Topological approach to differential inclusions

  • Lech Górniewicz
Part of the NATO ASI Series book series (ASIC, volume 472)


The purpose of these lectures is to show how the topological degree theory for (non-convex) multivalued mappings can be usefully applied to differential inclusions. Namely, we shall apply it to get a topological characterization of the set of solutions and periodic solutions for some differential inclusions. We discuss these problems in the case when the considered differential inclusions are defined on Banach spaces or on proximate retracts. Recall that a proximate retract is a compact subset A of the Euclidean space ℝn such that there exists an open neighbourhood U of A in ℝn and a metric retraction r: U → A. It is well known that any compact convex subset A of ℝn or any compact C 2-manifold M ⊂ ℝn is a proximate retract. Moreover, a topological degree method for implicit differential equations and differential inclusions is presented.


Open Neighbourhood Compact Convex Differential Inclusion Multivalued Mapping Topological Approach 
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  1. [ACZ1]
    Anichini,G., Conti,G. and Zecca, P., Approximation of nonconvex set valued mappings, Boll. Un. Mat. ltd. A(6) (1985), 145-153.Google Scholar
  2. [ACZ2]
    Anichini,G., Conti,G. and Zecca,P., Approximation and selection theorem for nonconvex multifunctions in infinite dimensional spaces, Boll. Un. Mat. Ital. B(7) 4 (1990), 411-422.MathSciNetzbMATHGoogle Scholar
  3. [AZ]
    Anichini,G. and Zecca,P., Multivalued differential equations in Banach space. An application to control theory, J. Optim. Theory Appl. 21 (1977), 477-486.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [AnC]
    Antosiewicz,H.A. and Cellina,A., Continuous selections and differential relations J. Differential Equation 19 (1975), 386-398.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [A]
    Aronszajn,N., Le correspondant topologique de l’unicité dans la théorie des équations différentielles, Ann. of Math. 43 (1942), 730-738.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [AC]
    Aubin, .P. and Cellina,A., Differential Inclusions, Springer-Verlag, Berlin-Heidel-Google Scholar
  7. berg-New York, 1984.Google Scholar
  8. [BK]
    Bader, R. and Kryszewski W., Fixed point index for compositions of set-valued maps with proximally ∞-connected values on arbitrary ANR’s, to appear in Set-Valued Anal.Google Scholar
  9. [Be]
    Beer,G., On a theorem of Cellina for set valued functions, Rocky Mountain J. 18 (1988), 37-47.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [Bi]
    Bielawski,R., Simplicial convexity and its applications, J. Math. Anal. Appl. 127 (1987), 155-171.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [BGo]
    Bielawski,R. and Górniewicz,L., A fixed point index approach to some differential equations, in: Topological Fixed Point Theory and Applications (Boju Jiang, ed.), Lecture Notes in Math. 1411, Springer-Verlag, Berlin-Heidelberg-New York, 1989, 9-14.CrossRefGoogle Scholar
  12. [BGP]
    Bielawski,R., Górniewicz,L. and Plaskacz,S., Topological approach to differential inclusions on closed subsets of ℝn, Dynam. Report. (N. S.) 1 (1992), 225-250.Google Scholar
  13. [Bl]
    Blagodatskikh,V.I, Local controlability of differential inclusions, Differentsial’nye Uravneniya 9 (1973), 361-362 (Russian); translation: Differential Equations 9 (1973), 277-278.Google Scholar
  14. [Bg]
    Bogatyrev,W.A., Fixed points and properties of solutions of differential inclusions, Izv. Akad. Nauk SSSR 47 (1983), 895-909 (Russian).MathSciNetzbMATHGoogle Scholar
  15. [Bo]
    Borsuk,K. Theory of Retracts, Monografie Matematyczne 4, PWN, Warszawa 1967.zbMATHGoogle Scholar
  16. [BGMOl]
