Monotonic solutions of Urysohn integral equation on unbounded interval (Q1767912)

The Urysohn integral equation \[ x(t)= f(t)+ \int_I u(t, s,x(s))\,ds, \] where \(I\) is an interval (bounded or not) in \(R\) and \(t\in I\), is a frequently used integral equation in nonlinear analysis. To avoid the major difficulty and complications in case of unbounded interval, the authors wisely focus on investigations for solvability of the integral equation (of Urysohn type) of the form \[ x(t)= f(t)+ \int^\infty_0 u(t, s,x(s))\,ds, \qquad t\geq 0. \] Measures of strong and weak non-compactness is used to prove the existence of solutions of the equation under consideration, in the space \(L^1(R_+)\) of Lebesgue integrable functions on the half axis \(R_+\). Operators of the form \((Ux)(t)\) and \((Fx)(t)\) are defined to obtain the solutions.