## 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.