Measures of noncompactness related to monotonicity. (English) Zbl 0999.47041

Banaś, Józef; Olszowy, Leszek

Ann. Soc. Math. Pol., Ser. I, Commentat. Math. 41, 13-23 (2001).

For a bounded set \(M\subset C[0,1]\), let \[ i(M)= \sup\{i(x): x\in M\},\quad d(M)= \sup\{d(x): x\in M\}, \] where \[ (x)= \sup\{|x(t)- x(s)|- x(t)+ x(s): t\leq s\} \] and \[ d(x)= \sup\{|x(t)- x(s)|- x(s)+ x(t): t\leq s\}. \] The authors study the characteristics \(i(M)\) and \(d(M)\) which are similar to measures of noncompactness in Sadovskij’s axiomatic sense. In particular, these characteristics may be expressed through the Hausdorff distance of \(M\) to the class of all sets \(N\) for which \(i(N)= 0\) resp. \(d(N)= 0\).

Reviewer: Jürgen Appell (Würzburg)

MSC:

47H09

Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.

Keywords:

monotonicity; measures of noncompactness; Hausdorff distance