Entire solutions of one certain type of nonlinear differential-difference equations

Abstract

We describe the entire solutions for two kinds of nonlinear differential-difference equations of the form

$$f^n(z)+\omega f^{n-1}(z)f^{\prime }(z)+q_1(z)e^{Q_1(z)}{f}_{c_1}+q_2(z)e^{Q_2(z)}{f}_{c_2}=u(z)e^{v(z)},n\ge 3$$

and

$$f^n(z)+q(z)e^{Q(z)}f(z+c)=u(z)e^{v(z)},n\ge 2,$$

where $q,Q,u,v,q_j(z),Q_j(z)$, for $j=1,2$, are polynomials such that $Q(z)$ and at least one of $Q_j(z)$ are not constants, $q(z)$ and $q_j(z)$ are not identically zero, and $\omega ,c_1,c_2,c$ are constants. Our results improve and generalize some previous results.