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Bi-step Method for Variational Inequalities on Riemannian Manifold of Non-negative Constant Curvature

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Abstract

In this paper, we have proposed a bi-step method for solving variational inequalities on Riemannian manifold with the use of auxiliary principle technique. A relaxed condition of monotonicity has been used to show the convergence of this newly defined method. Furthermore, several important characterizations for the convex functions on the strongly convex subset of Riemannian manifold have also been discussed.

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References

  1. Ansari, Q.H., Lalitha, C.S., Mehta, M.: Generalized Convexity, Non-smooth Variational Inequalities, and Non-smooth Optimization, 157–200. CRS Press Taylor and Frances Group, London (2014)

    Google Scholar 

  2. Boothby, W.M.: An introduction Differential Manifolds and Riemannian Geometry, 2nd edn. Academic Press, New York (1986)

    MATH  Google Scholar 

  3. DoCarmo, M.P.: Riemannian Geometry. Birkhauser, Boston (1992)

    Google Scholar 

  4. Hosseini, S., Pauryayevali, M.R.: Generalized gradients and characterization of epi-Lipschitz sets in Riemannian manifolds, 3884–3895. Nonlinear Anal, TMA (2011)

    Google Scholar 

  5. James, R.: Munkres. Addison-Wesley Publishing Company, Analysis on Manifold (1930)

    Google Scholar 

  6. Jayswal, A.,Kumari, B., Ahmad, I.: Vector Variational Inequalities on Riemannian manifolds with approximate geodesic star-shaped functions (2021)

  7. Long-Li, S., Li, C., Liou, Y.C., Yao, J.C.: Existence of solution of variational inequalities on Riemannian manifolds. Publ. Nonlinear Anal. 71, 5695–5706 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Nemeth, S.Z.: Monotonic vector fields, 437–449. Publ. Math. 54, 3–4 (1999)

    Google Scholar 

  9. Nemeth, S.Z.: Variational inequalities on Hadamard manifolds. Publ. Nonlinear Anal. 52, 1491–1498 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Nemeth, S.Z.: Five kind of monotone vector fields. Pure Appl. Math. 9, 417–428 (1999)

    MathSciNet  MATH  Google Scholar 

  11. Noor, M.A., Noor, K.I.: Auxiliary principle technique for variational inequalities. Appl. Math. Inf. Sci. 11(1), 165–169 (2017)

    Article  MathSciNet  Google Scholar 

  12. Noor, M.A., Noor, K.I.: Two-step methods for variational inequalities on Hadamard manifolds. Appl. Math. Inf. Sci. 9(4), 1863–1867 (2015)

    MathSciNet  Google Scholar 

  13. Noor, M.A.: New approximation schemes for general variational inequalities. J. Math. Appl. 251, 217–229 (2000)

    MathSciNet  MATH  Google Scholar 

  14. Sakai, T.: Riemannian Geometry in: Translations of Mathematical Monographs, vol. 149. American Mathematical Society (1992)

  15. Stampacchia, G.: Variational inequalities. In: Theory and Applications of Monotone Operators, A. Ghizzetti, (ed.), Proc. NATO Adv. Study Institute, Venice, Oderisi, Gubbio (1968)

  16. Tang, G., Zhou, L.W., Huang, N.J.: The proximal point algorithm for pseudo monotone variational inequalities on Hadamard Manifolds, Optim. Lett. https://doi.org/10.1007/s11590-0459-7

  17. Thamelt, W.: Directional derivatives and the generalized gradients on manifolds. Optimization, 97–115 (1992)

  18. Udristi, C.: Convex Functions and Optimization Methods on Riemannian Manifolds. Kluwer Academic Publishers, Dordrecht (1994)

    Book  Google Scholar 

  19. Udriste, C.: Continuity of convex functions on Riemannian manifolds. Bull. Math. Roum. 21, 215–218 (1977)

    MathSciNet  MATH  Google Scholar 

  20. Walter, R.: On the metric projections onto convex sets in Riemannian Spaces. Arch. Math. 96, 413–443 (1972)

    Google Scholar 

  21. Wang, J.H., Lopez, G., Martin-morquez, Li, C.: Monotonic and Accetive vector fields on Riemannian Manifolds. J. Optim. Theory Appl. 146, 691–708 (2010)

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  1. Department of Mathematics, Aligarh Muslim University, Aligarh, 202002, India

    Nagendra Singh, Akhlad Iqbal & Shahid Ali

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  1. Nagendra Singh
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  2. Akhlad Iqbal
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  3. Shahid Ali
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Correspondence to Akhlad Iqbal.

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Singh, N., Iqbal, A. & Ali, S. Bi-step Method for Variational Inequalities on Riemannian Manifold of Non-negative Constant Curvature. Int. J. Appl. Comput. Math 9, 25 (2023). https://doi.org/10.1007/s40819-023-01517-3

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  • DOI: https://doi.org/10.1007/s40819-023-01517-3

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