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 Δηf+a(z)P(f)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \Delta _{\eta }f+a(z)P(f)=0 $$\end{document} in C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathbb {C}} $$\end{document} and (ii) differential equation Dkf+h(z)g(f(z))=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ D^kf+h(z)g(f(z))=0 $$\end{document} in terms of total derivatives in Cn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathbb {C}}^n $$\end{document}. 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.