Uniqueness of meromorphic solutions sharing values with a meromorphic function to w(z + 1)w(z - 1) = h(z)w^{m}(z)

Abstract

For the nonlinear difference equations of the form \t\t\tw(z+1)w(z−1)=h(z)wm(z),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ w(z + 1)w(z - 1) = h(z)w^{m}(z), $$\\end{document} where h(z) is a nonzero rational function and m = pm 2, pm 1,0, we show that its transcendental meromorphic solution is mainly determined by its zeros, 1-value points and poles except for some special cases. Examples for the sharpness of these results are given.