Characterization of exponential polynomial as solution of certain type of non-linear delay-differential equation

TL;DR

This paper characterizes the solutions of a complex non-linear delay-differential equation, showing that exponential polynomial solutions are typical, and improves upon previous related results by providing a more comprehensive framework.

Contribution

It provides a unified characterization of solutions for a broad class of non-linear delay-differential equations, extending and improving prior specific results.

Findings

Exponential polynomial solutions are characterized for the given class of equations.

The results encompass and improve previous special case findings.

Examples demonstrate the applicability of the theoretical results.

Abstract

In this paper, we have characterized the nature and form of solutions of the following non-linear delay-differential equation: $$f^{n}(z)+\sum_{i=1}^{n-1}b_{i}f^{i}(z)+q(z)e^{Q(z)}L(z,f)=P(z),$$ where $b_i\in\mathbb{C}$, $L(z,f)$ be a linear delay-differential polynomial of $f$; $n$ be positive integers; $q$, $Q$ and $P$ respectively be non-zero, non-constant and any polynomials. Different special cases of our result will accommodate all the results of ([J. Math. Anal. Appl., 452(2017), 1128-1144.], [Mediterr. J. Math., 13(2016), 3015-3027], [Open Math., 18(2020), 1292-1301]). Thus our result can be considered as an improvement of all of them. We have also illustrated a handful number of examples to show that all the cases as demonstrated in our theorem actually occurs and consequently the same are automatically applicable to the previous results.