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In this study, we pay attention to a functional differential equation (FDE) of first order including N-variable delays. We construct new sufficient conditions in relation to the Hyers-Ulam stability (HUS) and the generalized Hyers-Ulam-Rassias stability (GHURS) of the FDE of first order including N-variable delays. By using Banach contraction principle (BCP), Picard operator and Gronwall's lemma, we confirm two new theorems with regard to the HUS and the GHURS. The results of this study are new, they extend and improve some earlier results of the HUS and the GHURS. [ABSTRACT FROM AUTHOR]

The aim of this paper is first to investigate the stability of the zero solution to a new Liénard type equation with multiple variable delays by two different methods. The methods to be used in the proofs involve the Lyapunov-Krasovskiĭ functional approach and the fixed point technique under an exponentially weighted metric, respectively. We make a comparison between the applications of these methods with the established conditions on the same stability problems. Then, we obtain three new results for uniformly stability and boundedness/uniformly boundedness of the solutions to the considered equation by the Lyapunov-Krasovskiĭ functional approach. An example is given to verify the results obtained by the Lyapunov-Krasovskiĭ functional approach. Our results complement and improve some recent ones in the literature. [ABSTRACT FROM AUTHOR]