Abstract

A notion of pseudo-Jacobian of continuous vector fields on Riemannian manifolds is presented. It is shown that the Clarke generalized Jacobian and Mordukhovich coderivative for locally Lipschitz vector fields are pseudo-Jacobians. Moreover, monotone vector fields are characterized in terms of pseudo-Jacobians.

Author information

Contact details are reproduced from the original publication and may be historical.

Mohamad Reza Pouryayevali

Dept. of Mathematics, University of Isfahan, Isfahan, Iran

pourya@math.ui.ac.ir

Keywords

  • Pseudo-Jacobian
  • Lipschitz vector fields
  • Monotone vector fields
  • Riemannian manifolds

Mathematics Subject Classification

58C20, 49J52, 47H05

References

  1. D. Azagra, J. Ferrera: Applications of proximal calculus to fixed point theory on Riemannian manifolds, Nonlinear. Anal. 67 (2007) 154–174.https://doi.org/10.1016/j.na.2006.04.019Google Scholar
  2. D. Azagra, J. Ferrera: Regularization by sup-inf convolutions on Riemannian manifolds: An extension of Lasry-Lions theorem to manifolds of bounded curvature, J. Math. Anal. Appl. 423 (2015) 994–1024.https://doi.org/10.1016/j.jmaa.2014.10.022Google Scholar
  3. D. Azagra, J. Ferrera, F. López-Mesas: Nonsmooth analysis and Hamilton-Jacobi equations on Riemannian manifolds, J. Funct. Anal. 220 (2005) 304–361.https://doi.org/10.1016/j.jfa.2004.10.008Google Scholar
  4. F. H. Clarke, Yu. S. Ledyaev, R. J. Stern, P. R. Wolenski: Nonsmooth Analysis and Control Theory, Graduate Texts in Mathematics 178, Springer, New York (1998).Google Scholar
  5. J. X. Da Cruz Neto, O. P. Ferreira, L. R. Lucambio Perez: Contributions to the study of monotone vector fields, Acta Math. Hung. 94 (2002) 307–320.https://doi.org/10.1023/A:1015643612729Google Scholar
  6. S. Hosseini, M. R. Pouryayevali: Euler characterization of epi-Lipschitz subsets of Riemannian manifolds, J. Convex. Analysis 20 (2013) 67–91.https://doi.org/10.68381/jca20005Google Scholar
  7. S. Hosseini, M. R. Pouryayevali: Generalized gradients and characterization of epi-Lipschitz sets in Riemannian manifolds, Nonlinear Anal. 74 (2011) 3884–3895.https://doi.org/10.1016/j.na.2011.02.023Google Scholar
  8. S. Hosseini, M. R. Pouryayevali: On the metric projection onto prox-regular subsets of Riemannian manifolds, Proc. Amer. Math. Soc. 141 (2013) 233–244.https://doi.org/10.1090/S0002-9939-2012-11828-3Google Scholar
  9. V. Jeyakumar, D. T. Luc: Approximate Jacobian matrices for nonsmooth continuous maps and C 1 -Optimization, SIAM J. Control Optim. 36 (1998) 1851–1832.https://doi.org/10.1137/S0363012996311745Google Scholar
  10. V. Jeykumar, D. T. Luc: Nonsmooth Vector Functions and Continuous Optimization, Springer Optimization and its Applications 10, Springer, New York (2008).Google Scholar
  11. V. Jeykumar, D. T. Luc: Characterizations of generalized monotone nonsmooth continuous maps using approximate Jacobians, J. Convex Analysis 5 (1998) 119–132.https://doi.org/10.68381/jca05-9Google Scholar
  12. S. Lang: Fundamentals of Differential Geometry, Graduate Texts in Mathematics 191, Springer, New York (1999).https://doi.org/10.1007/978-1-4612-0541-8Google Scholar
  13. Yu. S. Ledyaev, Q. J. Zhu: Nonsmooth analysis on smooth manifolds, Trans. Amer. Math. Soc. 359(8) (2007) 3687–3732.https://doi.org/10.1090/S0002-9947-07-04075-5Google Scholar
  14. C. Li, G. López, V. Martín-Márquez: Monotone vector fields and the proximal point algorithm on Hadamard manifolds, J. Lond. Math. Soc., II. Ser. 79 (2009) 663–683.https://doi.org/10.1112/jlms/jdn087Google Scholar
  15. B. S. Mordukhovich: Variational Analysis and Generalized Differentiation, I: Basic Theory, Grundlehren der Mathematischen Wissenschaften 331, Springer, New York (2006).https://doi.org/10.1007/3-540-31246-3Google Scholar
  16. S. Z. Németh: Monotone vector fields, Publ. Math. 54(3–4) (1999) 437–449.https://doi.org/10.5486/PMD.1999.2057Google Scholar
  17. F. Rampazzo, H. J. Sussmann, Commutators of flow maps of nonsmooth vector fields, J. Differ. Equations. 232 (2007) 134–175.https://doi.org/10.1016/j.jde.2006.04.016Google Scholar
  18. T. Sakai: Riemannian Geometry, Trans. Math. Monogor. 149, Amer. Math. Soc., Providence (1996).https://doi.org/10.1090/mmono/149Google Scholar
  19. H. J. Sussmann: Set separation, approximating multicones, and the Lipschitz maximum principle, J. Differ. Equations. 243 (2007) 448–488.https://doi.org/10.1016/j.jde.2007.05.041Google Scholar

Suggested citation

E. Ghahraei, S. Hosseini, M. R. Pouryayevali. “Pseudo-Jacobian and Characterization of Monotone Vector Fields on Riemannian Manifolds.” Journal of Convex Analysis 24 (2017), No. 1, 149–168. https://doi.org/10.68381/jca24012

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