×

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

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. \(d(N)= 0\).

MSC:

47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
PDF BibTeX XML Cite