Existence of Periodic Solutions for Neutral Nonlinear Dynamic Systems with Delay Using Shift Operators and Krasnoselskii's Fixed Point Theorem

Abstract

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.