On a new generalized symmetric vector equilibrium problem

Area of Science:

• Optimization Theory
• Mathematical Economics
• Game Theory

Background:

• Equilibrium problems are fundamental in various fields.
• Existing methods often rely on strong assumptions like lower semicontinuity.
• Ky Fan's lemma is a common tool for existence proofs in this area.

Purpose of the Study:

• Introduce a novel formulation of the symmetric vector equilibrium problem.
• Establish an existence theorem for this new problem formulation.
• Compare the utility of different mathematical lemmas for proving existence.

Main Methods:

• Development of a new symmetric vector equilibrium problem formulation.
• Application of nonlinear scalarization mapping techniques.
• Utilizing the maximal element lemma for existence proof.
• Demonstrating the limitations of Ky Fan's lemma for the main theorem.

Main Results:

• An existence theorem is established for the newly introduced symmetric vector equilibrium problem.
• The maximal element lemma is shown to be crucial, while Ky Fan's lemma is not applicable for the main theorem.
• The results extend and improve upon existing literature by relaxing conditions such as lower semicontinuity and quasiconvexity.

Conclusions:

• The new formulation and existence theorem offer a broader framework for studying vector equilibrium problems.
• The findings highlight the specific applicability of different mathematical tools in equilibrium problem theory.
• This work advances the understanding of equilibrium problems by relaxing stringent prior conditions.