Iterative methods of strong convergence theorems for the split feasibility problem in Hilbert spaces.

Area of Science:

• Optimization Theory
• Functional Analysis
• Numerical Analysis

Background:

• The split feasibility problem (SFP) is a fundamental problem in optimization and mathematical analysis.
• Existing iterative methods for SFP may have limitations in convergence speed or applicability.
• Hilbert spaces provide a rich mathematical framework for studying SFP.

Purpose of the Study:

• To develop and analyze new iterative algorithms for solving the split feasibility problem (SFP).
• To demonstrate the convergence properties of the proposed algorithms.
• To ensure the algorithms can find the minimum-norm solution.

Main Methods:

• Development of novel iterative schemes based on analytical techniques.
• Convergence analysis of the generated iterative sequences.
• Application of the methods to find the best close solution and minimum-norm solution.

Main Results:

• The proposed iterative algorithms are proven to converge to a solution of the SFP.
• The convergence is shown to be to the solution closest to a given point.
• The minimum-norm solution is attainable through the developed iteration method.

Conclusions:

• The new iterative algorithms offer an effective approach to solving the split feasibility problem.
• The analytical techniques employed provide rigorous convergence guarantees.
• These methods advance the capability for finding specific solutions, like the minimum-norm solution, in Hilbert spaces.