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On local attractivity and asymptotic stability of solutions of a quadratic Volterra integral equation. (English) Zbl 1175.45002

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
\[ x(t) =t\exp(-2t) +\frac{1}{2\pi}\arctan(\sqrt{t} +tx(t)) \int_{0}^{t} \left( \frac{2x(s)^{2/3}+x(s)}{(s+1)(t^2+1)} +\frac{1}{10(t^2+1)} \right)\,ds . \]

MSC:

45G10 Other nonlinear integral equations
47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)
45M05 Asymptotics of solutions to integral equations
45M10 Stability theory for integral equations
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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