PERIODIC SOLUTIONS FOR IMPULSIVE NEUTRAL DYNAMIC EQUATIONS WITH INFINITE DELAY ON TIME SCALES SPACE
DOI:
https://doi.org/10.37376/fesj.vi18.7293Keywords:
Periodic , dynamic equations, impulses, Krasnoselskii fixed point, time scales.Abstract
This study addresses the problem of determining the existence and uniqueness of periodic solutions to a class of impulsive neutral dynamic equations that incorporate infinite delay, defined over a periodic time scale denoted by .
The focal point of this work is a complex dynamic system that integrates multiple mathematical features: neutral terms, impulsive discontinuities at discrete instances, and an integral representation of the system’s historical behavior extending indefinitely into the past.
The dynamic model under consideration involves a delta derivative, multiplicative operator terms, and delayed functional components, and is governed by impulsive effects at specified time points. The analysis is grounded in a general framework that accommodates both discrete and continuous behavior through the unifying language of time scale calculus.
To establish the existence of periodic solutions, we utilize Krasnoselskii’s fixed point theorem—an essential tool in nonlinear operator theory known for its effectiveness in handling non-compact and non-linear mappings in Banach spaces. In contrast, the uniqueness of the solution is ensured by applying the Banach contraction principle, which demands more restrictive structural conditions on the system’s parameters but provides strong guarantees of solution distinctiveness.
The theoretical contributions presented herein not only address the inherent analytical challenges posed by the neutral and impulsive dynamics but also offer valuable insights into systems exhibiting long-term memory. Such systems are prevalent in various scientific domains, including automatic control mechanisms with feedback delays, macroeconomic models driven by historical trends, and biological oscillators subject to abrupt environmental perturbations.
By integrating advanced methods from time scale calculus, infinite-dimensional functional analysis, and fixed point theory, this work offers a comprehensive approach that enhances both the theoretical understanding and practical applicability of periodic solutions in delay-dominated dynamic systems.
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