Abstract generalized vector quasi-equilibrium problems in noncompact Hadamard manifolds

Area of Science:

• Mathematical Analysis
• Optimization Theory
• Differential Geometry

Background:

• The study addresses the abstract generalized vector quasi-equilibrium problem (GVQP).
• It focuses on noncompact Hadamard manifolds, a complex geometric setting.
• Existing literature lacks comprehensive solutions for GVQP in such spaces.

Purpose of the Study:

• To establish the existence of solutions for the abstract GVQP.
• To extend the applicability of equilibrium problem theory to noncompact Hadamard manifolds.
• To unify and generalize existing results in the field.

Main Methods:

• The research employs techniques from nonlinear analysis and variational inequalities.
• Existence theorems are proven using fixed-point theorems or related analytical tools.
• The study analyzes the properties of solution sets within the specified manifold.

Main Results:

• Existence of solutions is demonstrated for the abstract generalized vector quasi-equilibrium problem.
• Applications include abstract vector quasi-equilibrium problems and scalar equilibrium problems.
• New results on mixed variational inequalities are derived.

Conclusions:

• The findings significantly contribute to the theory of equilibrium problems on manifolds.
• The results generalize and unify numerous previously established theorems.
• This work opens avenues for further research in geometric analysis and optimization.