Remarks on sufficient conditions for fixed points of multivalued nonexpansive mappings

In recent papers S. Dhompongsa, T. Domínguez-Benavides, A. Kaewcharoen, A. Kaewkhao, B. Panyanak, The Jordan–von Neumann constants and fixed points for multivalued nonexpansive mappings, J. Math. Anal. Appl. 320 (2006) 916–927; S. Dhompongsa, A. Kaewcharoen, A. Kaewkhao, The Domínguez–Lorenzo condition and multivalued nonexpansive mappings, Nonlinear Anal. 64 (2006) 958–970], two sufficient conditions, namely the Domínguez–Lorenzo condition and property (D$D$), for fixed points of multivalued nonexpansive mappings are introduced. The authors also give some sufficient conditions for the Domínguez–Lorenzo condition and property (D$D$). In their proofs, it seems reasonable to use the James and von Neumann–Jordan constants. With slight modifications, significant improvements are obtained. Some new estimates are tight in Hilbert spaces.