A degree theory, fixed point theorems, and mapping theorems for multivalued noncompact mappings

Petryshyn, W. V.; Fitzpatrick, P. M.

doi:10.1090/S0002-9947-1974-2478129-5

A degree theory, fixed point theorems, and mapping theorems for multivalued noncompact mappings
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by W. V. Petryshyn and P. M. Fitzpatrick

Trans. Amer. Math. Soc. 194 (1974), 1-25

DOI: https://doi.org/10.1090/S0002-9947-1974-2478129-5

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Abstract:

We define and study the properties of a topological degree for ultimately compact, multivalued vector fields defined on the closures of open subsets of certain locally convex topological vector spaces. In addition to compact mappings, the class of ultimately compact mappings includes condensing mappings, generalized condensing mappings, perturbations of compact mappings by certain Lipschitz-type mappings, and others. Using this degree we obtain fixed point theorems and mapping theorems.

References

H. F. Bohnenblust and S. Karlin, On a theorem of Ville, Contributions to the Theory of Games, Annals of Mathematics Studies, no. 24, Princeton University Press, Princeton, N.J., 1950, pp. 155–160. MR 0041415
Felix E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. U.S.A. 54 (1965), 1041–1044. MR 187120, DOI 10.1073/pnas.54.4.1041
Felix E. Browder, Nonlinear operators and nonlinear equations of evolution in Banach spaces, Nonlinear functional analysis (Proc. Sympos. Pure Math., Vol. XVIII, Part 2, Chicago, Ill., 1968) Amer. Math. Soc., Providence, R.I., 1976, pp. 1–308. MR 0405188
Arrigo Cellina and Andrzej Lasota, A new approach to the definition of topological degree for multi-valued mappings, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 47 (1969), 434–440 (1970) (English, with Italian summary). MR 276937
Josef Daneš, Some fixed point theorems, Comment. Math. Univ. Carolinae 9 (1968), 223–235. MR 235435
Gabriele Darbo, Punti uniti in trasformazioni a codominio non compatto, Rend. Sem. Mat. Univ. Padova 24 (1955), 84–92 (Italian). MR 70164
D. G. de Figueiredo and L. A. Karlovitz, On the radial projection in normed spaces, Bull. Amer. Math. Soc. 73 (1967), 364–368. MR 211248, DOI 10.1090/S0002-9904-1967-11753-1
J. Dugundji, An extension of Tietze’s theorem, Pacific J. Math. 1 (1951), 353–367. MR 44116, DOI 10.2140/pjm.1951.1.353
Massimo Furi and Alfonso Vignoli, On $\alpha$-nonexpansive mappings and fixed points, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 48 (1970), 195–198 (English, with Italian summary). MR 279792
I. L. Glicksberg, A further generalization of the Kakutani fixed theorem, with application to Nash equilibrium points, Proc. Amer. Math. Soc. 3 (1952), 170–174. MR 46638, DOI 10.1090/S0002-9939-1952-0046638-5
Dietrich Göhde, Zum Prinzip der kontraktiven Abbildung, Math. Nachr. 30 (1965), 251–258 (German). MR 190718, DOI 10.1002/mana.19650300312

Investigations of some properties of bounded linear operators with their q-norms

29

A. Granas, Sur la notion du degré topologique pour une certaine classe de transformations multivalentes dans les espaces de Banach, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys. 7 (1959), 191–194 (unbound insert) (French, with Russian summary). MR 0108743
A. Granas, Theorem on antipodes and theorems on fixed points for a certain class of multi-valued mappings in Banach spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys. 7 (1959), 271–275 (unbound insert) (English, with Russian summary). MR 0117588
C. J. Himmelberg, J. R. Porter, and F. S. Van Vleck, Fixed point theorems for condensing multifunctions, Proc. Amer. Math. Soc. 23 (1969), 635–641. MR 246175, DOI 10.1090/S0002-9939-1969-0246175-1
W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1965), 1004–1006. MR 189009, DOI 10.2307/2313345

Sur les espaces complets

15

Ky. Fan, Fixed-point and minimax theorems in locally convex topological linear spaces, Proc. Nat. Acad. Sci. U.S.A. 38 (1952), 121–126. MR 47317, DOI 10.1073/pnas.38.2.121

Opérateurs nonexpansifs, P-compact, et propriétés géométriques de la norme

A. Lasota and Z. Opial, Fixed-point theorems for multi-valued mappings and optimal control problems, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 16 (1968), 645–649 (English, with Russian summary). MR 248580
E. A. Lifšic and B. N. Sadovskiĭ, A fixed point theorem for generalized condensing operators, Dokl. Akad. Nauk SSSR 183 (1968), 278–279 (Russian). MR 0236789
Tsoy-wo Ma, Topological degrees of set-valued compact fields in locally convex spaces, Dissertationes Math. (Rozprawy Mat.) 92 (1972), 43. MR 309103
Jack T. Markin, A fixed point theorem for set valued mappings, Bull. Amer. Math. Soc. 74 (1968), 639–640. MR 227825, DOI 10.1090/S0002-9904-1968-11971-8
Sam B. Nadler Jr., Multi-valued contraction mappings, Pacific J. Math. 30 (1969), 475–488. MR 254828, DOI 10.2140/pjm.1969.30.475

The fixed point index and fixed point theorems for k-set-contractions

Roger D. Nussbaum, The fixed point index for local condensing maps, Ann. Mat. Pura Appl. (4) 89 (1971), 217–258. MR 312341, DOI 10.1007/BF02414948
Zdzisław Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967), 591–597. MR 211301, DOI 10.1090/S0002-9904-1967-11761-0
W. V. Petryshyn, Structure of the fixed points sets of $k$-set-contractions, Arch. Rational Mech. Anal. 40 (1970/71), 312–328. MR 273480, DOI 10.1007/BF00252680
W. V. Petryshyn, A new fixed point theorem and its application, Bull. Amer. Math. Soc. 78 (1972), 225–229. MR 291920, DOI 10.1090/S0002-9904-1972-12930-6
W. V. Petryshyn, Fixed point theorems for various classes of $1$-set-contractive and $1$-ball-contractive mappings in Banach spaces, Trans. Amer. Math. Soc. 182 (1973), 323–352. MR 328688, DOI 10.1090/S0002-9947-1973-0328688-2
Simeon Reich, Fixed points in locally convex spaces, Math. Z. 125 (1972), 17–31. MR 306989, DOI 10.1007/BF01111112
B. N. Sadovskiĭ, Measures of noncompactness and condensing operators, Problemy Mat. Anal. Slož. Sistem 2 (1968), 89–119 (Russian). MR 0301582

Ultimately compact and condensing mappings

27

Helmut H. Schaefer, Topological vector spaces, Graduate Texts in Mathematics, Vol. 3, Springer-Verlag, New York-Berlin, 1971. Third printing corrected. MR 0342978, DOI 10.1007/978-1-4684-9928-5

On the degree of

ball

condensing vector fields

2

A method for producing new fixed point theorems

176

8

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