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Topological and approximation methods of degree theory of set-valued maps

Seria
Rozprawy Matematyczne tom/nr w serii: 336 wydano: 1994
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EN

Summary
The theory of topological degree of set-valued maps determined by morphisms, i.e. maps with values which are continuous images of almost acyclic sets, is presented, together with some of its applications.
In the first part, morphisms defined on finite-dimensional Euclidean manifolds are considered and the integer-valued degree is introduced by means of the Eilenberg-Montgomery-Górniewicz method based on the Vietoris-Begle-Sklyarenko theorem and using the approach of Dold in terms of the Alexander-Spanier cohomology theory.
In the second part, the degree theory is extended to a wide class of noncompact set-valued maps determined by morphisms acting in infinite-dimensional spaces. This new class, of the so-called A-morphisms, generalizing the compact set-valued vector fields, is studied from the general approximation viewpoint.
Topological and approximation methods are also used in the context of another class of set-valued maps whose values satisfy some geometrical, rather than topological, conditions. Moreover, the degree theory is extended to the class of Petryshyn's A-proper maps with nonconvex values. The class of A-proper maps contains many different types of maps considered elsewhere.
The degree theory is compared with the notion of essentiality, another useful tool in nonlinear analysis.
The paper proposes several results extending the well-known theorems on single-valued maps, such as the Borsuk antipodal theorem, the Bourgin-Yang theorem, nonlinear alternative, invariance of domain and others.
EN

CONTENTS
   Introduction................................................................................................................5
   Preliminaries...............................................................................................................7
A. Elements of homology theory.....................................................................................8
   1. Products.................................................................................................................8
   2. Orientation of manifolds........................................................................................10
I. Topology of morphisms..............................................................................................12
   1. Set-valued maps....................................................................................................12
   2. Vietoris maps.........................................................................................................14
   3. Category of morphisms..........................................................................................18
   4. Operations in the category of morphisms..............................................................21
   5. Homotopy and extension properties of morphisms................................................23
   6. Essentiality of morphisms.......................................................................................28
   7. Concluding remarks................................................................................................32
II. The topological degree theory of morphisms............................................................33
   1. Cohomological properties of morphisms.................................................................34
   2. The fundamental cohomology class........................................................................36
   3. The topological degree of morphisms.....................................................................38
   4. The degree of morphisms of spheres and open subsets of Euclidean space..........43
   5. Borsuk type theorems..............................................................................................48
   6. Applications.............................................................................................................56
III. The class of approximation-admissible morphisms......................................................59
   1. Filtrations.................................................................................................................60
   2. Approximation-admissible morphisms and maps......................................................63
   3. Approximation of A-maps.........................................................................................68
IV. Approximation degree theory for A-morphisms...........................................................73
   1. The degree of A-morphisms.....................................................................................73
   2. Properties of the degree of A-morphisms................................................................75
   3. Further properties of the degree. Applications.......................................................78
V. Other classes of set-valued maps..............................................................................83
   1. Single-valued approximations..................................................................................83
   2. Linear filtrations. AP-maps of Petryshyn...................................................................91
   References..................................................................................................................97
Miejsce publikacji
Warszawa
Copyright
Seria
Rozprawy Matematyczne tom/nr w serii: 336
Liczba stron
101
Liczba rozdzia³ów
Opis fizyczny
Dissertationes Mathematicae, Tom CCCXXXVI
Daty
wydano
1994
otrzymano
1991-11-26
poprawiono
1993-07-14
Twórcy
  • Instytut Matematyki, Uniwersytet Mikołaja Kopernika, Chopina 12/18, 87-100 Toruń, Poland, wkrysz@mat.uni.torun.pl
Bibliografia
  • [1] G. Albinus, Remarks on a theorem of S. N. Bernstein, Studia Math. 38 (1970), 227-234.
  • [2] P. S. Aleksandrov and B. A. Pasynkov, Introduction to Dimension Theory, Nauka, Moscow, 1973 (in Russian).
  • [3] G. Anichini, G. Conti and P. Zecca, Approximation of nonconvex set valued mappings, Boll. Un. Mat. Ital. C (6) 4 (1985), 145-154.
