Approximations and fixed points for condensing non-self-maps defined on a sphere

Lin, Tzu-Chu

doi:10.1090/S0002-9939-1989-0973838-5

Approximations and fixed points for condensing non-self-maps defined on a sphere
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Proc. Amer. Math. Soc. 105 (1989), 66-69 Request permission

Abstract:

In this paper, we investigate the validity of an interesting theorem of Ky Fan [Theorem 2, Math. Z. 112 (1969), 234-240] defined on a sphere (the boundary of a closed ball) in an infinite-dimensional Banach space. We will prove that it is true for a continuous condensing map with suitable conditions posed. As applications of our theorem, some fixed point theorems of continuous condensing non-self-maps defined on a sphere are derived. Our results generalize some results of R. Nussbaum [10] and P. Massatt [8].

References

M. Altman, A fixed point theorem in Hilbert space, Bull. Acad. Polon. Sci. Cl. III. 5 (1957), 19–22, III (English, with Russian summary). MR 0087064
Felix E. Browder, On a sharpened form of the Schauder fixed-point theorem, Proc. Nat. Acad. Sci. U.S.A. 74 (1977), no. 11, 4749–4751. MR 463982, DOI 10.1073/pnas.74.11.4749
Ky Fan, Extensions of two fixed point theorems of F. E. Browder, Math. Z. 112 (1969), 234–240. MR 251603, DOI 10.1007/BF01110225
Benjamin Halpern, Fixed-point theorems for set-valued maps in infinite dimensional spaces, Math. Ann. 189 (1970), 87–98. MR 273479, DOI 10.1007/BF01350295
Tzu Chu Lin, A note on a theorem of Ky Fan, Canad. Math. Bull. 22 (1979), no. 4, 513–515. MR 563767, DOI 10.4153/CMB-1979-067-x
T.-C. Lin and C. L. Yen, Applications of the proximity map to fixed point theory, Approximation theory and applications (St. John’s, Nfld., 1984) Res. Notes in Math., vol. 133, Pitman, Boston, MA, 1985, pp. 96–103. MR 879132
Tzu-Chu Lin and Ch’i Lin Yen, Applications of the proximity map to fixed point theorems in Hilbert space, J. Approx. Theory 52 (1988), no. 2, 141–148. MR 929300, DOI 10.1016/0021-9045(88)90053-6
Paul Massatt, A fixed point theorem for $\alpha$-condensing maps on a sphere, Proc. Roy. Soc. Edinburgh Sect. A 94 (1983), no. 3-4, 323–329. MR 709725, DOI 10.1017/S0308210500015687
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
Roger D. Nussbaum, Some fixed point theorems, Bull. Amer. Math. Soc. 77 (1971), 360–365. MR 284888, DOI 10.1090/S0002-9904-1971-12694-0