Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
R. Abraham and J. Robbin, Transversal mappings and flows, An appendix by Al Kelley (1967), W. A. Benjamin, Inc., New York-Amsterdam.
F. J. Almgren, Applications of multiple-valued functions, Geometric Modeling, SIAM, Philadelphia, PA, 1987, pp. 43–54.
_____, Multi-functions mod v, Geometric Analysis and Computer Graphics (Berkeley, CA, 1988), vol. 17, Math. Sci. Res. Inst. Publ., Springer, New York, 1991, pp. 1–17.
G. Anichini and G. Conti, About the existence of solutions of a boundary value problem for a Carathéodory differential system, Z. Anal. Anwendungen 16 (1997), 621–630.
R. Bader, G. Gabor and W. Kryszewski, On the extension of approximations for set-valued maps and the repulsive fixed points., Boll. Un. Mat. Ital. B (7) 10 (1996), 399–416.
J. C. Becker, A. Casson and D. H. Gottlieb, The Lefschetz number and fiber preserving maps, Bull. Amer. Math. Soc. 81 (1975), 425–427.
J. C. Becker and D. H. Gottlieb, The transfer map and fiber bundles, Topology 14 (1975), 1–12.
_____, Vector fields and transfers, Manuscripta Math. 72 (1991), 111–130.
J. C. Becker and R. E. Schultz, Fixed-point indices and left invariant framings, Geometric Applications of Homotopy Theory, Proc. Conf. Evanston (Ill., 1977), I, vol. 657, Lecture Notes in Math., Springer, Berlin, 1978, pp. 1–31.
_____, Axioms for bundle transfers and traces, Math. Z. 227 (1998), 583–605.
Yu. G. Borisovich, B. D. Gel'man, A. D. Myshkis and V. V. Obukhovskiĭ, New results in the theory of multivalued mappings I Topological characteristics and solvability of operator relations, Mathematical Analysis, vol. 25, Itogi Nauki i Tekhniki, pp. 123–197, 200 (Russian); Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform. Moscow, 1987; transl. in J. Soviet Math. 49 (1990), 800–855.
G. E. Bredon, Sheaf Theory, Graduate Texts in Mathematics, vol. 170, Springer-Verlag, New York, 1997.
R. Caccioppoli, Funzioni di variabili complesse. Equazioni funzionali, Opere, vol. II, Edizioni Cremonese, Rome, 1963.
R. Connelly, An extension of Brouwer's fixed-point theorem to nonacyclic, set valued functions, Proc. Amer. Math. Soc. 43 (1974), 214–218.
G. Conti, P. Nistri and P. Zecca, Systems of set-valued equations in Banach spaces, Delay Differential Equations and Dynamical Systems (Claremont, CA, 1990), Lecture Notes in Math., vol. 1475, Springer, Berlin, 1991, pp. 98–109.
_____, Systems of multivalued equations for solving controllability problems, Pure Math. Appl. Ser. B 2 (1992), 215–225.
G. Conti and J. Pejsachowicz, Fixed point theorems for multivalued weighted maps, Ann. Mat. Pura Appl. (4) 126 (1981), 319–341.
L. Dal Soglio, Grado topologico e teoremi di esistenza di punti uniti per trasformazioni plurivalenti di 3-celle., Rend. Sem. Mat. Univ. Padova 25 (1956), 386–405.
_____, Grado topologico e teoremi di esistenza di punti uniti per trasformazioni plurivalenti di n-celle, Rend. Sem. Mat. Univ. Padova 27 (1957), 103–121.
_____, Sulla nozione di grado e di coefficiente di allacciamento per mappe ponderate, Rend. Sem. Mat. Univ. Padova 28 (1958), 280–289.
G. Darbo, Grado topologico e teoremi di esistenza di punti uniti per trasformazioni plurivalenti di bicelle, Rend. Sem. Mat. Univ. Padova 19 (1950), 371–395.
_____, Teoria dell'omologia in una categoria di mappe plurivalenti ponderate, Rend. Sem. Mat. Univ. Padova 28 (1958), 188–220.
_____, Sulle coincidenze di mappe ponderate, Rend. Sem. Mat. Univ. Padova 29, 256–270, 1959..
_____, Estensione alle mappe ponderate del teorema di Lefschetz sui punti fissi, Rend. Sem. Mat. Univ. Padova 31 (1961), 46–57.
A. Dold, Lectures on Algebraic Topology, Springer—Verlag, New York yr 1972; Die Grundlehren der mathematischen Wissenschaften, Band 200.
