Equilibria on $L$-retracts in Riemannian manifolds

Seyedehsomayeh Hosseini, Mohamad R. Pouryayevali

DOI: http://dx.doi.org/10.12775/TMNA.2016.017

Abstract


We introduce a class of subsets of Riemannian manifolds called the $L$-retract. Next we consider a topological degree for set-valued upper semicontinuous maps defined on open sets of compact $L$-retracts in Riemannian manifolds. Then, we present a theorem on the existence of equilibria (or zeros) of an upper semicontinuous set-valued map with nonempty closed convex values satisfying the tangency condition defined on a compact $L$-retract in a Riemannian manifold.

Keywords


Set-valued map; degree theory; Euler characteristic; equilibrium

Full Text:

PREVIEW FULL TEXT

References


D. Azagra, J. Ferrera and F. López-Mesas, Nonsmooth analysis and Hamilton–Jacobi equations on Riemannian manifolds, J. Funct. Anal. 220 (2005), 304–361.

D. Azagra and J. Ferrera, Applications of proximal calculus to fixed point theory on Riemannian manifolds, Nonlinear. Anal. 67 (2007), 154–174.

H. Ben-El-Mechaiekh and W. Kryszewski, Equilibria of set-valued maps on nonconvex domains, Trans. Amer. Math. Soc. 346 (1997), 4159–4179.

G.M. Bergman and B.R. Halpern, A fixed point theorem for inward and outward maps, Trans. Amer. Math. Soc. 130 (1968), 353–358.

F. E. Browder, The fixed point theory of multivalued mappings in topological vector spaces, Math. Ann. 117 (1968), 283–301.

R. Brown, The Lefschetz Fixed Point Theorem, Scott, Foresman and Company, Glenview IL, London, 1971.

B. Cornet, Paris avec handicaps et théoremès de surjectivité de correspondances, C.R. Acad. Sci. Paris Sér. A 281 (1975), 479–482.

B. Cornet, Euler characteristic and fixed point theorems, Positivity 6 (2002), 243–260.

B. Cornet and M.-O. Czarnecki, Existence of generalized equilibria, Nonlinear. Anal. 44 (2001), 555–574.

A. Ćwiszewski and W. Kryszewski, Homotopy invariants for tangent vector fields on closed sets, Nonlinear. Anal. 65 (2006), 175–209.

M.P. do Carmo, Riemannian Geometry, Birkhäuser, Boston, 1992.

K. Fan, Extensions of two fixed point theorems of F.E. Browder, Math. Z. 112 (1969), 234–240.

A. Granas and J. Dugundji, Fixed Point Theory, Springer, Berlin 2004.

M.W. Hirsch, Differential Topology, Graduate Texts in Mathematics, Vol. 33, Springer, New York, 1976.

S. Hosseini and M.R. Pouryayevali, Generalized gradients and characterization of epiLipschitz sets in Riemannian manifolds, Nonlinear Anal. 74 (2011), 3884–3895.

S. Hosseini and M.R. Pouryayevali, Euler characterization of epi-Lipschitz subsets of Riemannian manifolds, J. Convex. Anal. 20 (2013), 67–91.

S. Hosseini and M.R. Pouryayevali, On the metric projection onto prox-regular subsets of Riemannian manifolds, Proc. Amer. Math. Soc. 141 (2013), 233–244.

J. M. Lasry and R. Robert, Analyse nonlinéaire multivoque, Cahiers de Math. de la Décision, Vol. 7611 (Université de Paris–Dauphine, Paris 1976).

J. M. Lee, Riemannian Manifolds, An Introuction to Curvature, Graduate Texts in Mathematics, Vol. 176, Springer, New York 1997.

J. Milnor, Topology from the Differentiable Viewpoint, Princton University Press, Princton, New Jersey, 1965.

T. Sakai, Riemannian Geometry, Translations of Mathematical Monographs, Vol. 149, American Mathematical Society, Providence, RI, 1996.

E. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966.


Refbacks

  • There are currently no refbacks.

Partnerzy platformy czasopism