    Borisovich,Yu. G., Gelman, B.D., Myshkis, A.D. and Obukhowskiĩ, V.V., Topological methods in the fixed point theory of multivalued mappings, Uspekhi. Mat. Nauk 35 (1980), 59-126 (Russian).MathSciNetGoogle Scholar
  17. [BGM02]
    Borisovich,Yu. G., Gelman,B.D., Myshkis,A.D. and Obukhowskiĩ,V.V., Introduction to the Theory of Multivalued Mappings, Voronezh. Gos. Univ., Voronezh, 1986 (Russian).Google Scholar
  18. [Br]
    Bressan,A., On the qualitative theory of lower semicontinuous differential inclusions, J. Differential Equations 31 (1989), 379-391.MathSciNetCrossRefGoogle Scholar
  19. [Brl]
    Bressan,A., Directionally continuous selections and differential inclusions, Funk- Google Scholar
  20. cial. Ekvac. 31 (1988), 459-470,Google Scholar
  21. [BC]
    Bressan,A. and Colombo,G., Extensions and selections of maps with decomposable values, Studia Math. 40 (1988), 69-86.MathSciNetGoogle Scholar
  22. [BCF]
    Bressan,A., Cellina,A. and Fryszkowski,A., A class of absolute retracts in spaces of integrable functions, Proc. Amer. Math. Soc. 112 (1991), 413-418.MathSciNetzbMATHCrossRefGoogle Scholar
  23. [BG]
    Browder,F. and Gupta,C.P., Topological degree and nonlinear mappings of analytic type in Banach spaces, J. Math. Anal. Appl. 26 (1969), 390-402.MathSciNetzbMATHCrossRefGoogle Scholar
  24. [CI]
    Cellina,A., On the set of solutions to Lipschitzean differential equations, Differential Integral Equations 1 (1988), 495-500.MathSciNetzbMATHGoogle Scholar
  25. [C2]
    Cellina,A., A selection theorem, Rend. Sem. Univ. Padova 55 (1976), 99-107.MathSciNetGoogle Scholar
  26. [CC]
    Cellina,A. and Colombo,G., An existence result for differential inclusions with non-convex right-hand side, Funkcial. Ekvac. 32 (1989), 407-416.MathSciNetzbMATHGoogle Scholar
  27. [CCo]
    Cellina,A. and Colombo,R.M., Some qualitative and quantitative results on differential inclusions, in: Set-Valued Analysis and Differential Inclusions (A.B. Kurzhanski and A. Lasota, eds.), Progr. Systems Control Theory 16, Birkhäuser, Basel-Boston 1993, 43-60.Google Scholar
  28. [CL]
    Cellina,A. and Lasota,A., A new approach to the definition of topological degree for multivalued mappings, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 8 (1969), 434-440.MathSciNetGoogle Scholar
  29. [Da]
    Davy,J.L., Properties of the solution set of a generalized differential equation, Bull. Austral. Math. Soc. 6 (1972), 379-398.MathSciNetzbMATHCrossRefGoogle Scholar
  30. [BM1]
    De Blasi,F.S. and Myjak,J., On the solution sets for differential inclusions, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys. 12 (1985), 17-23.Google Scholar
  31. [BM2]
    De Blasi,F.S. and Myjak,J., A remark on the definition of topological degree for set-valued mappings, J. Math. Anal. Appl. 92 (1983), 445-451.MathSciNetzbMATHCrossRefGoogle Scholar
  32. [BM3]
    De Blasi, F.S. and Myjak, J., On continuous approximation for multifunctions, Pacific J. Math. 123 (1986), 9-30.MathSciNetzbMATHGoogle Scholar
  33. [BP1]
    De Blasi,F.S. and Pianigiani,G., Topological properties of nonconvex differential inclusions, Nonlinear Anal. 20 (1993), 871-894.MathSciNetzbMATHCrossRefGoogle Scholar
  34. [BP2]
    De Blasi,F.S. and Pianigiani,G., A Baire category approach to the existence of solutions of multivalued differential equations in Banach spaces, Funkcial. Ekvac. 25 (1982), 153-162.MathSciNetzbMATHGoogle Scholar
  35. [BP3]
    De Blasi, F.S. and Pianigiani, G., On the density of extremal solutions of differential inclusions, Ann. Polon. Math. 56 (1992), 133-142.MathSciNetzbMATHGoogle Scholar
  36. [BP4]
    De Blasi,F.S. and Pianigiani,G., Differential inclusions in Banach spaces, J. Differential Equations 66 (1987), 208-229.MathSciNetzbMATHCrossRefGoogle Scholar
  37. [BP5]
