Picard-type theorem and the solutions of certain ODE and PDE

Picard’s theorem is among the most striking results in complex analysis and plays a decisive role in the development of the theory of entire and meromorphic functions. In this paper, we give an equivalence between Picard’s theorem and entire solutions of (i) difference equation Δ η f + a ( z ) P ( f ) = 0 in C and (ii) differential equation D k f + h ( z ) g ( f ( z ) ) = 0 in terms of total derivatives in C n . We prove two Picard-type theorems that generalize the results of Lu (Kodai Math J 26:221–229, 2003). In addition, examples have been exhibited to show that certain assumptions in the main result cannot be improved.