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Zbl 1175.45002
Banaś, Józef; Rzepka, Beata
On local attractivity and asymptotic stability of solutions of a quadratic Volterra integral equation. (English)
[J] Appl. Math. Comput. 213, No. 1, 102-111 (2009). ISSN 0096-3003

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 $$x(t) =t\exp(-2t) +\frac{1}{2\pi}\arctan(\sqrt{t} +tx(t)) \int_{0}^{t} \left( \frac{2x(s)^{2/3}+x(s)}{(s+1)(t^2+1)} +\frac{1}{10(t^2+1)} \right)\,ds .$$
[Anatoly Filip Grishin (Khar'kov)]

MSC 2000:: *45G10 Nonsingular nonlinear integral equations
47H30 Particular nonlinear operators
45M05 Asymptotic theory of integral equations
45M10 Stability theory of integral equations
47H09 Mappings defined by "shrinking" properties

Keywords: quadratic Volterra integral equation; measure of noncompactness; uniform local attractivity; global attractivity; asymptotic stability

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