  This paper discusses the variational principles on subsets for topological
pressure and topological entropy of non-autonomous dynamical systems. We define
the Pesin-Pitskel topological pressure (weighted topological pressure) and the
Bowen topological entropy (weighted Bowen topological entropy) for any subset.
Also, we define the measure-theoretic pressure and the measure-theoretic lower
entropy for all Borel probability measures. Then, we prove variational
principles for topological pressure (topological entropy) which links the
Pesin-Pitskel topological pressure (weighted topological pressure) on an
arbitrary nonempty compact subset to the measure-theoretic pressure of Borel
probability measures for non-autonomous dynamical systems (which links the
Bowen topological entropy (weighted Bowen topological entropy) on an arbitrary
nonempty compact subset to the measure-theoretic lower entropy of Borel
probability measures for non-autonomous dynamical systems).
  Moreover, we show that the Pesin-Pitskel topological pressure (weighted
topological pressure) and the Bowen topological entropy (weighted Bowen
topological entropy) can be determined by the measure-theoretic pressure and
the measure-theoretic lower entropy of Borel probability measures,
respectively. These results extend Feng and Huang's results (Variational
principles for topological entropies of subsets, J. Funct. Anal. (2012)), Ma
and Wen's results (A Billingsley type theorem for Bowen entropy, Comptes Rendus
Mathematique (2008)), and Tang et al. results (Variational principle for
topological pressures on subsets, J. Math. Anal. Appl. (2015)) for classical
dynamical systems to pressures and entropies of non-autonomous dynamical
systems.
