On the solvability of a Volterra integral equation of quadratic form in the class of continuous function (Q2853282)

By using the Darbo fixed point theorem, the authors prove the existence of solutions for the Volterra integral equation \(x(t)=1+Tx(t)\int_0^tk(t,s)\phi(s)x(s)\,ds\), \(t\in[0,a]\), in the class of continuous functions \(C([0,a])\), where \(\phi:[0,a]\to\mathbb R\) and \(k:[0,a]\times[0,a]\to\mathbb R\) are continuous, \(T:C([0,a])\to C([0,a])\) is a bounded and continuous operator satisfying the Darbo condition \(\chi(TX)\leq\alpha\chi(X)\) for any subset \(X\subset C([0,a])\) with \(\alpha\in[0,1)\), and \(\chi\) is the Hausdorff measure of noncompactness.