Skip to main content
SpringerLink
Log in

6. Some applications of multivalued mappings

  • Published:
Journal of Soviet Mathematics Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Literature cited

  1. E. A. Barbashin, “On the theory of generalized dynamical systems,” Uch. Zap. Mosk. Gos. Univ., Mat.,135, No. 2, 110–113 (1948).

    Google Scholar 

  2. N. Benkafadar and B. D. Gel'man, “On the local degree of multivalued vector fields with Fredholm principal part,” Voronezh Univ. (1982).

  3. N. Benkafadar and B. D. Gel'man, “On homotopy properties of spaces of subsets,” in: Topological and Geometric Methods in Mathematical Physics, Voronezh (1983), pp. 111–115.

  4. Yu. G. Borisovich, “Topology and nonlinear functional analysis,” Usp. Mat. Nauk,34, No. 6, 14–22 (1979).

    Google Scholar 

  5. Yu. G. Borisovich, “On the theory of topological degree of nonlinear Fredholm mappings perturbed by a multivalued operator,” Voronezh Univ. (1980).

  6. Yu. G. Borisovich, “On topological methods in the problem of solvability of nonlinear equations,” Proceedings of the Leningrad International Conference in Topology, 23–27 August, 1982, Nauka, Leningrad (1983), pp. 39–49.

    Google Scholar 

  7. Yu. G. Borisovich, B. D. Gel'man, E. Mukhamadiev, and V. V. Obukhovskii, “On the rotation of multivalued vector fields,” Dokl. Akad. Nauk SSSR,187, No. 5, 971–973 (1969).

    Google Scholar 

  8. Yu. G. Borisovich, B. D. Gel'man, É. Mukhamadiev, and V. V. Obukhovskii, “On the rotation of multivalued vector fields,” Proc. of the Seminar on Functional Analysis, Voronezh Univ., Issue 12 (1969), pp. 69–84.

  9. Yu. G. Borisovich, B. D. Gel'man, A. D. Myshkis, and V. V. Obukhovskii, “Topological methods in the theory of fixed points of multivalued mappings,” Usp. Mat. Nauk,35, No. 1, 59–126 (1980).

    Google Scholar 

  10. Yu. G. Borisovich, B. D. Gel'man, A. D. Myshkis, and V. V. Obukhovskii, “Multivalued mappings,” J. Sov. Math.,24, No. 6 (1984).

  11. Yu. G. Borisovich and Yu. E. Gliklikh, “On the Lefschetz number for a class of multivalued mappings,” in: Seventh Mathematics Summer School (1969), Kiev, 1970, pp. 283–294.

  12. Yu. G. Borisovich, V. G. Zvyagin, and Yu. I. Sapronov, “Nonlinear Fredholm mappings and Leray-Schauder theory,” Usp. Mat. Nauk,32, No. 4, 3–54 (1977).

    Google Scholar 

  13. Yu. G. Borisovich and V. V. Obukhovskii, “Homotopy properties, the theory of rotation, and fixed-point theorems for a class of noncompact multivalued mappings,” Voronezh Univ. (1980).

  14. A. I. Bulgakov, “Knezer's theorem for a class of integral inclusions,” Différents. Uravn.,16, No. 5, 894–900 (1980).

    Google Scholar 

  15. A. I. Bulgakov and L. N. Lyapin, “On the connectivity of sets of solutions of functional inclusions,” Mat. Sb.,119, No. 2, 295–300 (1982).

    Google Scholar 

  16. E. A. Gango and A. I. Povolotskii, “Regular directing functions for differential equations with multivalued right side,” in: Function Theory and Functional Analysis [in Russian], Leningrad (1975), pp. 35–41.

  17. B. D. Gel'man, “Multivalued integral operators and ω-periodic solutions,” Tr. Mat. Fak. Voronezh. Univ., Issue 4, 35–44 (1971).

  18. B. D. Gel'man, “A generalization of Kakutani's fixed-point theorem for multivalued mappings,” Dokl. Akad. Nauk SSSR,209, No. 1, 22–24 (1973).

    Google Scholar 

  19. B. D. Gel'man, “A topological characteristic of multivalued mappings and theorems of Kakutani type,” Tr. Mat. Fak. Voronezh. Univ., Issue 12, 12–21 (1974).