  • [4] G. Anichini, G. Conti and P. Zecca, Approximation and selection for nonconvex multifunctions in infinite dimensional spaces, Boll. Un. Mat. Ital. B (7) 4 (1990), 411-422.
  • [5] J. P. Aubin and A. Cellina, Differential Inclusions, Springer, Berlin, 1984.
  • [6] E. G. Begle, The Vietoris theorem for bicompact spaces, Ann. of Math. 51 (1950), 534-543, 544-550.
  • [7] C. Berge, Espaces topologiques. Fonctions multivoques, Dunod, Paris, 1959.
  • [8] C. Bessaga and A. Pełczyński, Selected Topics in Infinite-Dimensional Topology, Monograf. Mat. 58, PWN, Warszawa, 1975.
  • [9] R. Bielawski, The fixed point index for acyclic maps on ENR's, Bull. Polish Acad. Sci. Math. 35 (1987), 487-499.
  • [10] Yu. G. Borisovich, B. D. Gel'man, A. D. Myshkis and V. V. Obukhovskiĭ, Topological methods in the fixed point theory of multivalued mappings, Uspekhi Mat. Nauk 35 (1) (1980), 59-126 (in Russian).
  • [11] D. G. Bourgin, A generalization of the mapping degree, Canad. J. Math. 26 (1974), 1109-1117.
  • [12] G. E. Bredon, Introduction to Compact Transformation Groups, Academic Press, New York, 1972.
  • [13] F. E. Browder, Approximation-solvability of nonlinear equations in normed spaces, Arch. Rational Mech. Anal. 26 (1967), 33-42.
  • [14] F. E. Browder, The degree of mapping, and its generalizations, in: Contemp. Math. 21, Amer. Math. Soc., 1983, 15-40.
  • [15] F. E. Browder and W. V. Petryshyn, Approximation methods and the generalized topological degree for nonlinear mappings in Banach spaces, J. Funct. Anal. 3 (1969), 217-245.
  • [16] J. Bryszewski, On a class of multi-valued vector fields in Banach spaces, Fund. Math. 97 (1977), 79-94.
  • [17] J. Bryszewski and L. Górniewicz, Multi-valued maps of subsets of Euclidean spaces, ibid. 90 (1976), 233-251.
  • [18] J. Bryszewski and L. Górniewicz, A Poincaré type coincidence theorem for multivalued maps, Bull. Acad. Polon. Sci. 24 (1976), 593-598.
  • [19] J. Bryszewski, L. Górniewicz and T. Pruszko, An application of the topological degree theory to the study of the Darboux problem for hyperbolic equations, J. Math. Anal. Appl. 76 (1980), 107-115.
  • [20] B. D. Calvert, The local fixed point index for multivalued transformations in a Banach space, Math. Ann. 190 (1970), 119-128.
  • [21] A. Cellina and A. Lasota A new approach to the definition of topological degree for multivalued mappings, Atti Accad. Naz. Lincei 47 (1969), 434-440.
  • [22] P. E. Conner and E. E. Floyd, Fixed point free involutions and equivariant maps, Bull. Amer. Math. Soc. 66 (1960), 416-441.
  • [23] M. W. Davies, A note on Borsuk's antipodal-point theorem, Quart. J. Math. Oxford Ser. (2) 7 (1971), 293--300.
  • [24] A. Dold, Fixed-point index and fixed point theorem for euclidean neighbourhood retracts, Topology 1 (1965), 1-8.
  • [25] A. Dold, Lectures in Algebraic Topology, Springer, Berlin, 1972.
  • [26] J. Dugundji and A. Granas, Fixed Point Theory I, Monograf. Mat. 61, PWN, Warszawa, 1982.
  • [27] G. Dylawerski and L. Górniewicz, A remark on the Krasnosielskii's translation operator along trajectories of ordinary differential equations, Serdica 9 (1983), 102-107.
  • [28] Z. Dzedzej, Fixed point index theory for a class of nonacyclic multivalued maps, Dissertationes Math. 253 (1985).
  • [29] R. E. Edwards, Functional Analysis. Theory and Applications, Holt, Rinehart and Winston, New York, 1965.
  • [30] S. Eilenberg and D. Montgomery, Fixed point theorems for multivalued transformations, Amer. J. Math. 58 (1946), 214-222.
  • [31] S. Eilenberg and N. Steenrod, Foundations of Algebraic Topology, Princeton Univ. Press, 1952.
