Strong convergence of gradient projection method for generalized equilibrium problem in a Banach space

Area of Science:

• Optimization Theory
• Functional Analysis
• Applied Mathematics

Background:

• Generalized equilibrium problems (GEPs) and variational inequality problems (VIPs) are fundamental in optimization.
• Fixed-point theory is crucial for analyzing iterative processes and convergence.
• Real Banach spaces provide a general framework for studying these mathematical concepts.

Purpose of the Study:

• To propose a novel hybrid iterative method for solving GEPs, VIPs, and finding fixed points of relatively nonexpansive mappings simultaneously.
• To analyze the strong convergence properties of the proposed iterative scheme.
• To extend and improve upon existing results in the literature.

Main Methods:

• Development of a hybrid iterative algorithm combining techniques for GEPs, VIPs, and fixed-point problems.
• Application of convergence analysis techniques specific to real Banach spaces.
• Derivation of the strong convergence of generated sequences.

Main Results:

• The proposed hybrid iterative method successfully finds a common element for the solution sets of the three problem types.
• The sequences generated by the iterative scheme are proven to converge strongly.
• New theoretical results are established regarding the convergence of such iterative methods.

Conclusions:

• The developed method offers an effective approach for solving complex optimization and fixed-point problems.
• The strong convergence result provides a significant theoretical advancement.
• This work contributes to the broader understanding of iterative methods in functional analysis and optimization.