  20. B. D. Gel'man, “A topological characteristic of multivalued mappings and fixed-point theorems,” Dokl. Akad. Nauk SSSR,221, No. 3, 524–527 (1975).

    Google Scholar 

  21. B. D. Gel'man, “A topological characteristic of multivalued mappings in a Banach space,” Tr. Mat. Fak. Voronezh. Univ., Issue 16, 17–23 (1975).

  22. B. D. Gel'man, “On some classes of sections of multivalued mappings,” in: Application of Topology in Modern Analysis [in Russian], Voronezh (1985), pp. 42–62.

  23. Yu. E. Gliklikh, “Fixed points of multivalued mappings with convex images and the rotation of multivalued vector fields,” Sb. Tr. Aspirantov Mat. Fak. Voronezh. Univ., Issue 1, 30–38 (1971).

  24. P. P. Zabreiko, M. A. Krasnosel'skii, and V. V. Strygin, “On invariance principles of the rotations” Izv. Vyssh. Uchebn. Zaved., Mat., No. 5, 51–57 (1972).

    Google Scholar 

  25. A. D. Ioffe and V. L. Levin, “Subdifferentials of convex functions,” Tr. Mosk. Mat. Obshch.,26, 3–73 (1972).

    Google Scholar 

  26. A. D. Ioffe and V. M. Tikhomirov, “Duality of convex functions and extremal problems,” Usp. Mat. Nauk,23, No. 6, 51–116 (1968).

    Google Scholar 

  27. A. D. Ioffe and V. M. Tikhomirov, The Theory of Extremal Problems [in Russian], Nauka, Moscow (1974).

    Google Scholar 

  28. M. A. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations [in Russian], Gostekhizdat, Moscow (1956).

    Google Scholar 

  29. M. A. Krasnosel'skii, The Shift Operator along Trajectories of Differential Equations [in Russian], Nauka, Moscow (1966).

    Google Scholar 

  30. M. A. Krasnosel'skii and P. P. Zabreiko, Geometric Methods of Nonlinear Analysis [in Russian], Nauka, Moscow (1975).

    Google Scholar 

  31. A. D. Myshkis, “Generalizations of the theorem on a rest point of a dynamical system within a closed trajectory,” Mat. Sb.,34, No. 3, 525–540 (1954).

    Google Scholar 

  32. V. V. Obukhovskii, “On some fixed-point principles for multivalued impermeable operators,” Tr. Mat. Fak. Voronezh. Univ., Issue 4, 70–79 (1971).

    Google Scholar 

  33. V. V. Obukhovskii, “Periodic solutions of control systems,” Tr. Mat. Fak. Voronezh. Univ., Issue 7, 68–76 (1972).

  34. V. V. Obukhovskii, “On the question of periodic solutions of differential equations with multivalued right side,” Tr. Mat. Fak. Voronezh. Univ., Issue 10, 74–82 (1973).

  35. V. V. Obukhovskii, “Asymptotic theorems on a fixed point and a rest point of dynamical systems without uniqueness,” Sb. Rabot Aspirantov po Teorii Funktsii i Différents. Uravneniyam, Voronezh (1974), pp. 30–38.

  36. V. V. Obukhovskii, “On a condition of positive equilibrium in a model of competing economics,” Tr. of the Sixth Winter School on Mathematical Programming and Related Questions, Drogobych, 24 January-5 February 1973, Moscow (1975), pp. 183–189.

  37. V. V. Obukhovskii, “On the rotation of noncompact almost acyclic multivalued vector fields,” in: Equations on Manifolds [in Russian], Voronezh (1982), pp. 119–123.

  38. V. V. Obukhovskii, “On the topological degree for one class of noncompact multivalued mappings,” in: Functional Analysis, Ul'yanovsk, No. 23 (1984), pp. 82–93.

    Google Scholar 

  39. V. V. Obukhovskii, “On a generalization of the connectivity principle of M. A. Krasnosel'-skii-A. I. Perov,” in: Fifth Tiraspol' Symposium on General Topology and Its Applications [in Russian], Shtiintsa, Kishinev (1985), pp. 186–187.

