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A Study on Mixed Variational-like Inequality Under Generalized Quasimonotonicity in Banach Spaces

Author

Listed:
  • Gayatri Pany

    (Department of Mathematics, Institute of Technology, Nirma University, Ahmedabad 382481, Gujarat, India
    These authors contributed equally to this work.)

  • Bijaya K. Sahu

    (Department of Mathematics, Chandbali College, Chandbali, Bhadrak 756133, Odisha, India
    These authors contributed equally to this work.)

  • Ram N. Mohapatra

    (Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA
    These authors contributed equally to this work.)

Abstract

This paper studies nonlinear mixed variational-like inequalities under generalized quasimonotone mappings in Banach spaces. The main objective of this work is to relax the well-known KKM (Knaster–Kuratowski–Mazurkiewicz) condition which is extensively used in the literature to prove the existence of solutions for variational inequalities and equilibrium problems, and to establish the existence of solutions for the nonlinear mixed variational-like inequalities, which uncovers another approach of solving variational inequalities. Further, we propose an iterative algorithm to find approximate solutions to our problem and to study its convergence criteria. Finally, as an application, we find a gap function for nonlinear mixed variational-like inequalities, which uncovers another way of solving our variational-like inequalities using the methods of solutions and algorithms of the optimization problems.

Suggested Citation

  • Gayatri Pany & Bijaya K. Sahu & Ram N. Mohapatra, 2025. "A Study on Mixed Variational-like Inequality Under Generalized Quasimonotonicity in Banach Spaces," Mathematics, MDPI, vol. 13(3), pages 1-15, January.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:3:p:388-:d:1576460
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    References listed on IDEAS

    as
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    2. Guoqiang Tian, 1993. "Generalized Quasi-Variational-Like Inequality Problem," Mathematics of Operations Research, INFORMS, vol. 18(3), pages 752-764, August.
    3. Khan, Suhel Ahmad & Chen, Jia-wei, 2015. "Gap function and global error bounds for generalized mixed quasi variational inequalities," Applied Mathematics and Computation, Elsevier, vol. 260(C), pages 71-81.
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    5. Austine Efut Ofem & Jacob Ashiwere Abuchu & Hossam A. Nabwey & Godwin Chidi Ugwunnadi & Ojen Kumar Narain, 2023. "On Bilevel Monotone Inclusion and Variational Inequality Problems," Mathematics, MDPI, vol. 11(22), pages 1-28, November.
    6. Suhel Ahmad Khan & Jia-Wei Chen, 2015. "Gap Functions and Error Bounds for Generalized Mixed Vector Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 166(3), pages 767-776, September.
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