    De Blasi,F.S. and Pianigiani,G., Solution sets of boundary value problems for nonconvex differential inclusions, Topol. Meth. Nonlinear Anal. 1 (1993), 303-314.zbMATHGoogle Scholar
  38. [D]
    Dold,D., Lectures on Algebraic Topology, Springer-Verlag, Berlin-Heidelberg- New York, 1972.zbMATHGoogle Scholar
  39. [Del]
    Deimling,K., On solution sets of multivalued differential equations, Appl. Anal. 30 (1988), 129-135.MathSciNetzbMATHCrossRefGoogle Scholar
  40. [De2]
    Deimling,K., Ordinary Differential Equations in Banach Spaces, Lecture Notes in Math. 596, Springer-Verlag, Berlin-Heidelberg-New York, 596, 1977.Google Scholar
  41. [De3]
    Deimling,K., Multivalued differential equations on closed sets II, Differential Integral Equations 3 (1990), 639-642.MathSciNetzbMATHGoogle Scholar
  42. [DG]
    Dugundji,J. and Granas,A. Fixed Point Theory, Vol.1, PWN, Warszawa, 1981.Google Scholar
  43. [DyG]
    Dylawerski,G. and Górniewicz,L., A remark on the Krasnosielskii's translation operator, Serdica 9 (1983), 102-107.MathSciNetzbMATHGoogle Scholar
  44. [EM]
    Eilenberg,S. and Montgomery,D., Fixed point theorem for multivalued transformations, Amer. J. Math. 58 (1946), 214-222.MathSciNetCrossRefGoogle Scholar
  45. [Fi]
    Filippov,A.F., Differential Equations With Discontinuous Right Hand Sides, Math. Appl. (Soviet Ser.) 18, Kluwer, Dordrecht, 1988 (translation of the Russion edition, “Nauka”, Moscow, 1985).zbMATHGoogle Scholar
  46. [Fri]
    Frigon,M., Application de la théorie de la transversalité topologique à des problemes non lineaires pour des equations differentielles ordinaires, Dissertations Math. (Rozprawy Mat) 296 (1990), 1-71.MathSciNetGoogle Scholar
  47. [FR]
    Fryszkowski,A. and Rzezuchowski,T., Continuous version of Filippov-Ważewski theorem, J. Differential Equations 94 (1991), 254-265.MathSciNetCrossRefGoogle Scholar
  48. [FNPZ]
    Furi,M., Nistri,P., Pera,P. and Zecca,P., Topological methods for the global controllability of nonlinear systems, J. Optim. Theory Appl. 45 (1985), 231-256.MathSciNetzbMATHCrossRefGoogle Scholar
  49. [Gl]
    Górniewicz,L., Homological methods in fixed point theory of multivalued mappings, Dissertationes Math. (Rozprawy Mat.) 129 (1976), 1-71.Google Scholar
  50. [G2]
    Górniewicz,L., Topological degree of morphisms and its applications to differential inclusions, Race. Sem. Dip. Mat. Univ. Studi Calabria 5 (1985), 1-48.Google Scholar
  51. [G3]
    Górniewicz, L., On the solution set of differential inclusions, J. Math. Anal. Appl. 113 (1986), 235-244.MathSciNetzbMATHCrossRefGoogle Scholar
  52. [G4]
    Górniewicz,L., Recent results on the solution sets of differential inclusions, in: Méthodes topologiques en analyses convexe (Partie 3) (A. Granas, eds), Sem. de Math. Super. 110, Presses Univ. de Montréal, Montréal, 1990, 101-122.Google Scholar
  53. [GG]
    Górniewicz,L. and Granas,A., Some general theorems in coincidence theory, J. Math. Pure Appl. 60 (1981), 361-373.zbMATHGoogle Scholar
  54. [GGK1a]
    Górniewicz,L., Granas,A. and Kryszewski,W., Sur la méthode de l’homotopie dans la théorie des points fixes. Partie 1: Transversalité topologique; Partie 2: L'indice de point fixe, C. R. Acad. Sci. Paris 307 (1988), 489-492,zbMATHGoogle Scholar
  55. [GGK1b]
    Górniewicz,L., Granas,A. and Kryszewski,W., Sur la méthode de l’homotopie dans la théorie des points fixes. Partie 1: Transversalité topologique; Partie 2: L'indice de point fixe, C. R. Acad. Sci. Paris 308 (1989), 449-452.zbMATHGoogle Scholar
  56. [GGK2]
    Górniewicz,L., Granas,A. and Kryszewski,W., On the homotopy method in the fixed point index theory for multivalued mappings of compact ANR-s, J. Math. Anal. Appl. 161 (1991), 457-473.MathSciNetzbMATHCrossRefGoogle Scholar
  57. [GP]
    Górniewicz,L. and Plaskacz,S., Periodic solutions of differential inclusions in ℝn Boll. Un. Mat. Ital (7) 7A (1993), 409-420.Google Scholar
