K-essence in Horndeski models
  In this paper, we investigate a simple class of Horndeski models where the scalar field plays the role of a k-essence fluid. We present several solutions for early-time universe, namely inflation and cosmological bounce, by making use of some reconstruction technique. Moreover, we furnish the formalism to calculate perturbations in FRW space-time and we compute the spectral index and the tensor-to-scalar ratio during inflation. 
Shynaray Myrzakul, Ratbay Myrzakulov, Lorenzo Sebastiani
gr-qc