    Google Scholar 

  40. V. V. Obukhovskii and E. V. Gorokhov, “On the definition of the rotation of one class of compactly restrictable multivalued vector fields,” Tr. Mat. Fak. Voronezh. Univ., Issue 12, 45–54 (1974).

    Google Scholar 

  41. V. I. Ponomarev, “A new space of closed sets and multivalued continuous mappings of bicompact sets,” Mat. Sb.,48, No. 2, 191–212 (1959).

    Google Scholar 

  42. B. N. Sadovskii, “Limiting compact and impermeable operators,” Usp. Mat. Nauk,27, No. 1, 81–146 (1972).

    Google Scholar 

  43. Yu. I. Sapronov, “On the homotopy classification of impermeable mappings,” Tr. Mat. Fak. Voronezh. Univ., Issue 6, 78–80 (1972).

  44. E. G. Sklyarenko, “On some applications of the theory of sheaves in general topology,” Usp. Mat. Nauk,19, No. 6, 47–70 (1964).

    Google Scholar 

  45. A. F. Filippov, “On some questions of the theory of optimal control,” Vestn. Mosk. Univ., Ser. Mat., Mekh., Astron., Fiz., Khimii, No. 2, 25–32 (1959).

    Google Scholar 

  46. A. F. Filippov, Differential Equations with Discontinuous Right Side, [in Russian], Nauka, Moscow (1985).

    Google Scholar 

  47. N. Arronszajn, “Le correspondant topologique de l'unicité dans la théorie des équations différentielles,” Ann. Math.,43, No. 4, 326–350 (1942).

    Google Scholar 

  48. T. P. Aubin and A. Cellina, Differential Inclusions, Berlin-Heidelberg, Springer-Verlag (1984).

    Google Scholar 

  49. G. Berge, Théorie Generale des Jeux à n Personnes, Gauthier-Villars, Paris (1957).

    Google Scholar 

  50. C. Berge, Espaces Topologiques. Fonctions Multivoques, Dunod, Paris (1959).

    Google Scholar 

  51. F. E. Browder, “Nonlinear operators and nonlinear equations of evolution in Banach spaces,” Proc. Symp. in Pure Math., Vol. 18, Part 2, Am. Math. Soc., Providence, RI (1976).

    Google Scholar 

  52. F. E. Browder, “Fixed point theory and nonlinear problems,” Bull. Am. Math. Soc.,9, No. 1, 1–39 (1983).

    Google Scholar 

  53. J. Bryszewski, “On a class of multivalued vector fields in Banach spaces,” Fund. Math.,97, No. 2, 79–94 (1977).

    Google Scholar 

  54. C. Castaing, “Sur les équations differentielles multivoques,” C. R. Acad. Sci.,263, No. 2, A63-A66 (1966).

    Google Scholar 

  55. C. Castaing, “Sur les multiapplications measurables,” Rev. Franc. Inform, et Rech. Operat.,1, No. 1, 91–126 (1967).

    Google Scholar 

  56. A. Cellina and A. Lasota, “A new approach to the definition of topological degree for multivalued mappings,” Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis., Mat. e Natur.,47, No. 6, 434–440 (1969–70).

    Google Scholar 

  57. T. Eells, “Fredholm structures,” Proc. Symp. in Pure Math., Vol. 18, Am. Math. Soc., Providence, RI (1970), pp. 62–85.

    Google Scholar 

  58. S. Eilenberg and D. Montgomery, “Fixed point theorems for multivalued transformations,” Am. J. Math.,68, 214–222 (1946).

    Google Scholar 

  59. K. Fan, “Fixed-point and minimax theorems in locally convex topological linear spaces,” Proc. Nat. Acad. Sci. USA,38, 121–126 (1952).

    Google Scholar 

  60. I. L. Glicksberg, “A further generalization of the Kakutani fixed point theorem with application to Nash equilibrium points,” Proc. Am. Math. Soc.,3, No. 1, 170–174 (1952).

    Google Scholar 

  61. G. Haddad, “Topological properties of the sets of solutions for functional differential inclusions,” Nonlinear Analysis: Theory, Methods, Applications,5, No. 12, 1349–1366 (1981).

    Google Scholar 

  62. M. Q. Jacobs, “Measurable multivalued mappings and Lusin's theorem,” Trans. Am. Math. Soc.,134, No. 3, 471–481 (1968).