  58. [GS]
    Górniewicz,L. and Ślosarski,M., Topological essentiality and differential inclusions, Bull. Austral. Math. Soc. 45 (1992), 177-193.MathSciNetzbMATHCrossRefGoogle Scholar
  59. [Gr1]
    Granas,A., Sur la notion du degrée topologique pour une certaine classe de transformations multivalentes dans les espaces de Banach, Bull. Acad. Polon. Sci. Sér. Math. Astr. Phys. 7 (1959), 181-194.Google Scholar
  60. [Gr2]
    Granas,A., Theorem on antipodes and theorems on fixed points for a certain class of multivalued maps in Banach spaces, Bull. Acad. Polon. Sci. Ser. Math. Astr. Phys. 7 (1959), 271-275.MathSciNetzbMATHGoogle Scholar
  61. [Gr3]
    Granas,A., Topics in Infinite Dimensional Topology, Séminaire J. Leray, Paris, 1969/1970.Google Scholar
  62. [Gr4]
    Granas,A., The theory of compact vector fields and some of its applications, Dissertationes Math. (Rozprawy Mat.) 30 (1962), 1-136.MathSciNetGoogle Scholar
  63. [GGL]
    Granas,A., Guenther,R. and Lee,J., Nonlinear boundary value problems for ordinary differential equations, Dissertationes Math. (Rozprawy Mat.) 244 (1985), 1-132.MathSciNetGoogle Scholar
  64. [GJ]
    Granas,A. and Jaworowski,J., Some theorems of multivalued maps of subsets of the Euclidean space, Bull. Acad. Polon. Sci. Ser. Math. Astr. Phys. 7 (1959), 277-283.MathSciNetzbMATHGoogle Scholar
  65. [Ha]
    Haddad,G., Topological properties of the sets of solutions for functional differential equations, Nonlinear Anal. 5 (1981), 1349-1366.MathSciNetzbMATHCrossRefGoogle Scholar
  66. [HV]
    Himmelberg,C.J. and Van Vleck,F.S., A note on the solution sets of differential inclusions, Rocky Mountain J. Math. 12 (1982), 621-625.MathSciNetzbMATHCrossRefGoogle Scholar
  67. [Hu]
    Hukuhara,M., Sur l'application semi-continue dont la valeur est un compact convexe, Funkcial Ekvac. 10 (1967), 43-66.MathSciNetzbMATHGoogle Scholar
  68. [H]
    Hyman,D.M., On decreasing sequences of compact absolute retracts, Fund. Math. 64 (1959), 91-97.MathSciNetGoogle Scholar
  69. [JK]
    Jarnik,J. and Kurzweil,J., On conditions on right hand sides of differential relations, Časopis Pěst. Mat. 102 (1977), 334-349.MathSciNetzbMATHGoogle Scholar
  70. [Jl]
    Jaworowski,J., Some consequences of the Vietoris Mapping Theorem, Fund. Math. 45 (1958), 261-272.MathSciNetzbMATHGoogle Scholar
  71. [J2]
    Jaworowski,J., Set-valued fixed point theorems of approximative retracts, in: Set- Valued Mappings, Selections and Topological Properties of2 X (W.M. Fleischman, eds.) Lecture Notes in Math. 171, Springier-Verlag, Berlin-Heidelberg-New York, 1970, 34-39.Google Scholar
  72. [KO]
    Kamenskii,M.I. and Obukhovskii,V.V., On periodic solutions of differential inclusions with unbounded operators in Banach spaces, Univ. u Novom Sadu Zb. Rad. Prirod.-Mat. Fak. Ser. Mat. 21 (1991), 173-191.MathSciNetGoogle Scholar
  73. [Ki]
    Kisielewicz,M., Differential Inclusions and Optimal Control, PWN-Polish Scientific Publishers, Warszawa, and Kluwer Academic Publishers, Dordrecht-Boston-London, 1991.Google Scholar
  74. [KZ]
    Krasnoselskiĩ,M. and Zabreĩko,P., Geometrical Methods of Nonlinear Analysis, Springer-Verlag, Berlin-Heidelberg-New York, 1984.CrossRefGoogle Scholar
  75. [Kr]
    Kryszewski, W., Homotopy invariants for set-valued maps homotopy-approximation approach, in: Fixed Point Theory and Applications, (M.A.Thera and J-B.Baillon, eds.), Pitman Res. Notes Math. Ser. 252, Longman, Harlow, 1991, 269-284.Google Scholar
  76. [KR]
    Kurland,H. and Robin,J., Infinite Codimension and Transversality, Lecture Notes in Math. 468, Springer-Verlag, Berlin-Heidelberg-New York, 1974.Google Scholar
  77. [LR]
    Lasry, J.M. and Robert, R., Analyse non linéaire multivoque, Publ. no 7611, Centre Rech. Math. Décision (Ceremade), Université de Paris IX (Dauphine), 71-190.Google Scholar
