New Perspectives on the Oscillatory Behavior of First-Order Differential and Difference Equations With Several Delays: Counterexamples and Criteria

This paper investigates the oscillatory behavior of first-order linear differential and difference equations with several delays. The main objective is to determine whether classical liminf-type oscillation conditions for single-delay equations can be extended to the several-delay setting and to develop sharp oscillation criteria for this more general framework. We show, by constructing a counterexample, that there is no universal constant C>0 such that the condition lim inft⟶∞∫τmintt∑i=1mpisds>C guarantees oscillation of all solutions of x′t+∑i=1mpitxτit=0, where τmint:=min1≤i≤mτit, pi are nonnegative continuous functions, and τi are nondecreasing delay functions satisfying τit≤t and limt⟶∞τit=∞. Similarly, we prove that no universal constant C>0 exists such that lim infn⟶∞∑k=σminnn−1∑i=1mqik>C ensures oscillation of all solutions of the discrete analogue Δxn+∑i=1mqinxσin=0, with σminn:=min1≤i≤mσin, qin≥0, and nondecreasing delays σin≤n−1, and limn⟶∞σin=∞. Furthermore, new sharp oscillation criteria are obtained for both continuous and discrete equations. These results complement the above nonexistence results and contribute to the oscillation theory of equations with several delays. Several illustrative examples are presented to demonstrate the sharpness and applicability of the established criteria.