On attractivity and asymptotic stability of solutions of a quadratic Volterra integral equation of fractional order
The author considers the quadratic Volterra integral equation of fractional order \[ x(t)=p(t)+\frac{f(t,x(t))}{\Gamma(\alpha)}\int_0^t\frac{u(t,s,x(s))}{(t-s)^{1-\alpha}}ds,\;\;t\in \mathbb{R}_+,\tag{1} \] where \(\alpha\in (0,1)\) and \(p\), \(f\) and \(u\) are continuous functions. Using the measure of noncompactness along with Schauder's fixed point theorem, she shows, under some natural -- but technical -- assumptions too involved to be reproduced here, that the problem above has at least one bounded and continuous solution \(x:\mathbb{R}_+\to \mathbb{R}\). In addition, she proves that the solutions of the equation (1) are uniformly locally attractive.
quadratic Volterra integral equation of fractional order
scientific article; zbMATH DE number 5548696