Strong convergence theorems for a class of split feasibility problems and fixed point problem in Hilbert spaces

Area of Science:

• Optimization Theory
• Functional Analysis
• Numerical Analysis

Background:

• Split feasibility problems (SFP) are central in optimization, with applications in signal processing and medical imaging.
• Existing methods often struggle with convergence for complex operator sums and mappings.
• Hilbert spaces provide a robust framework for analyzing these problems.

Purpose of the Study:

• To develop a novel iterative algorithm for solving a class of split feasibility problems.
• To address the challenge of finding common solutions for zero points of monotone operators and fixed points of mappings.
• To establish strong convergence properties for the proposed algorithm.

Main Methods:

• The study employs a modified forward-backward splitting method.
• A viscosity iterative algorithm is proposed and analyzed.
• Convergence theorems are established under specific mathematical conditions.

Main Results:

• The proposed viscosity iterative algorithm exhibits strong convergence.
• The algorithm converges to a common solution of the considered problems.
• The theoretical findings are supported by discussions on applications and algorithm construction.

Conclusions:

• The developed algorithm offers an effective approach for solving complex split feasibility problems.
• The strong convergence guarantees enhance the reliability of the method.
• The research contributes to the advancement of iterative methods in optimization and functional analysis.