## Existence and asymptotic stability of solutions to a functional-integral equation.(English)Zbl 1145.45003

Using a fixed point theorem of Darbo type in the space of bounded continuous functions, the existence of a bounded solution of the equation $x(t)=f(t,x(t))+g(t,x(t))\int_{0}^{t}u(t,s,x(s))\,ds$
is obtained when $$| u(t,s,x)|\leq a(t)b(s)$$ with small $$a,b$$ and $$f(t,\cdot)$$ and $$g(t,\cdot)$$ are contractions with small constants $$k$$ and $$m(t)$$ where $$m(t)a(t)\to0$$ 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 $$x(t)-y(t)\to0$$ as $$t\to\infty$$.

### MSC:

 45G10 Other nonlinear integral equations 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 45M05 Asymptotics of solutions to integral equations 45M10 Stability theory for integral equations
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