Viscosity iterative algorithm for the zero point of monotone mappings in Banach spaces

Area of Science:

• Functional Analysis
• Numerical Analysis
• Optimization Theory

Background:

• Iterative algorithms are crucial for solving equations involving monotone mappings.
• Existing methods often rely on resolvent operators, which can be computationally intensive.
• Uniformly convex Banach spaces provide a rich framework for studying convergence properties.

Purpose of the Study:

• To introduce a novel viscosity iterative algorithm for approximating zeros of monotone mappings.
• To establish strong convergence guarantees for the proposed algorithm under simplified parameter conditions.
• To demonstrate the algorithm's utility through applications in convex minimization and Hammerstein integral equations.

Main Methods:

• A viscosity iterative algorithm is devised, specifically avoiding the use of resolvent operators.
• The algorithm is analyzed within the context of uniformly convex Banach spaces.
• Theoretical convergence is established under concise parameter constraints.

Main Results:

• The proposed algorithm is proven to exhibit strong convergence.
• Successful applications are demonstrated for constrained convex minimization problems.
• The algorithm is shown to be effective in solving Hammerstein integral equations.

Conclusions:

• The new algorithm provides an efficient and applicable method for finding zeros of monotone mappings.
• Its performance is validated through computational examples and comparisons with existing algorithms.
• The work extends existing iterative techniques by removing the dependency on resolvent operators.