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This paper focuses on the asymptotic stability (AS) problem of the delay dynamical system (DDS). An improved delay-product-type functional (DPTF) strategy is applied to construct a more general Lyapunov-Krasovskii functional (LKF). This method introduces more relevant information and improves the information capacity of LKF. In addition, the definition of the conditions <inline-formula> <tex-math notation="LaTeX">$V_{1} (x_{t}) $ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$V_{5} (x_{t}) $ </tex-math></inline-formula> are further relaxed, thereby developing new relaxed conditions. Then, based on the appropriate integral inequalities and the reciprocally convex combination lemma (RCCL). Finally, the feasibility of the method in this paper is verified based on two numerical examples.