Measures of noncompactness and asymptotic stability of solutions of a quadratic Hammerstein integral equation (Q658367)

The article deals with the nonlinear integral equation \[ x(t) = p(t) + f(t,x(t)) \int_0^\infty g(t,\tau) h(\tau,x(\tau)) \, d\tau, \quad t \geq 0, \tag{1} \] in the space of bounded and continuous real functions. Under some natural conditions, the existence of an asymptotically stable solution is proved. The proof is based on the Darbo fixed point principle. At the end of the article, the example \[ \begin{aligned} x(t) = &\sin (t^2 + 1) + \frac1{20}\left[\root3\of{t} x(t) \arctan x(t) + 5\root4\of{t + 1}\right] \\ &\int_0^\infty \frac\tau{1 + t^2 + \tau^2} \bigg(\frac{\tau^3}{1 + \tau^2} + \tau \root3\of{x^2(\tau)}\bigg) \, d\tau\end{aligned} \tag{2} \] is considered. Reviewers remark: It remains unclear why the authors call equations (1) and (2) quadratic.