具有分布式记忆的隐式系统分数格罗文沃尔-温德罗夫不等式
Fractional Grönwall--Wendroff Inequalities for Implicit Systems with Distributed Memory
AI总结：
本文提出了一种针对具有分布式记忆和时间延迟的隐式分数微分系统的格罗文沃尔-温德罗夫不等式，用于分析神经动力学系统中的极限环存在性和稳定性。
摘要：
本文建立了一个全面的分析框架，用于研究具有分布式记忆和时间延迟的隐式分数微分系统。我们开发了新的格罗文沃尔-温德罗夫类型的分数积分不等式，专门用于处理具有奇异核和隐式依赖的多变量函数。这些不等式为分析复杂的记忆依赖系统提供了必要的先验估计。基于这些结果，我们利用适当加权的巴拿赫空间中的不动点理论，证明了隐式分数微分方程的一般存在性和唯一性定理。此外，我们建立了乌拉-海尔斯稳定性准则，表明在支配方程中的小扰动会导致解中成比例的小偏差。理论进展应用于具有延迟的分数 FitzHugh-Nagumo 模型（FHN-α-τ），这是一个表现出亚扩散记忆和离散时间延迟的神经动力学系统。我们的分析给出了极限环存在的严格条件，对应于动作电位，以及渐近稳定性准则。结果揭示了分数阶α和延迟τ如何共同调节神经可兴奋性阈值和动态制度转换。这项工作将基本的分数微积分理论与数学神经科学中的应用联系起来，为那些当前演变依赖于历史状态和内在分数变化率的系统提供了分析工具。
英文摘要：
This work establishes a comprehensive analytical framework for studying implicit fractional differential systems with distributed memory and time delays. We develop novel fractional integral inequalities of Grönwall--Wendroff type that are specifically adapted to handle multivariate functions with singular kernels and implicit dependencies. These inequalities provide essential a priori estimates for analyzing complex memory-dependent systems. Building upon these results, we prove general existence and uniqueness theorems for implicit fractional differential equations using fixed-point theory in appropriately weighted Banach spaces. Furthermore, we establish Ulam--Hyers stability criteria, demonstrating that small perturbations in the governing equations lead to proportionally small deviations in solutions. The theoretical advances are applied to the fractional FitzHugh--Nagumo model with delay (FHN-$α$-$τ$), a neurodynamical system exhibiting both subdiffusive memory and discrete time delays. Our analysis yields rigorous conditions for the existence of limit cycles corresponding to action potentials, along with asymptotic stability criteria. The results reveal how fractional order $α$ and delay $τ$ jointly modulate neural excitability thresholds and dynamic regime transitions. This work bridges fundamental fractional calculus theory with applications in mathematical neuroscience, providing analytical tools for systems where present evolution depends non-trivially on historical states and intrinsic fractional rates of change.
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