On local attractivity and asymptotic stability of solutions of a quadratic Volterra integral equation
The authors study the integral equation \[ x(t) = p(t) +f(t, x(t))\int_0^t v(t, s, x(s))ds. \] They write the following: ``We will study the solvability of this equation in the space \(BC (\mathbb{R}_{+})\) consisting of all real functions defined, continuous and bounded on the interval \(\mathbb{R}_{+} = [0; \infty).\) More precisely, we will look for assumptions concerning the functions involved in this equation which guarantee that this equation has solution belonging to \(BC (\mathbb{R}_{+})\) and being locally attractive or asymptotic stable on \(\mathbb{R}_{+}\).'' As example the authors consider the equation \[ x(t) =t\exp(-2t) +\frac{1}{2\pi}\arctan(\sqrt{t} +tx(t)) \int_{0}^{t} \left( \frac{2x(s)^{2/3}+x(s)}{(s+1)(t^2+1)} +\frac{1}{10(t^2+1)} \right)\,ds . \]
scientific article; zbMATH DE number 5573823
scientific article