New generalization of non-autonomous Bernfeld–Haddock conjecture and its proof

Abstract

In this paper, the famous Bernfeld–Haddock conjecture is generalized to a broader form combining with a class of non-autonomous delay differential equations. With the help of differential inequality technique and Dini derivative theory, it is proved that each solution of the addressed equations has boundedness and tends to a constant without requiring the delay feedback function to be strictly increasing, which greatly refines and extends the corresponding results in the existing literature. In particular, an explanatory example is performed to substantiate the obtained analytical findings.