We consider Y-system functional equations of the form Yn(x+i)Yn(x-i)=∏m=1N(1+Ym(x))Gnmand the corresponding nonlinear integral equations of the thermodynamic Bethe ansatz. We prove an existence and uniqueness result for solutions of these equations, subject to appropriate conditions on the analytical properties of the Yn, in particular the absence of zeros in a strip around the real axis. The matrix Gnm must have non-negative real entries and be irreducible and diagonalisable over R with spectral radius less than 2. This includes the adjacency matrices of finite Dynkin diagrams, but covers much more as we do not require Gnm to be integers. Our results specialise to the constant Y-system, proving existence and uniqueness of a strictly positive solution in that case.