Linearized oscillation of first-order nonlinear neutral delay differential equations (Q5939793)

Here, the authors investigate the equivalence of oscillations for the first-order nonlinear neutral delay differential equation NEWLINE\[NEWLINE [x(t)-px(t-\tau)]'+ \sum_{i=1}^m q_i f_i (x(t-\sigma_i))=0, NEWLINE\]NEWLINE and its corresponding linear equation NEWLINE\[NEWLINE [x(t)-px(t-\tau)]'+ \sum_{i=1}^m q_i x(t-\sigma_i)=0, NEWLINE\]NEWLINE in two cases: a noncritical case and a critical case. Some new concepts are explained and supported by interesting theoretical developments. Simplified versions of several new integral criteria for oscillations are presented. Proofs which illustrate the various strategies and ideas involved are given. An example is given to illustrate the theory.