## On existence and asymptotic stability of solutions of a nonlinear integral equation.(English)Zbl 1029.45003

The authors prove an existence theorem for a nonlinear Volterra integral equation of a special type arising in traffic theory: $x(t)= f(t, x(t)) \int^1_0 u(t, s,x(s)) ds,\quad t\in t\in [0,1].\tag{1}$ 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.

### MSC:

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

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