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The global attractivity and asymptotic stability of solution of a nonlinear integral equation. (English) Zbl 1108.45006
The paper is devoted to the study of the nonlinear Volterra integral equations $$x(t)=f\left(t,x(t),\int^t_{0}u(t,s,x(s))ds\right),\ \ t\geq 0,\tag1$$ and $$x(t)=g(t,x(t))+x(t)\int^t_{0}u(t,s,x(s))ds,\ \ t\geq 0,\tag2$$ in the Banach space consisting of all real functions defined, bounded and continuous on $[0,+\infty)$. Using the measures of noncompactness and the fixed point theorem, conditions are given when equation (1) has at least one globally attractive solution and when equation (2) has at least one asymptotically stable solution. The last statement for the special case of equation (2) with $g(t,u)\equiv 0$ improves the corresponding result of {\it J. Banas} and {\it B. Rzepka} [J. Math. Anal. Appl. 284, No. 1, 165--173 (2003; Zbl 1029.45003)]. Three examples are given.

##### MSC:
 45M10 Stability theory of integral equations 45G10 Nonsingular nonlinear integral equations 47H09 Mappings defined by “shrinking” properties
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##### References:
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