The key conditions that need to be met to show the uniqueness of the periodic solution using the contraction mapping principle are outlined as follows:

  • Compactness and Continuity: The mapping Ψ: 𝑋 → 𝑋 is required to be compact and continuous. This property ensures that the mapping preserves the structure of the space 𝑋 and allows for the application of contraction mapping principles.
  • Contraction Mapping: The mapping Φ: 𝑋 → 𝑋 must be a contraction mapping. This condition is essential for demonstrating the existence of a unique fixed point in the space 𝑋, which corresponds to the periodic solution of the impulsive neutral dynamic equations.
  • Norm Bound: It is necessary to show that if 𝜑 and 𝜓 belong to the set 𝑀, then the norm of the mapping Φ𝜑 + Ψ𝜓 is bounded by a constant 𝐺. This condition ensures that the mapping does not lead to unbounded solutions and supports the uniqueness of the periodic solution.
  • By satisfying these key conditions, the study establishes the uniqueness of the periodic solution for the impulsive neutral dynamic equations with infinite delay on time scales using the contraction mapping principle. This rigorous approach ensures the reliability and robustness of the results obtained in the research.

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