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On a new approach of enriched operators
Teodor Turcanu1, Mihai Postolache1
1National University of Science and Technology Politehnica Bucharest, Department of Mathematics & Computer Science, 313 Splaiul Independentei, Bucharest, 060042, Romania.
Heliyon
|March 27, 2024
Summary
This study proves the existence and uniqueness of fixed points for generalized contractions and nonexpansive mappings in Banach spaces using Mann iteration. This novel approach expands applicability beyond traditional methods.
Area of Science:
- Fixed-point theory
- Functional analysis
- Numerical analysis
Background:
- Generalized contractions and nonexpansive mappings are crucial in various mathematical fields.
- Existing methods often rely on Krasnoselskij iteration, limiting applicability.
- Banach spaces provide a fundamental framework for studying convergence properties.
Purpose of the Study:
- To establish the existence and uniqueness of fixed points for generalized contractions in Banach spaces.
- To prove the convergence of Mann iteration for generalized contractions and nonexpansive mappings.
- To extend the applicability of Mann iteration to a broader class of mappings.
Main Methods:
- Utilizing fixed-point theorems in Banach spaces.
- Applying Mann iteration to generalized contraction mappings.
- Demonstrating convergence for generalized nonexpansive mappings.
Main Results:
- Existence and uniqueness of fixed points for generalized contractions proven.
- Convergence of Mann iteration established for generalized contractions and nonexpansive mappings.
- New approach extends applicability of Mann iteration, overcoming limitations of Krasnoselskij iteration.
Conclusions:
- The study successfully extends fixed-point theory to a wider class of mappings using Mann iteration.
- The findings have direct implications for numerical analysis and functional equations.
- This research offers a more versatile iterative technique for solving problems in analysis.
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