EIGENVALUES OF COUNTABLY CONDENSING MAPS Kim, In-Sook; Kim, Yun-Ho; Kwon, Sung-Hui;
Using an index theory for countably condensing maps, we show the existence of eigenvalues for countably k-set contractive maps and countably condensing maps in an infinite dimensional Banach space X, under certain condition that depends on the quantitative haracteristic, that is, the infimum of all for which there is a countably k-set-contractive retraction of the closed unit ball of X onto its boundary.
eigenvalues;countably condensing maps;fixed point index;
PS-TKA with more than 10° of Preoperative Posterior Tibial Slope, The Journal of Korean Knee Society, 2011, 23, 1, 7
J. M. Ayerbe Toledano, T. Domínguez Benavides, and G. López Acedo, Measures of Noncompactness in Metric Fixed Point Theory, Operator Theory: Advances and Ap-plications, 99. Birkhäuser Verlag, Basel, 1997.
F. E. Browder, Remarks on the paper: “Boundary conditions for condensing mappings” [Nonlinear Anal. 8 (1984), no. 3, 209–219] by Q. Y. Yu and Browder, Nonlinear Anal. 8 (1984), no. 9, 1113.
D. Caponetti, A. Trombetta, and G. Trombetta, Proper 1-ball contractive retractions in Banach spaces of measurable functions, Bull. Austral. Math. Soc. 72 (2005), no. 2, 299–315.
D. Caponetti, A. Trombetta, and G. Trombetta, An extension of Guo's theorem via k-Ψ-contractive retractions, Nonlinear Anal. 64 (2006), no. 9, 1897–1907.
D. Guo, Eigenvalues and eigenvectors of nonlinear operators, Chin. Ann. Math. 2 (1981), 65–80.
I.-S. Kim and M. Väth, Some remarks on measures of noncompactness and retractions onto spheres, Topology Appl. 154 (2007), no. 17, 3056–3069.
H. Mönch, Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces, Nonlinear Anal. 4 (1980), no. 5, 985–999.
R. D. Nussbaum, The fixed point index for local condensing maps, Ann. Mat. Pura Appl. (4) 89 (1971), 217–258.
Y. Sun, A generalization of Guo's theorem and applications, J. Math. Anal. Appl. 126 (1987), no. 2, 566–573.
M. Väth, Fixed point theorems and fixed point index for countably condensing maps, Topol. Methods Nonlinear Anal. 13 (1999), no. 2, 341–363.
J. Wośko, An example related to the retraction problem, Ann. Univ. Mariae Curie-Sk lodowska Sect. A 45 (1991), 127–130.
Q. Y. Yu and F. E. Browder, Boundary conditions for condensing mappings, Nonlinear Anal. 8 (1984), no. 3, 209–219.