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<p>In this paper, we investigate a model of abstract nonlocal differential equations involving bounded delay, nonlocal initial conditions, and weak nonlinearity that reduces regularity. We first establish the global existence and uniqueness of the mild solution by using the fixed point argument with certain conditions that describes the interaction of the nonlinearity and the nonlocality of the initial measurement. Then we prove the asymptotic stability (together with half of the linearized decaying rate) of the mild solution to the delayed problem by utilizing a novel nonlinear nonlocal Halanay-type inequality under a suitable assumption on the nonlocal initial measurement. Moreover, we show the optimal decaying rate of the mild solution to the non-delayed problem by combining a new version of the nonlocal Gronwall-type inequality and the Karamata Tauberian Theorem. Finally, we present examples to illustrate the effectiveness of the obtained results.</p>

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Stability analysis for nonlocal initial value problems involving nonlocal Caputo derivative and loss-of-regularity nonlinearities

  • 期刊论文

  • 发表日期：

    2026年4月10日

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  • Thi Thu Huong Nguyen,
  • Thi Lien Nguyen,
  • Nhu Thang Nguyen,
  • Anh Toan Pham

摘要

In this paper, we investigate a model of abstract nonlocal differential equations involving bounded delay, nonlocal initial conditions, and weak nonlinearity that reduces regularity. We first establish the global existence and uniqueness of the mild solution by using the fixed point argument with certain conditions that describes the interaction of the nonlinearity and the nonlocality of the initial measurement. Then we prove the asymptotic stability (together with half of the linearized decaying rate) of the mild solution to the delayed problem by utilizing a novel nonlinear nonlocal Halanay-type inequality under a suitable assumption on the nonlocal initial measurement. Moreover, we show the optimal decaying rate of the mild solution to the non-delayed problem by combining a new version of the nonlocal Gronwall-type inequality and the Karamata Tauberian Theorem. Finally, we present examples to illustrate the effectiveness of the obtained results.