A nonlinear operator functional equation of Volterra type. (Q1421262)

This paper is concerned with the existence of monotonic solutions in \(L^{1}[0,+\infty)\) of the following Volterra integral equation \[ x(t)=\int_{0}^{t}k_{1}(t,s)f\left(s,\int_{0}^{s}k_{2}(s,\theta)x(\phi(\theta))\,d\theta\right)\,ds, \;\;t\geq 0. \] The proofs rely on the measure of noncompactness and the Darboux fixed point theorem. As an application, the authors study the existence of almost everywhere nonincreasing solutions of the initial value problem \[ x'(t)=-ax(t)+f\left(t,\int_{0}^{t}k(t-\theta)x(\phi(\theta))\,d\theta\right), \quad t\geq 0,\qquad x(0)=0, \] where \(a>0\).