On existence and asymptotic stability of solutions of a nonlinear integral equation.

*(English)*Zbl 1029.45003The authors prove an existence theorem for a nonlinear Volterra integral equation of a special type arising in traffic theory:

$$x\left(t\right)=f(t,x\left(t\right)){\int}_{0}^{1}u(t,s,x\left(s\right))ds,\phantom{\rule{1.em}{0ex}}t\in t\in [0,1]\xb7\phantom{\rule{2.em}{0ex}}\left(1\right)$$

It is an example of a quadratic integral equation. Using measures of noncompactness, the authors show that (1) has continuous and bounded solutions on $[0,\infty )$. Fixed points results are used. Furthermore, for suitable measure of noncompactness the authors prove that those solutions are asymptotically stable in some sense defined in the paper.

Reviewer: Yves Cherruault (Paris)

##### MSC:

45G10 | Nonsingular nonlinear integral equations |

45M05 | Asymptotic theory of integral equations |

47H09 | Mappings defined by “shrinking” properties |

45M10 | Stability theory of integral equations |