In this paper, we propose an extension of the invariance principle for switched systems under dwell-time switched solutions. Our approach allows the derivative of an auxiliary function V along the solutions of the switched system to be positive on some bounded sets. The auxiliary function V, which plays the role of a Lyapunov function, is called a Lyapunov-like function in this paper. Our results are useful to estimate attractors of switched systems and basins of attraction. Results for a common Lyapunov-like function and multiple Lyapunov-like functions are given. Illustrative examples show the potential of the theoretical results in providing concrete information on the asymptotic behavior of nonlinear dynamical switched systems.