  • [32] R. Engelking, Topologia ogólna, Bibl. Mat. 47, PWN, Warszawa, 1976; English transl.: General Topology, Monograf. Mat. 60, PWN, Warszawa, 1977.
  • [33] R. Engelking, Teoria wymiaru, Bibl. Mat. 51, PWN, Warszawa, 1977; English transl.: Dimension Theory, PWN, Warszawa, 1978.
  • [34] A. I. Fet, Generalization of a theorem of Lusternik-Schnirelman on covering of spheres, Dokl. Akad. Nauk SSSR 95 (1954), 1149-1152.
  • [35] P. M. Fitzpatrick and W. V. Petryshyn, A degree theory, fixed point theorems, and mapping theorems for multivalued noncompact maps, Trans. Amer. Math. Soc. 194 (1975), 1-25.
  • [36] P. M. Fitzpatrick and W. V. Petryshyn, Fixed point theorems and the fixed point index for multivalued mappings in cones, J. London Math. Soc. 12 (1975), 75-85.
  • [37] G. Fournier and L. Górniewicz, The Lefschetz fixed point theorem for some noncompact multi-valued maps, Fund. Math. 94 (1977), 245-254.
  • [38] L. Fuchs, Abelian Groups, Pergamon Press, New York, 1960.
  • [39] M. Furi and I. Martelli, A degree for a class of acyclic-valued vector fields in Banach spaces, Ann. Scuola Norm. Sup. Pisa 1 (1975), 301-310.
  • [40] K. Gęba and L. Górniewicz, On the Bourgin-Yang theorem for multi-valued maps I, Bull. Polish Acad. Sci. Math. 34 (1986), 315-322.
  • [41] L. Górniewicz, Homological methods in fixed point theory of multivalued maps, Dissertationes Math. 129 (1976).
  • [42] L. Górniewicz, On non-acyclic multi-valued maps of subsets of Euclidean spaces, Bull. Acad. Polon. Sci. 20 (1972), 379-385.
  • [43] L. Górniewicz, Topological degree of morphisms and its applications to differential equations, Racc. di Seminari Mat. 5, Università di Calabria, 1983.
  • [44] L. Górniewicz and A. Granas, Some general theorems in coincidence theory I, J. Math. Pures Appl. 61 (1981), 361-373.
  • [45] L. Górniewicz and A. Granas, Fixed point theorems for multivalued mappings of absolute neighbourhood retracts, ibid. 49 (1970), 381-395.
  • [46] L. Górniewicz, A. Granas et W. Kryszewski, Sur la méthode de l'homotopie dans la théorie des points fixés pour les applications multivoques, partie I: Transversalité topologique, C. R. Acad. Sci. Paris 307 (1988), 489-492; partie II: L'indice pour les ANR's compactes, ibid. 309 (1989), 367-372.
  • [47] L. Górniewicz, A. Granas et W. Kryszewski, Approximation method in the fixed point index theory of multivalued mappings of compact ANR's, J. Math. Anal. Appl. 161 (1991), 457-473.
  • [48] L. Górniewicz and Z. Kucharski, Coincidence of k-set-contraction pairs, ibid. 107 (1985), 1-15.
  • [49] A. Granas, Sur la notion du degré topologique pour une classe de transformations multivalentes dans les espaces de Banach, Bull. Acad. Polon. Sci. 7 (1959), 191-194.
  • [50] A. Granas, Theorems on antipodes and theorems on fixed points for a certain class of multivalued mappings in Banach spaces, ibid., 271-275.
  • [51] A. Granas, The theory of compact vector fields and some of its applications to topology of functional spaces I, Dissertationes Math. 30 (1962).
  • [52] A. Granas, Sur la méthode de continuation de Poincaré, C. R. Acad. Sci. Paris 282 (1976), 978-985.
  • [53] A. Granas, The Leray-Schauder index and the fixed point theory for arbitrary ANR's, Bull. Soc. Math. France 100 (1972), 209-228.
  • [54] A. Granas and J. Jaworowski, Some theorems on multivalued mappings of subsets of the euclidean space, Bull. Acad. Polon. Sci. 7 (1959), 277-283.
  • [55] M. J. Greenberg, Lectures on Algebraic Topology, Benjamin, 1971.
