Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 03, pp. 1-10.

Title: Entire solutions for non-linear differential-difference equations

Authors: Harina P. Waghamore (Bangalore Univ., Jnana Bharathi, Bangalore, India)
         Manjunath Banagere Erajikkappa (Bangalore Univ., Jnana Bharathi, Bangalore, India)
  
Abstract:
In this article, we investigate the entire solutions of the
non-linear differential-difference equation
$$
f^n(z) + \omega f^{n-1}(z)f'(z) + q(z)e^{Q(z)}\mathcal{D}(z,f) 
= p_1(z)e^{\lambda z} + p_2(z)e^{-\lambda z},
$$
where $\mathcal{D}(z,f) = \sum_{i=0}^k b_if^{(t_i)}(z+c_i) \not\equiv 0$, 
with $b_i, c_i \in \mathbb{C}$, $t_i$ being non-negative integers, 
$c_0 = 0$, $t_0 = 0$. Here, $n$ is an integer, $\lambda, p_1, p_2$ are 
non-zero constants, $\omega$ is a constant, and 
$q \not\equiv 0$, $Q(z)$ are polynomials such that $Q(z)$ is non-constant. 
Our results improve upon and generalize some previously established 
findings in this area.


Submitted June 18, 2024. Published January 04, 2025.
Math Subject Classifications: 39A32, 30D35.
Key Words: Non-linear difference-differential equations; entire solution; Nevanlinna theory.