New Oscillation Criteria for Third-Order Non-Linear Neutral Delay Differential Equations

Abstract

This paper investigates the oscillatory behaviour of solutions to a class of third-order nonlinear neutral delay differential equations of the form:

where the neutral term is defined as:

                                                          Q(t) = x(t) + R(t)x(l(t)),

We operate under the assumption that the canonical integral condition 

holds, ensuring the divergence of the potential function. By employing Riccati transformation techniques and integral inequality methods, we establish new sufficient conditions that guarantee all solutions are either oscillatory or converge to zero. This work accounts for the nonlinear functional f, assuming only that f(u)  u   for u  The results further incorporate the commutativity of delay operators l(t) & b(t) satisfies l  to simplify the estimation of the neutral coefficient R(t). Our findings extend and complement existing criteria in the literature.