In this paper, we consider an abstract non-autonomous evolution equation with multiple delays in a Hilbert space H:  u'(t) + Au(t) = F(u(t-r₁),...,u((t-r_n)) + g(t), where A: D(A)?H→H is a positive definite selfadjoint operator,  F: Hⁿ_a → H is a nonlinear mapping,  r₁,...,r_n are nonnegative constants, and  g(t)∈ C(□;H) is bounded. Motivated by [1] [2], we obtain the existence and stability of synchronizing solution under some convergence condition. By this result, we provide a general approach for guaranteeing the existence and stability of periodic, quasiperiodic or almost periodic solution of the equation.