Entire solutions for nonlinear differential-difference equations

Abstract

In this article, we study entire solutions of the nonlinear differential-difference equation $$ q(z)f^{n}(z)+a(z)f^{(k)}(z+1)=p_1(z)e^{q_1(z)}+p_2(z)e^{q_2(z)} $$ where $p_1(z)$, $p_2(z)$ are nonzero polynomials, $q_1(z)$, $q_2(z)$ are nonconstant polynomials, $q(z)$, $a(z)$ are nonzero entire functions of finite order, $n\geq2$ is an integer. We obtain additional results for case: $n=3$, $q_1(z)=-q_2(z)$, and $p_1(z)$, $p_2(z)$ nonzero constants.

Keywords

• Nevanlinna theory
• differential-difference equation
• entire function