Existence and asymptotic stability of solutions to a functional-integral equation.

*(English)*Zbl 1145.45003Using a fixed point theorem of Darbo type in the space of bounded continuous functions, the existence of a bounded solution of the equation

$$x\left(t\right)=f(t,x\left(t\right))+g(t,x\left(t\right)){\int}_{0}^{t}u(t,s,x\left(s\right))\phantom{\rule{0.166667em}{0ex}}ds$$

is obtained when $\left|u\right(t,s,x\left)\right|\le a\left(t\right)b\left(s\right)$ with small $a,b$ and $f(t,\xb7)$ and $g(t,\xb7)$ are contractions with small constants $k$ and $m\left(t\right)$ where $m\left(t\right)a\left(t\right)\to 0$ sufficiently fast as $t\to \infty $.

It is also claimed that the solution $x$ is asymptotically stable; however, the authors mean by this apparently only that the solution is asymptotically unique, i.e. any solution $y$ of the *same* equation (with not too large norm) satisfies $x\left(t\right)-y\left(t\right)\to 0$ as $t\to \infty $.

Reviewer: Martin Väth (Gießen)