Explicit upper bound approximations for the distance between successive zeros of all solutions to first-order differential equations with several monotone delays

This paper examines the distribution of zeros of all solutions to first-order delay differential equations with several increasing delay arguments (DDEs). By advancing and generalizing the existing frameworks of the oscillation theory of delay and neutral differential equations, new explicit upper bounds are established for the separation between successive zeros of solutions. These bounds are constructed by analyzing the properties of positive solutions over a closed interval, providing sufficient conditions to prevent the persistence of such positivity. This approach provides a general analytical framework for identifying conditions under which all nontrivial solutions oscillate. Moreover, we determine the locations of the zeros of solutions to equations with several delays and oscillatory coefficients. Furthermore, the developed approach allows future extensions to functional and difference equations with non-monotone or distributed delays. Several illustrative examples and comparisons with existing criteria are included to demonstrate the accuracy and effectiveness of the proposed results.