_____, The fixed point transfer of fibre-preserving maps, Math. Z. 148(3) (1976), 215–244.
_____, A coincidence-fixed-point index., Enseign. Math. (2) 24 (1978), 41–53.
_____, Ramified coverings, orbit projections and symmetric powers, Math. Proc. Cambridge Philos. Soc. 99 (1986), 65–72.
J. Dugundji, Remark on homotopy inverses, Portugal. Math. 14 (1955), 39–41.
S. Eilenberg and D. Montgomery, Fixed point theorems for multi-valued transformations, Amer. J. Math. 68 (1946), 214–222.
P. L. Felmer, Periodic solutions of spatially periodic Hamiltonian systems, J. Differential Equations 98 (1992), 143–168.
_____, Rotation type solutions for spatially periodic Hamiltonian systems, Nonlinear Anal. 19 (1992), 409–425.
P. M. Fitzpatrick, I. Massabò and J. Pejsachowicz, Global several-parameter bifurcation and continuation theorems: a unified approach via complementing maps, Math. Ann. 263 (1983), 61–73.
_____, On the covering dimension of the set of solutions of some nonlinear equations, Trans. Amer. Math. Soc. 296 (1986), 777–798.
G. Fournier and L. Górniewicz, The Lefschetz fixed point theorem for some non-compact multi-valued maps, Fund. Math. 94) (1977), 245–254.
E. M. Friedlander and H. Blaine Lawson Jr., A theory of algebraic cocycles, Ann. of Math. (2) 136 (1992), 361–428.
K. Fukaya and K. Ono, Arnold conjecture and Gromov-Witten invariant for general symplectic manifolds, The Arnoldfest (Toronto, ON, 1997), Fields Inst. Commun. 24 (1999), Amer. Math. Soc., Providence, RI, 173–190.
F. B. Fuller, Fixed points of multiple-valued transformations, Bull. Amer. Math. Soc. 67 (1961), 165–169.
M. Furi, M. Martelli and A. Vignoli, On the solvability of nonlinear operator equations in normed spaces, Ann. Mat. Pura Appl. (4) 124 (1980), 321–343.
L. Górniewicz, Homological methods in fixed-point theory of multi-valued maps, Dissertationes Math. (Rozprawy Mat.) 129 (1976), 71.
_____, Topological fixed point theory of multivalued mappings, Mathematics and its Applications, vol. 495, Kluwer Academic Publishers, Dordrecht, 1999.
L. Górniewicz, A. Granas and W. Kryszewski, Sur la méthode de l'homotopie dans la théorie des points fixes pour les applications multivoques II, L'indice dans les ANR-s compacts, C. R. Acad. Sci. Paris Sér. I Math. 308 (1989), 449–452.
_____, On the homotopy method in the fixed point index theory of multi-valued mappings of compact absolute neighborhood retracts, J. Math. Anal. Appl. 161 (1991), 457–473.
L. Górniewicz and M. Lassonde, Approximation and fixed points for compositions of R δ -maps, Topology Appl. 55 (1994), 239–250.
L. Górniewicz and P. Nistri, Topological essentiality and nonlinear boundary value control problems, Topol. Methods Nonlinear Anal. 13 (1999), 53–72.
L. Górniewicz and M Ślosarski, Topological essentiality and differential inclusions, Bull. Austral. Math. Soc. 45 (1992), 177–193.
D. Gottlieb, On realizing Nakaoka's coincidence point transfer as an S-map, Illinois J. Math. 30 (1986), 689–695.
D. H. Gottlieb and M. Özaydin, Intersection numbers, transfers, and group actions, Topology Appl. 55 (1994), 87–100.
A. Granas, The theory of compact vector fields and some of its applications to topology of functional spacesI, Rozprawy Mat. 30 (1962), 93.
_____, Generalizing the Hopf-Lefschetz fixed point theorem for non-compact ANRs, Symposium on Infinite-Dimensional Topology (Louisiana State Univ., Baton Rouge, La., 1967), pp. 119–130; Ann. of Math. Studies 69 (1972), Princeton Univ. Press, Princeton, N. J.
A. Granas and J. Dugundji, Fixed Point Theory, Springer Monographs in Mathematics, Springer—Verlag, New York, 2003.
V. Guillemin and A. Pollack, Differential Topology, Prentice-Hall Inc., Englewood Cliffs, N.J., 1974.
B. Halpern, Algebraic topology and set-valued maps, Set-Valued Mappings, Selections and Topological Properties of 2X (Proc. Conf., SUNY, Buffalo, N.Y., 1969), Lecture Notes in Math., vol. 171, Springer, Berlin, 1970, pp. 27–33.
H. Hofer and E. Zehnder, Symplectic invariants and Hamiltonian dynamics, Birkhäuser Advanced Texts: Basler Lehrbücher (Birkhäuser Advanced Texts: Basel Textbooks), Birkhäuser Verlag, Basel, 1994.
