This paper aims to derive more generalized Lyapunov inequality conditions and establish new fixed-time stability lemmas for Filippov systems. Using the definition of fixed-time stability and advanced inequality techniques, we rigorously prove that the zero solution is fixed-time stable. Furthermore, we provide detailed theoretical analysis showing that setting \(r=1\) in the generalized economical inequality conditions yields the minimal settling time. These results not only improve upon existing fixed-time stability lemmas, but also validate previous related hypotheses. As a key application, the newly established fixed-time stability lemmas are employed to study synchronization and anti-synchronization in fixed-time for master–slave discontinuous leakage-delayed competitive neural networks modeled by Filippov systems. Leveraging differential inclusion theory and delay-free non-chattering controllers, we derive leakage-delay-dependent criteria to ensure synchronization, marking the first theoretical results in this domain. In addition, the derived settling times are also leakage-delay-dependent, revealing the explicit influence of leakage delays on convergence speed. Finally, numerical simulations are presented to verify the correctness of the main theoretical findings.
