Existence and asymptotically stable solution of a Hammerstein type integral equation in a Hölder space

Abstract

The following nonlinear quadratic integral equation of Hammerstein type is studied. $$x(t)=p(t)+x(t)\int_0^{q(t)} H(t,\tau,x(\tau)){\rm d}\tau.$$ The methodology relies on the measure of noncompactness in the space of functions with tempered increments, namely the space of $\alpha$-Hölder continuous functions. The results follow from the Darbo fixed point theorem. Some examples are included to show the applicability of the main results.