H. Hopf, On some properties of one-valued transformations of manifolds, Proc. Nat. Acad. Sci. 14 (1928), 206–214.
W. Hurewicz and H. Wallman, Dimension Theory, Princeton Math. Ser., vol. 4, Princeton University Press, Princeton, N. J., 1941.
R. P. Jerrard, On Knaster's conjecture, Trans. Amer. Math. Soc. 170 (1972), 385–402.
_____, Homology with multiple-valued functions applied to fixed points, Trans. Amer. Math. Soc. 213 (1975), 407–427.
_____, Erratum to: “Homology with multiple-valued functions applied to fixed points”, (Trans. Amer. Math. Soc., vol. 213 (1975), 407–427), Trans. Amer. Math. Soc. 218 (1976), 406.
_____, A stronger invariant for homology theory., Michigan Math. J. 26 (1979), 33–46.
_____, Fixed points and product spaces, Houston J. Math. 11 (1985), 191–198.
R. Jerrard and M. D. Meyerson, Homotopy with m-functions, Pacific J. Math. 84(2) (1979), 305–318.
S. Jodko-Narkiewicz, Topological degree of multivalued weighted mappings, Ph. D. thesis (1989), Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Toruń, Poland. (Polish)
R. J. Knill, On the homology of a fixed point set, Bull. Amer. Math. Soc. 77 (1971), 184–190.
_____, On the index of isolated closed tori, Differential Topology, Foliations and Gelfand-Fuks Cohomology, (Proc. Sympos., Pontifícia Univ. Católica, Rio de Janeiro, 1976), Lecture Notes in Math., vol. 652, Springer, Berlin, 1978, pp. 196–211.
______, The Lefschetz fixed point theorem for compact groups, Proc. Amer. Math. Soc. 66 (1977), 148–152.
Y. Kodama, On ANR for metric spaces, Sci. Rep. Tokyo Kyoiku Daigaku. Sect. A 5 (1955), 96–98.
W. Kryszewski, Some homotopy classification and extension theorems for the class of compositions of acyclic set-valued maps, Bull. Sci. Math. 119(1) (1995), 21–48.
_____, The fixed-point index for the class of compositions of acyclic set-valued maps on ANRs, Bull. Sci. Math. 120 (1996), 129–151.
_____, Remarks to the Vietoris theorem, Topol. Methods Nonlinear Anal. 8 (1997), 383–405.
_____, Homotopy properties of set-valued mappings, Ph. D. thesis (1997), Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Toruń, habilitation dissertation.
_____, Graph-approximation of set-valued maps on noncompact domains, Topology Appl. 83 (1998), 1–21.
_____, Graph-approximation of set valued maps, a survey, Differential Inclusions and Optimal Control, (J. Andres, L. Góorniewicz, P. Nistri), eds.), vol. 2, Lecture Notes in Nonlinear Analysis, Juliusz Schauder Center for Nonlinear Studies, Nicolaus Copernicus University, Toruń, 1998, pp. 223–235.
H. Kurland and J. Robbin, Infinite codimension and transversality, Dynamical Systems — Warwick 1974, (Proc. Sympos. Appl. Topology and Dynamical Systems, Univ. Warwick, Coventry, 1973/1974; presented to E. C. Zeeman on his fiftieth birthday), vol. 468, Lecture Notes in Math., Springer, Berlin, 1975, pp. 135–150.
S. Lefschetz, Intersections and transformations of complexes and manifolds, Trans. Amer. Math. Soc. 28 (1926), 1–49.
_____, On the fixed point formula, Ann. of Math. (2) 38 (1937), 819–822.
_____, On coincidences of transformations, Bol. Soc. Mat. Mexicana (2) 2 (1957), 16–25.
J. Leray, Sur les équations et les transformations, J. Math. Pures Appl. (9) 24 (1946), 201–248.
_____, Théorie des points fixes: indice total et nombre de Lefschetz, Bull. Soc. Math. France 87 (1959), 221–233.
E. Magenes, Un'osservazione sui teoremi di esistenza di punti uniti in trasformazioni plurivalenti di una N-cella, Rend. Sem. Mat. Univ. Padova 19 (1950), 108–113.
S. Mardešić, Equivalence of singular and Čech homology for ANRs. Application to unicoherence, Fund. Math. 46 (1958), 29–45.
S. Mardešić and J. Segal, Shape theory, North-Holland Mathematical Library, vol. 26, North-Holland Publishing Co., Amsterdam, 1982.
I. Massabo, P. Nistri and J. Pejsachowicz, On the solvability of nonlinear equations in Banach spaces, Fixed Point Theory (Sherbrooke, Que., 1980), Lecture Notes in Math., vol. 886, Springer, Berlin, 1981, pp. 270–299.
I. Massabò and J. Pejsachowicz, On the connectivity properties of the solution set of parametrized families of compact vector fields, J. Funct. Anal. 59 (1984), 151–166.
