Generalizations of Darbo's fixed point theorem for new condensing operators with application to a functional integral equation (Q1999611)

The authors define a new class of nonlinear operators which are condensing in a weak sense, depending on certain auxiliary functions \(\alpha\) and \(\beta\). They obtain a corresponding fixed point theorem which contains Darbo's fixed point principle for a special choice of \(\alpha\) and \(\beta\), and apply this to the functional-integral equation \[x(t)= A(t)+ h(t,x(\xi(t)))+ f(t, x(\xi(t)), \varphi(Gx(t)))\qquad (t\ge 0),\] where \(G\) is the functional-integral operator of Uryson type \[Gx(t):= \int^{\beta(t)}_0 g(t,s,x(\eta(s)))\,ds.\] The paper contains some inexact points; for instance, Example 1.7 does not make sense.