Using a model-independent analysis method which is based on the Lagrange mean value theorem for obtaining the derivative of the Hubble function, we analyze H(z) parameter data with some restrictive conditions. We find that: (a) the Universe may experience an accelerated expansion with a confidence level greater than 5σ at redshift z101 ∈ (0, 0.36) and greater than 1.9σ at redshifts z3835 ∈ (1.3, 1.53) and z3836 ∈ (1.43, 1.53), where zj < zij < zi and i marks the ith Hubble parameter data we consider; (b) the Universe may experience a decelerated expansion with a confidence level greater than 1.5σ at redshift z2012 ∈ (0.40, 0.52); (c) wx ≤ wt < −1 with confidence level greater than 1.6σ at redshift z3836 ∈ (1.43, 1.53). These results indicate that the evolution of dark energy may be oscillatory.