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Abstract

This paper is devoted to value distribution and uniqueness problems for difference polynomials of entire functions such as $f^n(f-1)f(z+c)$. We also consider sharing value problems for $f(z)$ and its shifts $f(z+c)$, and improve some recent results of Heittokangas et al. [J. Math. Anal. Appl. 355 (2009), 352–363]. Finally, we obtain some results on the existence of entire solutions of a difference equation of the form $f^{n}+P(z)(\Delta _cf)^m=Q(z).$