Solving the multiple-set split equality common fixed-point problem of firmly quasi-nonexpansive operators.

Area of Science:

• Optimization Theory
• Functional Analysis
• Numerical Analysis

Background:

• Fixed-point problems are central to many areas of mathematics and applied sciences.
• Existing iterative methods often require prior knowledge of operator norms, limiting their applicability.
• The multiple-set split equality common fixed-point problem presents a significant computational challenge.

Purpose of the Study:

• To develop new iterative algorithms for solving the multiple-set split equality common fixed-point problem.
• To introduce parallel and cyclic iterative schemes for firmly quasi-nonexpansive operators.
• To establish weak convergence results for these novel algorithms under mild conditions.

Main Methods:

• Development of parallel iterative algorithms.
• Formulation of cyclic iterative algorithms.
• Combination of parallel and cyclic approaches into mixed iterative algorithms.
• Analysis of weak convergence in Hilbert spaces.

Main Results:

• Proposed several novel iterative algorithms for the targeted problem.
• Demonstrated that the algorithms do not require prior knowledge of operator norms.
• Proved weak convergence of the iterative sequences under mild assumptions.
• Derived new iterative algorithms for the multiple-set split equality problem as applications.

Conclusions:

• The proposed parallel, cyclic, and mixed iterative algorithms are effective for solving the multiple-set split equality common fixed-point problem.
• The convergence analysis provides theoretical support for the practical application of these algorithms.
• These algorithms offer a valuable tool for researchers and practitioners in relevant fields.