(untitled) (Al-Mukhtar Journal of Basic Sciences, 2026)
In this research, we investigate the existence of periodic solutions for a class of neutral-type nonlinear dynamic systems with delay, described by the equation:. To address this problem, we adopt a contemporary framework for periodicity based on shift operators, which extends traditional periodic concepts to a broader class of time scales. This modern shift-based perspective proves particularly advantageous for time scales where the additivity condition for all and for a fixed may not hold—a limitation that precludes the use of classical periodicity in non-uniform or non-additive time domains. Notably, this generalized notion of periodicity is well-suited for non-standard time scales such as the quantum time scale and the Cantor-like union , which do not admit a regular periodic structure in the conventional sense. To effectively examine the periodic behavior of such systems, particularly those involving q-difference dynamics, we construct a technical apparatus capable of analyzing periodic solutions under the shift-based setting. Central to this approach is the transformation of the differential system into an equivalent integral form, a step that necessitates consideration of the transition matrix associated with the homogeneous Floquet-type system: This integral reformulation enables the application of Krasnoselskii’s fixed point theorem, a foundational result in nonlinear operator theory, which facilitates the demonstration of fixed-point existence—and thereby confirms the presence of nontrivial periodic solutions within the system.