    Google Scholar 

  63. S. Kakutani, “A generalization of Brower's fixed point theorem,” Duke Math. J.,8, 457–459 (1941).

    Google Scholar 

  64. S. Karlin, Mathematical Methods and Theory in Games, Programming, and Economics. I, II, Pergamon Press, London-Paris (1959).

    Google Scholar 

  65. N. Kikuchi, “Control problems of contingent equation,” Publ. Res. Inst. Math. Sci., Ser. A,3, No. 1, 85–99 (1967).

    Google Scholar 

  66. N. Kikuchi, “On contingent equations satisfying the Caratheodory type conditions,” Publ. Res. Inst. Math. Sci., Ser. A,3, No. 3, 361–371 (1968).

    Google Scholar 

  67. K. Kuratowski, Topology, Vol. 1, Academic Press, New York-London (1966).

    Google Scholar 

  68. K. Kuratowski, Topology, Vol. 2, Academic Press, New York-London (1968).

    Google Scholar 

  69. A. Lasota and Z. Opial, “An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations,” Bull. Acad. Pol. Sci., Ser. Sci. Math., Astron. Phys.,13, Nos. 11–12, 781–786 (1965).

    Google Scholar 

  70. A. Lasota and Z. Opial, “An approximation theorem for multivalued mappings,” Podst. Sterow.,1, No. 1, 71–75 (1971).

    Google Scholar 

  71. J. H. Lasry and R. Robert, “Acycliceté de l'ensemble des solutions de certaines équations fonctionelles,” C. R. Acad. Sci.,282, No. 22, A1283-A1286 (1976).

    Google Scholar 

  72. E. A. Michael, “Topologies on spaces of subsets,” Trans. Am. Math. Soc.,71, 152–183 (1951).

    Google Scholar 

  73. E. A. Michael, “Continuous selections. I,” Ann. Math.,63, No. 2, 361–381 (1956).

    Google Scholar 

  74. P. Momal, “Théorèmes de maximum,” C. R. Acad. Sci.,A278, No. 13, 905–907 (1974).

    Google Scholar 

  75. S. B. Nadler, “Multivalued contraction mappings,” Pac. J. Math.,30, No. 2, 475–488 (1969).

    Google Scholar 

  76. H. Nikaido, Convex Structures and Economic Theory, Academic Press, New York-London (1968).

    Google Scholar 

  77. L. Nirenberg, Topics in Nonlinear Functional Analysis, Courant Institute of Mathematical Sciences, New York University (1974).

  78. V. V. Obukhovskii, “A degree and fixed points for a class of noncompact multivalued maps,” Leningrad International Topology Conference Reports, Leningrad Division, Nauka, Leningrad (1982), p. 116.

    Google Scholar 

  79. R. S. Palais, Seminar on the Atiyah-Singer Index Theorem, Princeton Univ. Press (1965).

  80. A. Plis, “Remark on measurable set-valued functions,” Bull. Acad. Polen. Sci., Ser. Sci. Math., Astron. Phys.,9, No. 12, 857–859 (1961).

    Google Scholar 

  81. R. L. Plunkett, “A fixed point theorem for continuous multivalued transformations,” Proc. Am. Math. Soc.,7, No. 1, 160–163 (1956).

    Google Scholar 

  82. R. E. Smithson, “Fixed point theorems for certain classes of multifunctions,” Proc. Am. Math. Soc.,31, No. 2, 595–600 (1972).

    Google Scholar 

  83. G. P. Szegö and G. Treccani, Semigruppi di Transformazioni Multivoche, Lect. Notes Math.,101 (1969).

  84. L. Vietoris, “Über den. höheren Zusammenhang kompakter Raume und eine Klasse von zusammen-hangstreuen Abbildungen,” Math, Ann.,97, 454–472 (1927).

    Google Scholar 

  85. A. D. Wallace, “A fixed point theorem for trees,” Bull. Am. Math. Soc.,47, 757–760 (1941).

    Google Scholar 

Download references

Rights and permissions

Reprints and permissions

About this article

Cite this article

6. Some applications of multivalued mappings. J Math Sci 39, 2804–2811 (1987). https://doi.org/10.1007/BF01127065

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01127065

Keywords

Search

Navigation