Relationship between the value-sharing behavior of an entire function and its derivative, and the analytic structure of a nonlinear differential equation

TL;DR

This paper investigates the relationship between entire functions sharing values with their derivatives and the structure of certain nonlinear differential equations, extending previous results and providing new insights into their interconnected behavior.

Contribution

It generalizes and improves existing results on value-sharing between entire functions and derivatives, revealing a deep link with nonlinear differential equations.

Findings

Enhanced conditions for value-sharing properties

Established connections between function behavior and differential equation structure

Provided examples illustrating the necessity of conditions

Abstract

In this paper, we study uniqueness problems for entire functions that partially share two values with their higher-order derivatives. The results obtained here both improve and generalize the related results of Li and Yi \cite{LYi}, L\"{u} et al. \cite{LXY1} and Sauer and Schweizer \cite{SS1}. Furthermore, we show that our results reveal a deep relationship between the value-sharing behavior of an entire function $f$ and its $k$-th derivative $f^{(k)}$, and the analytic structure of a particular type of nonlinear differential equation. Several examples are provided to illustrate the necessity of the conditions used in our results.