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

For the nonlinear difference equations of the form $$ w(z + 1)w(z - 1) = h(z)w^{m}(z), $$ 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.