Về Chỉ Số Tốpô của Giải Pháp cho Bất Đẳng Thức Biến Thiên Trên Các Manifol Riemann

Alvarez, F., Bolte, J., Munier, J.: A unifying local convergence result for Newton’s method in Riemannian manifolds. Found. Comput. Math. 8, 197–226 (2008) Azagra, D., Ferrera, J., López-Mesas, F.: Nonsmooth analysis and Hamilton–Jacobi equations on Riemannian manifolds. J. Funct. Anal. 220, 304–361 (2005) Benedetti, I., Obukhovskii, V.: On the index of solvability for variational inequalities in Banach spaces. Set-Valued Anal. 16(1), 67–92 (2008) Borisovich, Yu.G., Gelman, B.D., Myshkis, A.D., Obukhovskii, V.V.: Topological methods in the theory of fixed points of multivalued mappings, Uspekhi Mat. Nauk 35(1, 211), 59–126 (1980) (in Russian). English transl.: Russian Math. Surveys 35, 65–143 (1980) Borisovich, Yu.G., Gelman B.D., Myshkis, A.D., Obukhovskii, V.V.: Introduction to the Theory of Multivalued Maps and Differential Inclusions, 2nd edn. Librokom, Moscow (2011) (in Russian) Borisovich, Yu.G., Gliklikh, Yu.E.: On the Lefschetz number for a class of multivalued maps. In: Seventh Math. Summer School (Katsiveli, 1969), pp. 283–294. Izd. Akad. Nauk Ukrain SSR, Kiev (1970) (in Russian) Borsuk, K.: Theory of Retracts. Monografie Mathematyczne, Tom 44. Panstwowe Wydawnictwo Naukowe, Warsaw (1967) Brown, R.F.: The Lefschetz Fixed Point Theorem. Scott, Foresman and Co., Glenview (1971) do Carmo, M.P.: Riemannian Geometry, Mathematics: Theory & Applications. Birkhauser, Boston (1992) Cheeger, J., Gromoll, D.: On the structure of complete manifolds of nonnegative curvature. Ann. Math. 96(2), 413–443 (1972) Ćwiszewski, A., Kryszewski, W.: The constrained degree and fixed-point index theory for set-valued maps. Nonlinear Anal. 64(12), 2643–2664 (2006) Dedieu, J.P., Priouret, P., Malajovich, G.: Newton’s method on Riemannian manifolds: covariant alpha theory. IMA J. Numer. Anal. 23, 395–419 (2003) Dold, A.: Lectures on Algebraic Topology, 2nd edn. Grundlehren der Mathematischen Wissenschaften 200. Springer, Berlin (1980) Ferreira, O.P., Oliveira, P.R.: Proximal point algorithm on Riemannian manifolds. Optimization 51, 257–270 (2002) Ferreira, O.P., Svaiter, B.F.: Kantorovich’s theorem on Newton’s method in Riemannian manifolds. J. Complex. 18, 304–329 (2002) Górniewicz, L.: Topological Fixed Point Theory of Multivalued Mappings, 2nd edn. Topological Fixed Point Theory and its Applications 4. Springer, Dordrecht (2006) Gromoll, D., Klingenberg, W., Meyer, W.: Riemannsche geometrie im Grossen. In: Lecture Notes in Mathematics 55. Springer, Berlin (1968) Harker, P.T., Pang, J.S.: Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications. Math. Program. 48, 161–220 (1990) Hirsch, M.: Differential Topology. Springer, Berlin (1976) Hu, S., Papageorgiou, N.S.: Handbook of Multivalued Analysis, vol. I: Theory. Kluwer Academic, Dordrecht (1997) Hyman, D.M.: On decreasing sequences of compact absolute retracts. Fundam. Math. 64, 91–97 (1969) Kamenskii, M., Obukhovskii, V., Zecca, P.: Condensing multivalued maps and semilinear differential inclusions. In: Banach Spaces, de Gruyter Series in Nonlinear Analysis and Applications 7. Walter de Gruyter, Berlin (2001) Ledyaev, Y.S., Zhu, Q.J.: Nonsmooth analysis on smooth manifolds. Trans. Am. Math. Soc. 359, 3687–3732 (2007) Li, C., López, G., Martín-Máquez V.: Monotone vector fields and the proximal point algorithm on Hadamard manifolds. J. Lond. Math. Soc. 79, 663–683 (2009) Li, C., López, G., Martín-Márquez, V., Wang, J.H.: Resolvents of set-valued monotone vector fields in Hadamard manifolds. Set-Valued Var. Anal. 19(3), 361–383 (2011) Li, C., Mordukhovich, B.S., Wang, J.H., Yao, J.C.: Weak sharp minima on Riemannian manifolds. SIAM J. Optim. (2011, in press) Li, C., Wang, J.H.: Newton’s method on Riemannian manifolds: Smale’s point estimate theory under the γ-condition. IMA J. Numer. Anal. 26 228–251 (2006) Li, S.L., Li, C., Liou, Y.C., Yao, J.C.: Existence of solutions for variational inequalities on Riemannian manifolds. Nonlinear Anal. 71 5695–5706 (2009) Myshkis, A.D.: Generalizations of the theorem on a stationary point of a dynamical system inside a closed trajectory. Math. Sb. 34, 525–540 (1954). In Russian Németh, S.Z.: Five kinds of monotone vector fields. Pure Appl. Math. 9 417–428 (1999) Németh, S.Z.: Monotone vector fields. Publ. Math. 54, 437–449 (1999) Németh, S.Z.: Geodesic monotone vector fields. Lobachevskii J. Math. 5, 13–28 (1999) Németh, S.Z.: Monotonicity of the complementary vector field of a non-expansive map. Acta Math. Hung. 84, 189–197 (1999) Németh, S.Z.: Homeomorphisms and monotone vector fields. Publ. Math. 58(4), 707–716 (2001) Németh, S.Z.: Variational inequalities on Hadamard manifolds. Nonlinear Anal. 52(5), 1491–1498 (2003) Patriksson, M.: Nonlinear Programming and Variational Inequality Problems: A Unified Approach. Kluwer Academic, Dordrecht (1999) Sakai, T.: Riemannian Geometry. Translations of Mathematical Monographs, vol. 149. American Mathematical Society, Providence (1992) Udrişte, C.: Convex functions and optimization methods on Riemannian manifolds. In: Mathematics and Its Applications 297. Kluwer Academic Publishers Group, Dordrecht (1994) Wang, J.H., López, G., Martín-Márquez, V., Li, C.: Monotone and accretive vector fields on Riemannian manifolds. J. Optim. Theory Appl. 146, 691–708 (2010) Walter, R.: On the metric projections onto convex sets in Riemannian spaces. Arch. Math. (Basel) 25, 91–98 (1974)