Auxiliary principle technique and iterative algorithm for a perturbed system of generalized multi-valued mixed
Mijanur Rahaman1, Chin-Tzong Pang2, Mohd Ishtyak1
1Department of Mathematics, Aligarh Muslim University, Aligarh, 202002 India.
Journal of Inequalities and Applications
|March 21, 2017
Summary
This study introduces an iterative algorithm for solving perturbed generalized multi-valued mixed quasi-equilibrium-like problems in Hilbert spaces. The method ensures strong convergence of approximate solutions under monotonicity conditions.
Area of Science:
- Mathematical Analysis
- Optimization Theory
- Functional Analysis
Background:
- Generalized mixed quasi-equilibrium-like problems are crucial in applied mathematics and economics.
- Multi-valued mappings introduce complexity in solving equilibrium problems.
- Hilbert spaces provide a robust framework for analyzing such problems.
Purpose of the Study:
- To introduce and analyze a perturbed system of generalized multi-valued mixed quasi-equilibrium-like problems.
- To develop an iterative algorithm for finding approximate solutions.
- To establish the strong convergence of the proposed algorithm.
Main Methods:
- Development of a perturbed auxiliary system.
- Application of the Fan-KKM technique for existence and uniqueness.
- Utilization of an auxiliary principle technique for algorithm formulation.
- Monotonicity and mild conditions for convergence analysis.
Main Results:
- Existence and uniqueness of solutions for the auxiliary system.
- Formulation of an iterative algorithm for the main problem.
- Demonstration of strong convergence of iterative sequences.
- Generalization of existing results in the field.
Conclusions:
- The proposed iterative algorithm effectively solves the perturbed system.
- The convergence analysis provides theoretical guarantees for the method.
- This work extends the understanding of equilibrium problems with multi-valued mappings.
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