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Convergence analysis of Suzuki's generalized nonexpansive mappings using the Picard-Abbas iteration process
Bashir Nawaz1, Krzysztof Gdawiec2, Kifayat Ullah1
1Department of Mathematics, University of Lakki Marwat, Lakki Marwat, Khyber Pakhtunkhwa, Pakistan.
Plos One
|October 15, 2025
Summary
This study explores Suzuki generalized nonexpansive mappings convergence using the Picard-Abbas iteration. The research establishes convergence results and demonstrates its effectiveness through numerical examples and visual polynomiographs.
Area of Science:
- Nonlinear analysis
- Fixed-point theory
- Iterative methods
Background:
- Suzuki generalized nonexpansive mappings are crucial in nonlinear analysis.
- Understanding their convergence properties is essential for solving various mathematical problems.
- Existing iterative methods may have limitations in convergence speed or applicability.
Purpose of the Study:
- To investigate the convergence behavior of Suzuki generalized nonexpansive mappings.
- To introduce and analyze the Picard-Abbas iteration process for these mappings.
- To establish both weak and strong convergence theorems.
Main Methods:
- Application of the Picard-Abbas iteration process.
- Development of theoretical convergence analysis.
- Numerical examples to validate theoretical findings.
- Generation and comparison of polynomiographs.
Main Results:
- Established weak and strong convergence results for Suzuki generalized nonexpansive mappings using the Picard-Abbas iteration.
- Demonstrated the practical effectiveness of the Picard-Abbas iteration through a numerical example.
- Generated novel polynomiographs offering visual insights into the iteration process.
Conclusions:
- The Picard-Abbas iteration process is effective for approximating fixed points of Suzuki generalized nonexpansive mappings.
- The proposed method offers advantages in terms of convergence and visual representation compared to existing schemes.
- This research contributes to the advancement of fixed-point theory and iterative methods.
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