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Bielecki-Ulam stability of a hammerstein-type difference system
Gul Rahmat1, Sohail Ahmad2, Muhammad Sarwar3,4
1Department of Mathematics, Islamia College University Peshawar, Khyber Pakhtoonkhwa Pakistan.
Methodsx
|August 27, 2025
Summary
This study analyzes Bielecki-Ulam (B-U) stability for two Hammerstein-type difference systems (HT-DS). Conditions for unique solutions and B-U stability were established using the Banach contraction principle and Bielecki norm.
Area of Science:
- Differential Equations
- Numerical Analysis
- Dynamical Systems
Background:
- Hammerstein-type difference systems (HT-DS) are crucial in modeling various phenomena.
- Investigating the stability of these systems is essential for understanding their long-term behavior and reliability.
- Bielecki-Ulam (B-U) stability provides a robust framework for analyzing the sensitivity of solutions to perturbations.
Purpose of the Study:
- To investigate the Bielecki-Ulam (B-U) stabilities of two distinct forms of Hammerstein-type difference systems (HT-DS).
- To establish conditions ensuring the existence and uniqueness of solutions for these systems.
- To derive sufficient criteria for B-U stability in the considered HT-DS.
Main Methods:
- Analysis of two specific Hammerstein-type difference systems with different nonlinear structures.
- Application of the Banach contraction principle in conjunction with the Bielecki norm.
- Verification of boundedness and Lipschitz continuity for nonlinear terms within the systems.
- Derivation of conditions based on operator contractivity under the Bielecki norm.
Main Results:
- Sufficient conditions for the existence and uniqueness of solutions were derived (Theorems 2 and 3).
- Conditions guaranteeing Bielecki-Ulam stability for both HT-DS forms were established (Theorems 4 and 5).
- Key conditions include boundedness of coefficients, Lipschitz continuity of nonlinear functions, and satisfaction of a contraction inequality.
Conclusions:
- The study successfully established criteria for the existence, uniqueness, and Bielecki-Ulam stability of solutions for the investigated Hammerstein-type difference systems.
- The findings provide a theoretical foundation for analyzing the stability of such systems in various applications.
- An illustrative example confirmed the practical applicability and validity of the derived theoretical results.
Keywords:
Bielecki stabilityDifference equationsHammerstein type difference system (HT-DE)Ulam stabilityMore Related Videos
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