  • [56] C. J. Himmelberg, Fixed points of compact multifunctions, J. Math. Anal. Appl. 38 (1972), 205-207.
  • [57] J. W. Jaworowski, Involutions of compact spaces and a generalization of Borsuk's theorem on antipodes, Bull. Acad. Polon. Sci. 3 (1955), 289-292.
  • [58] J. W. Jaworowski, Theorems on antipodes for multivalued mappings and a fixed point theorem, ibid., 187-191.
  • [59] W. B. Johnson, Markushevich bases and duality theory, Trans. Amer. Math. Soc. 149 (1970), 171-177.
  • [60] T. Kaczyński, Topological transversality of set-valued condensing maps, doctoral dissertation, McGill Univ., Montreal, 1986.
  • [61] M. A. Krasnosel'skiĭ, Topological Methods in the Theory of Nonlinear Integral Equations, New York, 1964.
  • [62] W. Kryszewski, Some remarks on the nonlinear eigenvalue problems of Birkhoff-Kellogg type, Bull. Polish Acad. Sci. Math. 32 (1984), 455-462.
  • [63] W. Kryszewski, The Lefschetz type theorem for a class of noncompact maps, Rend. Circ. Mat. Palermo Suppl. 14 (1987), 365-384.
  • [64] W. Kryszewski, An approximation method in the theory of nonlinear noncompact operators, in: Méthodes topologiques en analyse convexe, C. R. OTAN, Partie 3, A. Granas (ed.), Sém. Math. Sup. 110, Presses Univ. Montréal, 1990.
  • [65] W. Kryszewski, Some remarks on the continuation method in the fixed point theory, Zeszyty Nauk. Uniw. Gdańsk. 6 (1987), 43-53.
  • [66] W. Kryszewski, An application of A-mapping theory to boundary value problems for ordinary differential equations, Nonlinear Anal. 15 (1990), 697-717.
  • [67] W. Kryszewski, The coincidence index on non-compact manifolds, in preparation.
  • [68] W. Kryszewski, H. Ben-El-Mechaiekh and P. Deguire, 0-separating maps and their applications, in preparation.
  • [69] W. Kryszewski and B. Przeradzki, The topological degree and fixed points of DC-mappings, Fund. Math. 126 (1985), 15-26.
  • [70] W. Kryszewski, B. Przeradzki and S. Wereński, Remarks on approximation methods in degree theory, Trans. Amer. Math. Soc. 316 (1989), 97-114.
  • [71] Z. Kucharski, A coincidence index, Bull. Acad. Polon. Sci. 24 (1976), 245-252.
  • [72] Z. Kucharski, Two consequences of the coincidence index, ibid., 437-444.
  • [73] K. Kuratowski, Topologie I, Monograf. Mat. 3, Warszawa, 1933.
  • [74] K. Kuratowski, Wstęp do teorii mnogości i topologii, Bibl. Mat. 9, PWN, Warszawa, 1980; English transl.: Introduction to Set Theory and Topology, PWN-Pergamon Press, Warszawa-Oxford, 1977.
  • [75] M. Landsberg, Lineare beschränkte Abbildungen von einem Produkt in einen lokal radial beschränkten Raum und ihre Filter, Math. Ann. 146 (1962), 232-248.
  • [76] A. Lasota, Applications of generalized functions to contingent equations and control theory, in: Inst. Fluid Dynamics Appl. Math. Lecture Ser. 51, Univ. of Maryland, 1970-1971, 41-52.
  • [77] A. Lasota and Z. Opial, Fixed-point theorems for multi-valued mappings and optimal control problems, Bull. Acad. Polon. Sci. 16 (1968), 645-649.
  • [78] J. Leray et J. P. Schauder, Topologie et équations fonctionnelles, Ann. Sci. École Norm. Sup. 65 (1934), 45-78.
  • [79] N. G. Lloyd, Degree Theory, Cambridge Univ. Press, 1978.
  • [80] T.-W. Ma, Topological degrees of set-valued compact fields in locally convex spaces, Dissertationes Math. 92 (1972).
  • [81] R. H. Martin, Nonlinear Operators and Differential Equations in Banach Spaces, Wiley, New York, 1976.
