Meromorphic Solutions of Nonlinear Differential-Difference Equations Involving Periodic Functions

Abstract

We investigate the following two types of nonlinear differential-difference equations

$$ L(z,f)+H(z,f)=\sum _{k=1}^r\alpha _k(z)e^{\beta _k z}; \ \ \ \ $$$$L(z,f)+H(z,f)=\sum _{k=1}^rF_k(z), \ \ \ \ \ \ \ \ \ $$

where \(\alpha _1, \ldots , \alpha _r\) are meromorphic functions of order \(<1,\) and \(F_1,\ldots , F_r\) are periodic transcendental entire functions, and L, H are defined by \(L(z,f)=\sum _{k=1}^pa_k(z)f^{(m_k)}(z+\tau _k)\not \equiv 0,\) \(H(z,f)=\sum _{k=1}^qb_k(z)\big [f^{(n_k)}(z+\zeta _k)\big ]^{s_k} \ \ \) with small meromorphic coefficients \(a_i, b_j.\) By introducing a new method, we obtain the exact forms of the solutions of these two equations under certain growth conditions.