In the paper two nonlocal, nonlinear problems for a system of parabolic equations are considered: to find a solution of the system \vec u_t (x,t) = D\vec u_{xx}(x, t) + \vec f (x, t, \vec u (x, t)) subject to the conditions \vec u (0, t) = \vec \varphi (t), \quad t \in (0, T), \vec u (x, 0) = \vec \psi (x), \quad x \in (0, 1), \vec u (1, t) - \vec u (x_0, t) = \vec h (x_0, t, \vec u (x_0, t)) or \int ^1_0 \vec u (x, t) dx = \vec g (t). For this an operator L: C(\bar \Omega) \to C(\bar \Omega) being a sum of four potentials is constructed. It is shown that the operator L has only one fixed point. Moreover it is proved that the fixed point is the only solution of the considered problem.