In the paper the following optimization problem is considered: Let Xbe a Banach space is a vector performance index. We are looking for a point such that where U(x) denotes some neighbourhood of x. Making use of the results of Walczak [10] and Censor [1] we generalize the well-known Dubovicki-Milutin Theorem and them we apply it to the problem (*) obtaining a necessary condition for local Pareto optimum. For convex problems a local Pareto optimum in also a global Pareto one. If additonal assumptions are fulfilled for convex problems i.e. are continuous and Ponstein convex and the so-called Slater's condition is satisfied, then the Euler-Lagrange equation gives a sufficient condition for global Pareto optimum. In the paper an example of a Pareto optimal control problem is also given. A dynamical cascade system is described by two partial differential equations of parabolic type in a Sobolev space.