In this paper, we study the oscillation of a class of second-order nonlinear differential equations with mixed neutral terms in the non-canonical case. New criteria are derived that ensure the oscillation of the studied equation. The results obtained here greatly improve and extend some of the results reported in previous studies. To illustrate this, we present some examples.

This paper develops new unified fixed point theorems by extending the classical Ćirić contractions through the iterative framework introduced by Singh in his generalization of Kannan-type mappings. We first revisit the two well-known definitions of Ćirić contractions in their standard forms and then introduce the concepts of $\rho$-Ćirić contraction and generalized $\rho$-Ćirić contraction, in which the contractive condition is imposed on higher iterates of the mapping. These extended classes retain the intrinsic structure of Círic-type mappings while significantly enlarging the family of operators that admit a unique fixed points. For each class, we establish comprehensive fixed point theorems ensuring existence, uniqueness, and global convergence of the associated Picard iteration. Our results unify and generalize several classical fixed-point theorems, including those of Banach, Kannan, Chatterjea, and the original Ćirić contractions, which arise as special cases within this broader framework. Illustrative examples are provided to demonstrate the applicability and sharpness of the proposed generalizations.

In this study, we aim to contribute to the increasing interest in functional differential equations by obtaining new theorems for the oscillation of second-order neutral differential equations of mixed type in a non-canonical form. The results obtained here improve and extend those reported in the literature. The applicability of the results is illustrated by several examples.