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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 Nonsingular nonlinear integral equations 45M05 Asymptotic theory of integral equations 47H09 Mappings defined by “shrinking” properties 45M10 Stability theory of integral equations
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##### References:
 [1] Argyros, I. K.: Quadratic equations and applications to Chandrasekhar’s and related equations. Bull. austral. Math. soc. 32, 275-292 (1985) · Zbl 0607.47063 [2] Banaś, J.: Measures of noncompactness in the space of continuous tempered functions. Demonstratio math. 14, 127-133 (1981) [3] Banaś, J.; Goebel, K.: Measures of noncompactness in Banach spaces. Lecture notes in pure and applied mathematics 60 (1980) [4] Deimling, K.: Nonlinear functional analysis. (1985) · Zbl 0559.47040