  • [82] I. Massabò and P. Nistri, A topological degree for multivalued A-proper maps in Banach spaces, Boll. Un. Mat. Ital. B 13 (1976), 672-685.
  • [83] W. S. Massey, Homology and Cohomology Theory, Marcel Dekker, New York, 1978.
  • [84] E. Michael, Continuous selections I, Ann. of Math. 63 (1956), 361-382.
  • [85] P. S. Milojević, Multivalued mappings of A-proper and condensing type and boundary value problems, Ph.D. thesis, Rutgers Univ., New Brunswick, N.J., 1975.
  • [86] P. S. Milojević and W. V. Petryshyn, Continuation theorems and the approximation-solvability of equations involving multivalued A-proper mappings, J. Math. Anal. Appl. 60 (1977), 658-692.
  • [87] V. N. Nikol'skiĭ, Best approximation and bases in a Fréchet space, Dokl. Akad. Nauk SSSR 59 (1948), 639-642 (in Russian).
  • [88] L. Nirenberg, Topics in Nonlinear Functional Analysis, New York Univ. Press, 1975.
  • [89] B. Nowak, DJ-maps and their homotopies, Acta Univ. Lodz. 1981 (in Polish).
  • [90] B. O'Neill, Induced homology homomorphism for set-valued maps, Pacific J. Math. 7 (1957), 497-509.
  • [91] E. de Pascale and R. Guzzardi, On the boundary values dependence for the topological degree of multivalued noncompact maps, Boll. Un. Mat. Ital. A 13 (1976), 110-116.
  • [92] W. V. Petryshyn, On the approximation solvability of equations involving A-proper and pseudo-A-proper mappings, Bull. Amer. Math. Soc. 81 (1975), 223-312.
  • [93] M. J. Powers, Multi-valued mappings and Lefschetz fixed point theorems, Proc. Cambridge Philos. Soc. 68 (1970), 619-630.
  • [94] T. Pruszko, Some applications of the topological degree theory to multivalued boundary value problems, Dissertationes Math. 229 (1984).
  • [95] B. Przeradzki, Topological degree of DC-maps, Ph.D. thesis, Univ. of Łódź, 1984 (in Polish).
  • [96] W. Robertson, Completions of topological vector spaces, Proc. London Math. Soc. 8 (1958), 242-257.
  • [97] B. N. Sadovskiĭ, Limit-compact and condensing operators, Uspekhi Mat. Nauk 27 (1971), 81-146 (in Russian).
  • [98] W. Segiet, Nonsymmetric Borsuk-Ulam theorem for multivalued mappings, Bull. Polish Acad. Sci. Math. 32 (1984), 113-119.
  • [99] W. Segiet, Local coincidence index for morphisms, ibid. 30 (1982), 261-267.
  • [100] H. S. Shapiro, Some negative theorems of approximation theory, Michigan Math. J. 11 (1964), 211-217.
  • [101] H. W. Siegberg and G. Skordev, Fixed point index and chain approximations, Pacific J. Math. 102 (1982), 455-486.
  • [102] E. G. Sklyarenko, On some applications of sheaf theory in general topology, Uspekhi Mat. Nauk 19 (6) (1964), 47-70 (in Russian).
  • [103] G. Skordev, Fixed point index for open sets in euclidean spaces, Fund. Math. 121 (1984), 41-58.
  • [104] E. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966.
  • [105] J. R. Webb, On degree theory for multivalued mappings and applications, Boll. Un. Mat. Ital. 9 (1974), 137-158.
  • [106] S. Wereński, On the fixed point index of non-compact mappings, Studia Math. 78 (1984), 155-160.
  • [107] S. A. Williams, An index for set-valued maps in infinite-dimensional spaces, Proc. Amer. Math. Soc. 31 (1972), 557-563.
  • [108] L. Vietoris, Über den höheren Zusammenhang kompakter Räume und eine Klasse von zusammenhangstreuen Abbildungen, Math. Ann. 97 (1927), 454-472.
  • [109] C.-T. Yang, On theorems of Borsuk-Ulam, Kakutani-Yamabe-Yujobô and Dyson, I, Ann. of Math. 60 (1954), 262-282.
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Uwagi
1991 Mathematics Subject Classification: 54C60, 47H04, 55M25, 47H11, 47H10, 47H17, 54H25, 55M20.
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