Generalized concentrative mappings and their fixed points
Josef Daneš · Czech digital mathematics library · 1970
0« Introduction.The notion of the measure of noncompactness w»s introduced by G. Darbo [7] and Sadovskii[ 13] • By means of this notion G. Darbo defined a Jk -setcontraction and Sadovskii a concentrative mapping.Darbo and Sadovskii proved for their classes of mappings the fixed point theorems.We observe that the Sadovskii 8 class of mappinga is broader than the Darbo's one.But the sum of a completely continuous mapping and a Jk -contraction is the to/ -set-contraction of Darbo 11].Hence the most important case is already covered by Darbo (implicitly).Let us note that the classes CL (Jk) (0 £ Ms < A ) and Cv of Frum-Ketkov C8J are near to that of Darbo.Further development of concentrative mappings (reap.Jfc-set-contractions) is contained in Badoev, Sadovskii 12], BorisoviS, Sepronov L3J, Danes £4,5,6], and Nussbaum [12]• Index and rotation notions for this class of mappings are developed in t3, 121.The notion of a generalised concentrative mapping was introduced by LifSic -Sadovskii in [ 11], and in more general fashion in our