Ch. N. Maxwell, Fixed points of symmetric product mappings, Proc. Amer. Math. Soc. 8 (1957), 808–815.
M. C. McCord, Classifying spaces and infinite symmetric products, Trans. Amer. Math. Soc. 146 (1969), 273–298.
M. Nakaoka, Coincidence Lefschetz number for a pair of fibre preserving maps, J. Math. Soc. Japan 32(4) (1980), 751–779.
P. Nistri, V. V. Obukhovskiĭ and P. Zecca, On the solvability of systems of inclusions involving noncompact operators, Trans. Amer. Math. Soc. 342(2) (1994), 543–562.
R. Oliver, A transfer homomorphism for compact Lie group actions, Math. Ann. 260 (1982), 351–374.
B. O'Neill, Induced homology homomorphisms for set-valued maps, Pacific J. Math. 7 (1957), 1179–1184.
B. O'Neill and E. G. Straus, A fixed point theorem, Proc. Mmer. Math. Soc. 8 (1957), 1148–1151.
J. Pejsachowicz, The homotopy theory of weighted mappings, Boll. Un. Mat. Ital. B (5) 14 (1977), 702–721.
_____, A Lefschetz fixed point theorem for multivalued weighted mappings, Boll. Un. Mat. Ital. A (5) 14 (1977), 391–397.
_____, Teoria di omotopia di mappe ponderate, Ph. D. thesis (1973), Tesi di Perfezionamento SNS Pisa, Pisa, Italy.
_____, Relation between the homotopy and the homology theory of weighted mappings, Boll. Un. Mat. Ital. B (5) 15 (1978), 285–302.
B. Ralph, A homotopy transfer for finite group actions, Pacific J. Math. 160 (1993), 133–141.
R. Schultz, Homological transfers for orbit space projections, Manuscripta Math. 24 (1978), 229–238.
G. Scorza Dragoni, Il lemma di Sperner: sue implicazioni negli spazi euclidei equestioni connesse, Contributi del CLISMA 43(2) (1978), 160–170.
B. Segre, Recouvrements de sphères et correspondances entre variétés topologiques, Colloque sur les Questions de Réalité en Géométrie, Liège, 1955, Georges Thone, Liège, 1956, pp. 149–175.
H. W. Siegberg and G. Skordev, Fixed point index and chain approximations, Pacific J. Math. 102 (1982), 455–486.
R. Skiba, On the Lefschetz fixed point theorem for multivalued weighted mappings, Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math. 40 (2001), 201–214.
_____, Topological essentiality for multivalued weighted mappings, Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math. 41 (2002), 131–145.
R. Skiba, Graph-approximation of multivalued weighted maps (to appear).
G. S. Skordev, The multiplicativity property of the fixed point index for multivalued maps, Serdica 15 (1989), 160–170.
L. Smith, Transfer and ramified coverings, Math. Proc. Cambridge Philos. Soc. 93 (1983), 485–493.
C. Viterbo, Exact Lagrange submanifolds, periodic orbits and the cohomology of free loop spaces, J. Differential Geom. 47 (1997), 420–468.
F. von Haeseler, H.-O. Peitgen and G. Skordev, Lefschetz fixed point theorem for acyclic maps with multiplicity, Topol. Methods Nonlinear Anal. 19 (2002), 339–374.
F. von Haeseler and G. Skordev, Borsuk-Ulam theorem, fixed point index and chain approximations for maps with multiplicity, Pacific J. Math. 153 (1992), 369–396.
Ch. S. Vora, Fixed point theorems for certain compact weighted maps of a manifold, Math. Student 46 (1979), 81–86.
_____, Some applications of the fixed point theorems for compact weighted maps, Math. Student 46 (1979), 87–99.
T. Watanabe, The continuity axiom and the Čech homology, Geometric Topology and Shape Theory (Dubrovnik, 1986), Lecture Notes in Math., vol. 1283, Springer, Berlin, 1987, pp. 221–239.
Th. J. Worosz, Multiple-valued functions and their application to the fixed point properties of product spaces, Bull. Malaysian Math. Soc. (2) 1 (1978), 101–109.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer
About this chapter
Cite this chapter
Pejsachowicz, J., Skiba, R. (2005). Fixed Point Theory of Multivalued Weighted Maps. In: Brown, R.F., Furi, M., Górniewicz, L., Jiang, B. (eds) Handbook of Topological Fixed Point Theory. Springer, Dordrecht. https://doi.org/10.1007/1-4020-3222-6_6
Download citation
DOI: https://doi.org/10.1007/1-4020-3222-6_6
Publisher Name: Springer, Dordrecht
Print ISBN: 978-1-4020-3221-9
Online ISBN: 978-1-4020-3222-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)