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An algorithm for equilibrium problems with mixed-type fixed point constraints.

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This study introduces novel nonlinear mappings (ϑ-strictly asymptotically pseudocontractive-type multivalued mapping) and a new iterative method. The method converges to solutions for equilibrium problems and common fixed points, enhancing existing literature.

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Area of Science:

  • Nonlinear analysis
  • Functional analysis
  • Optimization theory

Background:

  • Equilibrium problems (EP) are fundamental in applied mathematics and economics.
  • Fixed point theory for multivalued mappings is crucial for solving various mathematical models.
  • Existing methods often lack convergence guarantees for complex mapping types.

Purpose of the Study:

  • Introduce a new class of mappings: ϑ-strictly asymptotically pseudocontractive-type multivalued mapping (ϑ-SAPM).
  • Develop and analyze a novel iterative method for solving equilibrium problems and finding common fixed points.
  • Extend and generalize existing results in nonlinear analysis and fixed point theory.

Main Methods:

  • Definition and analysis of ϑ-SAPM in Hilbert spaces.
  • Initiation of a new iterative algorithm.
  • Convergence analysis using standard techniques in nonlinear functional analysis.
  • Demonstration of mapping class independence.
  • Numerical illustration of the proposed method.

Main Results:

  • Strong convergence of the iterative method to the solution set of an EP.
  • Strong convergence to the set of common fixed points for two families of mappings (ϑ-SAPM and ϑ-strictly pseudocontractive-type multivalued mapping - ϑ-SPM).
  • Proof of the independence of the considered mapping classes.
  • Validation of the method's implementability through a numerical example.

Conclusions:

  • The proposed method provides a robust tool for solving equilibrium problems and finding common fixed points.
  • The newly introduced ϑ-SAPM class expands the scope of nonlinear mappings studied.
  • The findings generalize and improve upon several existing results in the literature.