  78. [MNZ]
    Macki,J.W., Nistri,P. and Zecca,P., An existence of periodic solutions to nonau-tonomus differential inclusions, Proc. Amer. Math. Soc. 104 (1988), 840-844.MathSciNetzbMATHCrossRefGoogle Scholar
  79. [MC]
    Mas-Colell,A., A note on a theorem of F.Browder, Math. Programming 6 (1974), 229-233.MathSciNetzbMATHCrossRefGoogle Scholar
  80. [O1]
    Olech,C., Existence of solutions of non-convex orientor field, Boll. Un. Mat. Ital. (4) 11 (1975), 189-197.MathSciNetzbMATHGoogle Scholar
  81. [Pa]
    Papageorgiou,N.S., A property of the solution set of differential inclusions in Banach spaces with Caratheodory orientor field, Appl. Anal. 27 (1988), 279-287.MathSciNetzbMATHCrossRefGoogle Scholar
  82. [P1]
    Plaskacz,S., Periodic solutions of differential inclusions on compact subsets of ℝn J. Math. Anal. Appl. 148 (1990), 202-212.MathSciNetzbMATHCrossRefGoogle Scholar
  83. [P2]
    Plaskacz,S., On the solution sets for differential inclusions, Boll. Un. Mat. Ital. (7) 6A (1992), 387-394.MathSciNetGoogle Scholar
  84. [P3]
    Plaskacz,S., Periodic Solutions of Differential Inclusions on Closed Subsets of Eu- clidean Spaces, Ph. D. Thesis, Uniwersytet Mikolaja Kopernika, Toruń, 1990 (Polish).Google Scholar
  85. [PI]
    Pliś,A., Measurable orientor fields, Bull. Acad. Polon. Sci. Sér. Math. Astr. Phys. 13 (1965), 565-569.zbMATHGoogle Scholar
  86. [Pr]
    Pruszko,T., Some applications of the degree theory to multi-valued boundary value problems, Dissertationes Math. (Rozprawy Mat. 229 (1984), 1-48.MathSciNetGoogle Scholar
  87. [Rl]
    Ricceri,B., Une propriété topologique de l’ensemble des points fixes d'une con- traction multivoque à valeures convexes, Atti Accad. Naz. Lincei Rend. CI. Sci. Fis. Mat. Natur. (8) 81 (1987), 283-286.MathSciNetzbMATHGoogle Scholar
  88. [R2]
    Ricceri,B., Existence theorems for nonlinear problems, Rend. Acad. Naz. Sci. XL Mem. Mat. (5) 11 (1987), 77-99.MathSciNetzbMATHGoogle Scholar
  89. [R3]
    Ricceri, B., On the Cauchy problem for the differential equation Ft,x,x’,ldots, x (k) ) = 0, Glasgow Math. J. 33 (1991), 343-348.MathSciNetzbMATHCrossRefGoogle Scholar
  90. [Rz]
    Rzeżuchowski,T., Scorza-Dragoni type theorem for upper semicontinuous multi-valued functions, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys. 28 (1980), 61-66.zbMATHGoogle Scholar
  91. [S]
    Ślosarski,M., Some Applications of the Topological Essentiality and Characterization of the Fixed Point Set to Differential Inclusions, Ph. D. Thesis, Uniwersytet Mikolaja Kopernika, Toruri, 1993 (Polish).Google Scholar
  92. [Sz]
    Szufla,S., Sets of fixed points of nonlinear mappings in function spaces, Funkcial. Ekvac. 22 (1979), 121-126.MathSciNetzbMATHGoogle Scholar
  93. [Ta]
    Tallos,P., Viability problems for nonautonomus differential inclusions, SI AM J. Control and Opt. 29 (1991), 253-263.MathSciNetzbMATHCrossRefGoogle Scholar
  94. [Tol]
    Tolstonogov,A.A., On the structure of the set of solution for differential inclusions in a Banach space, Mat. Sb. (N. S.) 118 (160) (1982) no. 1, 3-18 (Russian); translation: Math. USSR Sb. 46 (1983), 1-15.Google Scholar
  95. [To2]
    Tolstonogov,A.A., Differential Inclusions in a Banach Space, “Nauka” Sibirsk. Otdel., Novosibirsk, 1986 (Russian).zbMATHGoogle Scholar
  96. [Neu]
    Von Neumann,J., A Model of General Economic Equilibrium, Collected Works, Vol. VI, Pergamon Press, Oxford, 1963, 29-37.Google Scholar

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© Springer Science+Business Media Dordrecht 1995

Authors and Affiliations

  • Lech Górniewicz
    • 1
  1. 1.Wydzial Matematyki i InformatykiUniwersytet Mikołaja KopernikaToruńPoland

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