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A maximal element theorem in FWC-spaces and its applications.

Haishu Lu1, Qingwen Hu2, Yulin Miao1

  • 1School of Business, Jiangsu University of Technology, Changzhou, Jiangsu 213001, China.

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|May 1, 2014
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Summary

A new maximal element theorem is established in finite weakly convex spaces (FWC-spaces). This theorem is then applied to prove existence theorems for various equilibrium and optimization problems within these spaces.

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Area of Science:

  • Optimization Theory
  • Mathematical Analysis
  • Convex Analysis

Background:

  • Finite weakly convex spaces (FWC-spaces) lack standard linear or topological structures.
  • Existing theorems often rely on specific topological or convex properties not present in FWC-spaces.

Purpose of the Study:

  • To establish a maximal element theorem in the context of FWC-spaces.
  • To develop new existence theorems for several classes of mathematical problems in FWC-spaces.
  • To unify and extend existing results in the literature.

Main Methods:

  • A novel maximal element theorem is proven for FWC-spaces.
  • The established theorem is utilized to derive existence results for variational relation problems.
  • The theorem is applied to generalized equilibrium problems, equilibrium problems with bounds, and minimax problems.

Main Results:

  • A maximal element theorem is successfully established for FWC-spaces.
  • New existence theorems are demonstrated for variational relation, generalized equilibrium, equilibrium with bounds, and minimax problems.
  • The findings unify and extend previously known results.

Conclusions:

  • The maximal element theorem provides a foundational tool for FWC-spaces.
  • The study expands the applicability of existence theorems to a broader class of spaces.
  • This research contributes to the advancement of optimization and equilibrium theory.