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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)) 0 t u(t,s,x(s))ds

is obtained when |u(t,s,x)|a(t)b(s) with small a,b and f(t,·) and g(t,·) are contractions with small constants k and m(t) where m(t)a(t)0 sufficiently fast as t.

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)0 as t.

45G10Nonsingular nonlinear integral equations
47H09Mappings defined by “shrinking” properties
45M05Asymptotic theory of integral equations
45M10Stability theory of integral equations