal stresses at deformation of macromolecules and relaxation equations for internal variables. However, the general form of the constitutive equation is rather complicated. The constitutive equation contains some mesoscopic parameters, which appear to be small for the fluid polymer system. Therefore, one can make use of it and write down different approximations of the constitutive equation with different accuracy. The simplest constitutive equation coincides with the Vinogradov phenomenological equation.