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Zbl 1029.45003
Banaś, Józef; Rzepka, Beata
On existence and asymptotic stability of solutions of a nonlinear integral equation. (English)
[J] J. Math. Anal. Appl. 284, No.1, 165-173 (2003). ISSN 0022-247X

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.
[Yves Cherruault (Paris)]

MSC 2000:: *45G10 Nonsingular nonlinear integral equations
45M05 Asymptotic theory of integral equations
47H09 Mappings defined by "shrinking" properties
45M10 Stability theory of integral equations

Keywords: asymptotic stability; fixed points; nonlinear Volterra integral equation; traffic theory; quadratic integral equation; measures of noncompactness; continuous and bounded solutions

Cited in: Zbl 1190.47090 Zbl 1108.45006

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