Canonical Transformation and Oscillatory Behavior in Functional Differential Equations with Distributed Delays

This paper establishes new oscillation criteria ensuring that all solutions of a class of third-order nonlinear semi-canonical differential equations with a neutral term exhibit oscillatory behavior. The main objective of this paper is to establish a transformation of the considered semi-canonical equation into an equivalent canonical form without requiring additional restrictive assumptions. By employing this transformation, the equation is reduced to a more tractable form, allowing a clearer analytical treatment. Furthermore, by combining the comparison principle with an integral averaging technique, new sufficient conditions are derived to guarantee the oscillation of the associated canonical equation. These results are then extended to the original semi-canonical framework, thereby ensuring the oscillatory behavior of its solutions. The significance of the proposed approach lies in extending the applicability of known results for canonical equations to a broader class of semi-canonical problems. Finally, several illustrative examples are presented to demonstrate the effectiveness and applicability of the obtained theoretical results.